Applications of quantum mechanical techniques to areas outside of quantum mechanics

Haven, Emmanuel

Khrennikov, Andrei

2018

162

Frontiers Media SA

This book deals with applications of quantum mechanical techniques to areas outside of quantum mechanics, so-called quantum-like modeling. Research in this area has grown over the last 15 years. But even already more than 50 years ago, the interaction between Physics Nobelist Pauli and the psychologist Carl Jung in the 1950's on seeking to find analogous uses of the complementarity principle from quantum mechanics in psychology needs noting. This book does NOT want to advance that society is quantum mechanical! The macroscopic world is manifestly not quantum mechanical. But this rules not out that one can use concepts and the mathematical apparatus from quantum physics in a macroscopic environment. A mainstay ingredient of quantum mechanics, is 'quantum probability' and this tool has been proven to be useful in the mathematical modelling of decision making. In the most basic experiment of quantum physics, the double slit experiment, it is known (from the works of A. Khrennikov) that the law of total probability is violated. It is now well documented that several decision making paradoxes in psychology and economics (such as the Ellsberg paradox) do exhibit this violation of the law of total probability. When data is collected with experiments which test 'non-rational' decision making behaviour, one can observe that such data often exhibits a complex non-commutative structure, which may be even more complex than if one considers the structure allied to the basic two slit experiment. The community exploring quantum-like models has tried to address how quantum probability can help in better explaining those paradoxes. Research has now been published in very high standing journals on resolving some of the paradoxes with the mathematics of quantum physics. The aim of this book is to collect the contributions of world's leading experts in quantum like modeling in decision making, psychology, cognition, economics, and finance.

Science (General)

Physics (General)

English

9782889454273

Syntax Warning: Invalid Font Weight
APPLICATIONS OF QUANTUM
MECHANICAL TECHNIQUES TO
AREAS OUTSIDE OF QUANTUM
MECHANICS
EDITED BY : Emmanuel Haven and Andrei Khrennikov
PUBLISHED IN : Frontiers in Physics

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ISSN 1664-8714
ISBN 978-2-88945-427-3
DOI 10.3389/978-2-88945-427-3

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1

February 2018  |  Quantum-Like Modeling

APPLICATIONS OF QUANTUM
MECHANICAL TECHNIQUES TO AREAS
OUTSIDE OF QUANTUM MECHANICS
Topic Editors:
Emmanuel Haven, Memorial University, Canada
Andrei Khrennikov, Linnaeus University, Sweden

Image by Philippe Haven.

This book deals with applications of quantum mechanical techniques to areas outside of quantum
mechanics, so-called quantum-like modeling. Research in this area has grown over the last 15
years. But even already more than 50 years ago, the interaction between Physics Nobelist Pauli
and the psychologist Carl Jung in the 1950’s on seeking to find analogous uses of the complementarity principle from quantum mechanics in psychology needs noting.

Frontiers in Physics

2

February 2018  |  Quantum-Like Modeling

This book does NOT want to advance that society is quantum mechanical! The macroscopic
world is manifestly not quantum mechanical. But this rules not out that one can use concepts
and the mathematical apparatus from quantum physics in a macroscopic environment.
A mainstay ingredient of quantum mechanics, is ‘quantum probability’ and this tool has been
proven to be useful in the mathematical modelling of decision making. In the most basic
experiment of quantum physics, the double slit experiment, it is known (from the works of A.
Khrennikov) that the law of total probability is violated. It is now well documented that several
decision making paradoxes in psychology and economics (such as the Ellsberg paradox) do
exhibit this violation of the law of total probability. When data is collected with experiments
which test ‘non-rational’ decision making behaviour, one can observe that such data often exhibits
a complex non-commutative structure, which may be even more complex than if one considers
the structure allied to the basic two slit experiment. The community exploring quantum-like
models has tried to address how quantum probability can help in better explaining those paradoxes. Research has now been published in very high standing journals on resolving some of
the paradoxes with the mathematics of quantum physics. The aim of this book is to collect the
contributions of world’s leading experts in quantum like modeling in decision making, psychology,
cognition, economics, and finance.
Citation: Haven, E., Khrennikov, A., eds. (2018). Applications of Quantum Mechanical Techniques
to Areas Outside of Quantum Mechanics. Lausanne: Frontiers Media.
doi: 10.3389/978-2-88945-427-3

Frontiers in Physics

3

February 2018  |  Quantum-Like Modeling

Table of Contents

05

07

22

36
47
57
75

89
97

116
127

140

152

Frontiers in Physics

Editorial: Applications of Quantum Mechanical Techniques to Areas Outside of
Quantum Mechanics
Emmanuel Haven and Andrei Khrennikov
A Hamiltonian Driven Quantum-Like Model for Overdistribution in Episodic
Memory Recollection
Jan B. Broekaert and Jerome R. Busemeyer
On the Foundations of the Brussels Operational-Realistic Approach to
Cognition
Diederik Aerts, Massimiliano Sassoli de Bianchi and Sandro Sozzo
Information and Temporality
Christian Flender
Toward a Quantum Theory of Humor
Liane Gabora and Kirsty Kitto
Quantum-like modeling of cognition
Andrei Khrennikov
The Physics of Teams: Interdependence, Measurable Entropy, and
Computational Emotion
William F. Lawless
Nilpotent Quantum Mechanics: Analogs and Applications
Peter Marcer and Peter Rowlands
Quantum Probabilistic Models Revisited: The Case of Disjunction Effects in
Cognition
Catarina Moreira and Andreas Wichert
Topological and Orthomodular Modeling of Context in Behavioral Science
Louis Narens
The Real and the Mathematical in Quantum Modeling: From Principles to
Models and from Models to Principles
Arkady Plotnitsky
Quantization, Frobenius and Bi Algebras from the Categorical Framework of
Quantum Mechanics to Natural Language Semantics
Mehrnoosh Sadrzadeh
Inconclusive Quantum Measurements and Decisions under Uncertainty
Vyacheslav I. Yukalov and Didier Sornette

4

February 2018  |  Quantum-Like Modeling

EDITORIAL
published: 24 November 2017
doi: 10.3389/fphy.2017.00060

Editorial: Applications of Quantum
Mechanical Techniques to Areas
Outside of Quantum Mechanics
Emmanuel Haven 1* and Andrei Khrennikov 2
1

Faculty of Business Administration, Memorial University, St. John’s, NL, Canada, 2 Department of Mathematics, International
Center for Mathematical Modeling, Linnaeus University, Vaxjo, Sweden
Keywords: quantum-like paradigm, quantum field theory, quantum probability, quantum probability cognition
models, quantum information

Editorial on the Research Topic
Applications of Quantum Mechanical Techniques to Areas Outside of Quantum Mechanics

Edited and reviewed by:
Alex Hansen,
Norwegian University of Science and
Technology, Norway
*Correspondence:
Emmanuel Haven
ehaven@mun.ca
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 12 September 2017
Accepted: 09 November 2017
Published: 24 November 2017
Citation:
Haven E and Khrennikov A (2017)
Editorial: Applications of Quantum
Mechanical Techniques to Areas
Outside of Quantum Mechanics.
Front. Phys. 5:60.
doi: 10.3389/fphy.2017.00060

Frontiers in Physics | www.frontiersin.org

The recent quantum information revolution has tremendous consequences not only for physics.
It stimulates the use of quantum formalisms in various areas outside of quantum physics:
cognition, psychology, economics and finance, microbiology and genetics. This approach is known
as quantum-like modeling. For cognition, this modeling should be sharply distinguished from
attempts to represent information processing by the brain through quantum physical processes
(cf. with works of Penrose and Hameroff). For microbiology and genetics, quantum-like modeling
should be distinguished from quantum biophysics. In the quantum-like approach a biological
system (brain, cell) is considered as a black box processing information in accordance with the
laws of quantum information and probability.
In psychology one can now claim that quantum probability has reached the mainstream. Ideas
from quantum field theory now reach into applications to biology and medicine and economics
and finance. One of the papers in this special issue, by Marcer and Rowlands, does look at so called
“nilpotent quantum mechanics” a form of quantum field theory. The use of a functor in natural
language semantics, as proposed in the work of Sadrzadeh derives also from quantum field theory.
The overarching theme in the applications considered here in this special issue, is the use of the
so called “quantum-like paradigm.” As pointed out in the article by Khrennikov, social science is
confronted with probabilistic and “entangled” systems. In this special issue, the paper by Lawless
looks specifically at entanglement in his treatment of the interdependence of teams.
Each of the papers accepted for publication under our research topic “Applications of quantum
mechanical techniques to areas outside of quantum mechanics,” highlights a particular facet of this
new multi-faceted area of research.
Plotnitsky’s paper in our collection of papers is maybe the contribution which provides for an
overarching thinking template on all the work published here. The papers in our special issue
assume that the mathematical modeling of a social science bound phenomenon is possible. But
interestingly enough, as Plotnitsky remarks, even if we were to question such an assumption, it will
not necessarily lead to halting the use of mathematics in such modeling but rather it may result in
new modeling and even, maybe, new mathematics? The paper by Aerts et al. far from claiming that
mathematical modeling is impossible, does propose that new mathematical structures may well be
needed (structures which go beyond quantum structures) to model cognition.

November 2017 | Volume 5 | Article 60 | 5

Haven and Khrennikov

The idea of “probability waves,” a novel intuitive concept when
quantum mechanics was being formulated, was born out of the
double slit experiment. This brings us neatly to think about the
multiple cases of violations of the law of total probability. The
wave function is a fantastic device which helps us to understand
that there is no unique position, until a measurement is made.
As the paper by Flender carefully lays out, understanding this
uncertainty lies at the heart of so called temporality. Temporality
is a key ingredient in non-chronological time (what Flender calls
“time of acausality”) and it seems to define also information. The
paper by Yukalov and Sornette comes back to the interference
effect which is the result, from what they call, an inconclusive
event. Inconclusive events, as they rightly point out, underlie
many human decisions too. One may argue that their paper
transcends some of the results presented in this special issue,
as the model they propose can be used for both quantum
measurements AND decision making.
In the paper by Broekaert and Busemeyer the authors propose
a Hamiltonian based quantum-like model which allows for the
temporal evolution of memory states. Time is not the usual
physical time variable, but it is rather used for the temporal
ordering of states. The paper also carefully spells out the issue
of closed and open systems. Open systems have now also been
considered in areas other than psychology, such as political
science and economics. The paper by Khrennikov provides for
an overview.
Narens considers orthomodular lattice modeling in behavioral
science. The article queries why there may be a link between

Frontiers in Physics | www.frontiersin.org

Quantum-Like Modeling

the conservation principle and psychology and it also wonders
why Hilbert space based quantum probability may be relevant to
psychology.
Gabora and Kitto’s contribution develops a so-called quantum
theory of humor (i.e., the cognitive aspect of humor is
considered). An experimental study is set up to start defining the
state space of “humor.”
The work by Moreira and Wichert compare several models
which explain violations of the well know sure-thing principle in
expected utility.
We hope this special issue provides for a rich addition to the
problem of modeling social science phenomena with the help of
the quantum-like paradigm.

AUTHOR CONTRIBUTIONS
All authors listed have made a substantial, direct and intellectual
contribution to the work, and approved it for publication.
Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Haven and Khrennikov. This is an open-access article distributed
under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original
author(s) or licensor are credited and that the original publication in this journal
is cited, in accordance with accepted academic practice. No use, distribution or
reproduction is permitted which does not comply with these terms.

November 2017 | Volume 5 | Article 60 | 6

ORIGINAL RESEARCH
published: 23 June 2017
doi: 10.3389/fphy.2017.00023

A Hamiltonian Driven Quantum-Like
Model for Overdistribution in
Episodic Memory Recollection
Jan B. Broekaert 1, 2* and Jerome R. Busemeyer 3
1
Center Leo Apostel for Interdisciplinary Studies, Vrije Universiteit Brussel, Brussels, Belgium, 2 Department of Psychology,
City, University of London, London, United Kingdom, 3 Department of Psychological and Brain Sciences, Indiana University,
Bloomington, IN, United States

Edited by:
Emmanuel E. Haven,
University of Leicester,
United Kingdom
Reviewed by:
Kirsty Kitto,
Queensland University of Technology,
Australia
Yoshiharu Tanaka,
Tokyo University of Science, Japan
*Correspondence:
Jan B. Broekaert
jan.broekaert@vub.ac.be
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 12 November 2016
Accepted: 30 May 2017
Published: 23 June 2017
Citation:
Broekaert JB and Busemeyer JR
(2017) A Hamiltonian Driven
Quantum-Like Model for
Overdistribution in Episodic Memory
Recollection. Front. Phys. 5:23.
doi: 10.3389/fphy.2017.00023

Frontiers in Physics | www.frontiersin.org

While people famously forget genuine memories over time, they also tend to mistakenly
over-recall equivalent memories concerning a given event. The memory phenomenon
is known by the name of episodic overdistribution and occurs both in memories of
disjunctions and partitions of mutually exclusive events and has been tested, modeled
and documented in the literature. The total classical probability of recalling exclusive
sub-events most often exceeds the probability of recalling the composed event, i.e., a
subadditive total. We present a Hamiltonian driven propagation for the Quantum Episodic
Memory model developed by Brainerd et al. [1] for the episodic memory overdistribution
in the experimental immediate item false memory paradigm [1–3]. Following the
Hamiltonian method of Busemeyer and Bruza [4] our model adds time-evolution of
the perceived memory state through the stages of the experimental process based
on psychologically interpretable parameters—γc for recollection capability of cues, κp
for bias or description-dependence by probes and β for the average gist component
in the memory state at start. With seven parameters the Hamiltonian model shows
good accuracy of predictions both in the EOD-disjunction and in the EOD-subadditivity
paradigm. We noticed either an outspoken preponderance of the gist over verbatim
trace, or the opposite, in the initial memory state when β is real. Only for complex β a
mix of both traces is present in the initial state for the EOD-subadditivity paradigm.
Keywords: episodic over distribution, disjunction fallacy, subadditivity, quantum cognition, Hamiltonian operator

1. INTRODUCTION - THE EPISODIC MEMORY
In an early effort to systematize the developing science of memory, Tulving [5] aimed to provide
operative definitions for presumed various categories of memory. Continuing a dichotomic
approach, he proposed to complement the previously coined “semantic” memory with the
“episodic” memory. While our “semantic” memory would allow us to regain facts and abstract
knowledge about our world, our “episodic” memory would let us recall personally lived events in a
specific spatio-temporal context from our past. While distinct, both were still considered partially
overlapping information processing systems. With Mandler’s [6] dual process approach it became
more clear to distinguish the more contrived recollection by details with respect to the recall of
facts [7]. In the dual recollection-familiarity process models a cue is processed respectively either
in terms of remembering an event’s details up to its genuine recollection, or by retrieving a feature
which is associated to the cue so it becomes familiar and conflated with a truly episodic memory.

June 2017 | Volume 5 | Article 23 | 7

Broekaert and Busemeyer

Jacoby [8] pointed out a confusion of the recollectionfamiliarity process with the retrieval task itself. He urged for
explicit process dissociation providing two separate parameters
for the aspects of recollection—or intentional memory use—
and for familiarity—or automatic memory use—in the dual
process. In a further developed dual process approach the
“conjoint recognition” model of Brainerd et al. [9] proposes
separate parameters for the processes of; identity judgment,
similarity judgment, and response bias. The latter model
is able to implement the “fuzzy trace” theory (FTT) with
its identity vs. similarity distinction. Reyna and Brainerd
[10] crucially distinguishes verbatim and gist dimensions to
memories. Verbatim traces hold the detailed contextual features
of a past event, while gist traces hold its semantic—“fuzzy”—
details. Our brain would analyze a past event by accessing
its stored verbatim and gist trace in parallel. On the one
hand the verbatim trace of a verbal cue handles it “surface”
content—i.e., orthography and phonology for words—with its
contextual features, while the verbal cue’s gist trace will encode
“relational” content—i.e., semantic content for words—with its
contextual features. In more recent work the FTT model has
received a quantum probabilistic formalization to cope with
overdistribution in memory tests [1, 4, 11, 12]. While we
are essentially connecting to this line of research with our
present quantum model, a wide variety of recollection memory
models have been developed in the literature that are not
discussed here. We do refer to one specific semantic network
approach by Nelson et al. [13] and Bruza et al. [14] which also
infers quantum structures for its development. In essence their
model proposes a semantically associated network, in which
a target word is adjacent to all associated terms according
the natural language of its users. It has been shown best
prediction of memory performance is obtained by implementing
the network in a quantum superposition state of either complete
activation—amplitude 1—or non-activation—amplitude 0. The
model provides weighed directional word associations, and a
quantumlike entanglement between nodes is invoked to predict
parallel instead of serial activation of neighbors. We have not
included the Nelson and McEvoy model in our present discussion
since it has not been developed to explicitly implement a gistverbatim distinction with respect to which the EOD effect we
target here is developed.
The EOD effect. One striking phenomenon concerning
memory is the Episodic Over-distribution effect—or EOD. More
or less this effect expresses a person’s proneness to conflate
memories of distinct events. More precisely the effect points
out we tend to affiliate “alien” memories to true memories
concerning a given event, leading to an “exaggeration” of
memories concerning that event.
In Brainerd et al. [3] the disjunction fallacy is modeled
for the item false memory paradigm while the source false
memory version is covered in Brainerd et al. [15]. Brainerd et al.
[1] exposes the more common case of subadditivity of episodic
memory.
These EOD effects are shown in specifically designed
experiments: the item false memory experiment in 2015 is a
modification (also [2, 3, 9]) of a classical paradigm in which

Frontiers in Physics | www.frontiersin.org

A Hamiltonian Quantum Like Model for EOD

a single “instruction” (or probe) would be given to measure
whether “a given cue is a target (or not).”
EOD–subadditivity. In the item false memory (IFM)
experiment three possible cues “old”—or o, “new-similar”—or
ns, and “new-dissimilar”—or nd are presented. These cues are
crossed with three “probes” namely o?, ns?, and nd?. These probes
“o?, ns?, and nd?,” respectively, enquire the participant “is this
probe old?” (studied before), “is this probe new but similar?”
(semantically related to the old cues but not literally among
them) and “is this probe new and dissimilar?” (has nothing to
do with with the studied cues, even not semantically). In this
experimental paradigm after exposure to an unidentified cue the
participant is enquired by one of three distinct probes.
In practice most of the IFM experiments turn out subadditive
acceptance probabilities:
p(o?) + p(ns?) + p(nd?) > 1.

(1)

That is, the disjoint partial features are over recalled with respect
to its encompassing event. Notice that even if the law of total
probability would be satisfied Brainerd mentions the possible
issue of compensations; a systematic change in remembering ns
as such may compensate a reverse change to remember ns as o,
restoring the classical addition to 1.
EOD–disjunction fallacy. In the 2010-version of the IFM
experiment a disjunctive probe was presented to the participants
instead of the nd? probe. This probe questioned whether the
cue was either “old” or “new-similar,” leaving unnecessary the
answer to the question which one of both types the cue really was.
A comparison of acceptance probability under the disjunctive
probe o or ns? and the summed acceptance probabilities under
the separate probes o? and ns? revealed a subadditive relation
p(o?) + p(ns?) > p(o or ns?).

(2)

This relation amounts to a disjunction fallacy since both cue types
are mutually exclusive categories. The EOD effect was further
identified using the unpacking factor of [16]
p(o?) + p(ns?)
> 1.
p(o or ns?)

(3)

the excess value of the fraction above one gives a measure of the
amplitude of the effect.
Explanations of EOD effects. A number of theoretical
explanations have been provided to interpret this phenomenon:
The fuzzy trace theory was implemented in QEM—the
Quantum Episodic Memory model—by Brainerd et al. [1, 11].
By processing the perception of the verbatim and gist memory
trace as separate components of a state vector, QEM allows to
encompass the non-classical EOD probability effects of episodic
memory. This capacity, we will see in the next section, is
essentially due to the ubiquity of the gist component and its
implementation in the corresponding outcome projectors for
acceptance. Another quantum-like modeling perspective has
been proposed by Busemeyer and Bruza [4, Ch.6] which provides
unitary transformation matrices based on Feynman path analysis

June 2017 | Volume 5 | Article 23 | 8

Broekaert and Busemeyer

and ordering of the gist/verbatim processing of cues, which we
will discuss below. Finally a complementarity based quantumlike development was done by Denolf and Subadditivity [17], and
Denolf and Lambert-Mogiliansky [18]. Bohr’s complementarity
provides the gist–verbatim features by implementing for each
an alternative basis of the Hilbert space. Our Hamiltonian
development follows more closely the outline of the QEM model
for FTT.
We will at present not make comparisons to either Markovian
models [4, 19, 20] but focus on the possibility of Hamiltoniandriven time propagation in QEM, and compare to the Feynman
path model and the original QEM model itself.
Experimental paradigms. A number of experimental
paradigms have been proposed to test the over-distribution effect
and the episodic disjunction effect. In this paper we mainly refer
to Brainerd et al. [1, 3] which build and expand on “item false
memory” and “source false memory” experimental paradigms,
but only the former IFM paradigm will be modeled here. We
shortly describe both paradigms of 2010 and 2015 for the IFM
case. As we have mentioned, each of these paradigms consist of
two consecutive stages.
2010 experimental paradigm. In the first stage participants
studied a set {o} of memory targets consisting of words from a
list of the Deese–Roediger–McDermott paradigm (DRM). The
presented DRM lists are abbreviated sequences of the original
15 semantically related words that all associate forward to one
common word. That latter word does not appear in the list and
is therefore known as the distractor [21, 22]. We will in our
approach not include the issue of the preliminary orienting task
based on qualifying adjectives as positive, neutral, or negative,
which “increase the processing of semantic content during
subsequent exposure to word lists” [3]. After this memorization
stage either immediate testing ensues or a time delay of a week is
inserted. Subsequently a cue is presented to the participant and
finally an instruction to respond to the cue is given.
Three possible types of cue are used; a studied target from
{o} consisting of a word from one of the 24 lists, a related nontarget from set {ns} consisting of words on the list but not learned
({o} ∩ {ns} = φ) and finally a new-dissimilar non-target from set
{nd} with words not related in any sense to the selected DRM lists
({nd} ∩ {ns} = φ = {nd} ∩ {o}).
These cues are crossed with one of three instructions per
participant1 . Either; the first instruction o? (or old?) to accept
only an exact target from {o} and otherwise reject, or the second
instruction ns? (new-similar?) to accept only a related nontarget from {ns} otherwise reject, or a third instruction. The third
instruction is o or ns? (or old or new-similar? ) to accept either
an exact target from {o} or a related non-target from {ns} and
otherwise reject.
2015 experimental paradigm. The alternative version of the
IFM paradigm of Brainerd et al. [1] follows precisely the two
stages of the 2010-version except for the final stage. First the
1 We adopted the notation of Brainerd et al. [1] in the context of Brainerd et al.
[3] as well. Always cues will be denoted o, ns, and nd for old, new-similar, and
new-dissimilar, and their respective enquiring probes are o?, ns?, and nd?. Memory
traces are denoted by V, G, and N for verbatim, gist, and neither.

Frontiers in Physics | www.frontiersin.org

A Hamiltonian Quantum Like Model for EOD

participants studied cues c of memory target words (24 times 6 in
total). Then a time delay is either inserted or not. in the test phase
participants are first exposed to a cue which is either a studied
target from o, a new-similar non-target ns, or a new-dissimilar
non-target nd. Finally the participant is asked to respond to one
of three probes querying to which category the cue belongs; that
is o?, ns?, or nd?. In comparison to the 2010-paradigm the o or ns?
probe has been replaced by the nd? probe.
About the source false memory paradigm. In source false
memory experiments the experimental paradigm focuses on the
origin of the cue. It probes the source recollection in memory
of cues originating from either List 1 or List 2 and crossed with
probes List1?, List2?, and nd?. We will only focus our present
Hamiltonian based quantum model on the IFM setting, it is
however very possible to adapt the model to SFM requirements
as well.
Besides the QEM model, this specific paradigm has been
alternatively modeled by Denolf and Lambert–Mogiliansky using
Bohr’s quantum approach to consider gist and verbatim traces
as complementary properties, each trace represented by an
alternative bases of the same Hilbert space [18].
Experimental data in 2010 and 2015 paradigms. Since
Brainerd et al. [1] focuses on subadditivity with the probes o?,
ns?, or nd?, there is no interest in o or ns? thus it is not measured
nor reported. While vice versa [3] has a focus on the disjunction
fallacy, which reports o? , ns?, and o or ns? but does no reporting
of nd cue data. We therefore take the data of Brainerd et al.
[9] from which a full 3 × 3 grid of data can be reconstructed
using an intervention proposed by Busemeyer and Bruza
[4, p. 171].
In sum we have no data set which shows the subadditivity and
the disjunction effect at the same time. We will thus adapt the
parameters of the Hamiltonian model to each data set separately
(see Tables 1, 2). We adopt Busemeyer’s solution to complete
the data set in the paradigm for the EOD disjunction effect
by supplementing the nd probe data in the set through their
response bias measures bT , bR , and bT+R ([9]–Table 6). Moreover,
we will fit to the average values over six experimental conditions
here2 . While for the EOD subadditivity effect we will use ([1],
Table 3, p. 233) in which we take the values for ns cue as
the averages of ns-critical and ns-related cues, distinguished by
Brainerd et al. [1].

2. QUANTUM MODELS
Probabilistic anomalies with respect to classical probability law
have in many cases been successfully covered by models using
quantum formalism, likewise the anomalies of EOD have been
modeled in quantum-like manner. We shortly present some of
these developments, mainly focussing on QEM.

2.1. The QEM Model by Brainerd et al.
Memory state vectors. QEM provides three orthogonal vectors
in Hilbert space, respectively one for (verbal) surface form,
2 In

the conjoint recognition model the probabilities for acceptance for unrelated
distractors are: pu,T = bT , pu,R = bR , and pu,T+R = bT+R , ([9], Equations 19–21).

June 2017 | Volume 5 | Article 23 | 9

Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

TABLE 1 | EOD-disjunction fallacy: Experimental and predicted acceptance probabilities by probe and cue in item false memory paradigm - immediate test [9].
Cue

Source

Probe
o?

o

ns

nd

Unpack.

ns?

Conj.

RMSE

b?

Exp

0.70

0.13

0.78

1.06

0.05

Pred β > 0

0.7137

0.1325

0.7770

1.09

0.07

0.0073

Pred β < 0

0.7302

0.1486

0.7180

1.22

0.16

0.0341

Pred-BB2012

0.70

0.16

0.83

1.04

0.03

0.0298

Pred-Br2015

0.7400

0.1300

0.7400

1.18

0.13

0.1112

Exp

0.32

0.66

0.71

1.38

0.27

Pred β > 0

0.3172

0.6609

0.7113

1.38

0.27

0.0073

Pred β < 0

0.3254

0.6431

0.7797

1.24

0.19

0.0341

Pred-BB2012

0.35

0.66

0.71

1.04

0.03

0.0298

Pred-Br2015

0.5633

0.5633

0.5633

2.00

0.56

0.1112

Exp

0.13

0.22

0.32

1.09

0.03

Pred β > 0

0.1459

0.2184

0.3183

1.14

0.05

Pred β < 0

0.1343

0.2287

0.3308

1.10

0.03

0.0341

Pred-BB2012

0.07

0.22

0.33

0.88

0.0298

Pred-Br2015

0.2250

0.2200

0.2250

1.98

−0.04

0.22

–

–

–
0.0073

0.1112

The experimental p(p?|c) are averages over six experimental conditions. Corresponding unpacking factor and “conjunction probability” values are listed. The Hamiltonian model has a
very good prediction RMSE = 0.0073 for β > 0, and a less good RMSE = 0.0341 for β < 0. Fitting attempts for a β ∈ C gravitated toward the β < 0 solution, i.e., phase(β) → π
and have therefore not been included. The predictions of the Feynman path based model [4] have an RMSE = 0.0298 and are indicated with Pred-BB2012. The re-calculated QEM
predictions [1] are indicated with Pred-Br2015 and have an RMSE = 0.1112.

one for semantical relatedness, and one for the case when
neither of both previous are relatedly present. In line with the
FTT these respective dimensions acquire probability amplitudes
that represent the participant’s mental state on the cue in the
experiment, which we order as (vc , gc , nc )τ 3 . The fact that these
features are expressed by orthogonal vectors, reflects that these
are perceived distinct properties of a word in memory. This
orthogonality property should be differentiated from associative
relationships of words like e.g., for a target word in a semantic
memory network [13], which dominantly hinges on related gist
but mostly leaves out related verbatim features. Brainerd et al.
[1] and Brainerd and Reyna [2] describe the “perceived memory
state” spanned by vectors in three-dimensional Hilbert space
corresponding to verbatim, gist, and non-matching dimensions
of the respective fuzzy traces for the set of words in the
experimental paradigm in the brain:
|Sc i = vc |Vi + gc |Gi + nc |Ni

(4)

Where c can be any cue type, o, ns, or nd, and each basis vector
corresponds to respectively the fuzzy trace of form (V), semantic
relation (G), and complete unrelatedness (N)4 . According the
3 We use the symbol τ to designate the transpose of a vector or matrix. Basically
transposition turns columns into rows and vice versa.
4 We recall that state functions or vectors in quantum-like models for cognitive
processes will always represent averages of the participant group. Individual
memory state vectors are not envisaged in this approach: as all humans are
allegedly equal but rather existentially different the average state function does
not reflect the individual’s memory state. We emphasize the difference with the
situation in the micro-physical realm; e.g., the state function of an ensemble of

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model requirements—exhaustiveness and exclusiveness of the
cues—the respective probabilities add up to unity
|vc |2 + |gc |2 + |nc |2 = 1.

(5)

With these three normalizations constraints QEM requires the
parameters {vo , go , no , vns , gns , nns , vnd , gnd , nnd } of which six are
independent. We discuss some related fitting issues in QEM at
the end of Section (2).
Acceptance projectors. A probe o?, ns?, or nd? is
affirmatively—“yes”
(y)—answered
by
applying
the
corresponding projection operator







1 0 0
0 0 0
0 0 0
My,o? =  0 1 0  , My,ns? =  0 1 0  , My,nd? =  0 0 0 
0 0 0
0 0 0
0 0 1
(6)
on the state |Sc i. These respective projector matrices are simply
obtained by considering the final outcome vectors which they
need to produce. In VGN space the projector My,nd? should
lead to a vector proportional to (0, 0, 1), representing perception
of no related verbatim nor gist of the nd cue. The form of
this expected outcome vector (0, 0, 1) is directly related to the
projector expression Equation (6, c). Similarly the projector
My,ns? , Equation (6, b), is constructed from the expected outcome
vector (0, 1, 0) representing only perception of related gist in
the ns cue. For the projector matrix My,o? the outcome should
identically prepared electrons does reflect the behavior of an individual electron
since all electrons are equal, not just allegedly.

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A Hamiltonian Quantum Like Model for EOD

TABLE 2 | EOD-subadditivity: Experimental and predicted acceptance probabilities by probe and cue in item false memory paradigm—immediate test [1].
Cue

o

Source

RMSE

ns?

Exp

0.53(0.19)

0.43(0.19)

0.26(0.16)

1.22

Pred β > 0

0.5626

0.4165

0.2237

1.20

0.0565

nd?
–

Pred β < 0

0.5523

0.4454

0.2966

1.29

0.0191

Pred β ∈ C

0.5273

0.4336

0.2540

1.21

0.0032

0.6350

0.4300

0.3650

1.43

0.0963

Exp

0.42(0.26)

0.57(0.22)

0.31(0.22)

1.30

Pred β > 0

0.3415

0.6348

0.3338

1.31

0.0489

–

Pred β < 0

0.4184

0.5508

0.2899

1.26

0.0191

Pred β ∈ C

0.4226

0.5674

0.3130

1.30

0.0032

0.560

0.560

0.440

1.56

0.0963

Pred-Br2015
nd

Sum

o?

Pred-Br2015
ns

Probe

Exp

0.17(0.18)

0.34(0.22)

0.62(0.19)

1.13

Pred β > 0

0.2393

0.2765

0.5400

1.06

0.0489

–

Pred β < 0

0.1677

0.3244

0.6059

1.10

0.0191

Pred β ∈ C

0.1676

0.3403

0.6222

1.13

0.0032

0.2967

0.2967

0.7033

1.30

0.0963

Pred-Br2015

The experimental p(p?|ns) are the averages of critical and related distractor cues. The Hamiltonian model with β > 0 constraint has a prediction accuracy RMSE = 0.0489, the β < 0
constraint gives slightly improved RMSE = 0.0191, while for β ∈ C we have a strongly improved RMSE = 0.0032. The re-calculated QEM predictions [1] are indicated with Pred-Br2015
and have an RMSE = 0.0963.

lead into the space spanned by both related verbatim and gist
components for the perception of the o cue. The latter is a
two dimensional space spanned by the basis vectors (1, 0, 0) and
(0, 1, 0), and is the outcome space of projector Equation (6, a).
The β-parameter—present in the vector |oi for old cues—will
allow to navigate such vectors in this two-dimensional subspace,
altering the relative weight of the verbatim to gist components
(see functioning of β in the description of the initial state, below).
In the experimental paradigm for the EOD disjunction fallacy
the operator My,b? for the probe o or ns ? is used;
My,b? = My,o? + My,ns? − My,o? My,ns?
= My,o?

(7)

since My,o? My,ns? = My,ns? . In QEM the memory state for
the experimental paradigm is posited to be |Sc i following the
exposure to cue c. After providing the probe p the state collapses
to the eigenvector of the projection operators My,p : “the cue elicits
the memory state, and the probe determines the projector used to
answer [affirmatively to] the question.” ([1], p. 243).
The origin of EOD in QEM. Notice that the form of
the projectors My,o? and My,ns? show that subadditivity is
an immediate consequence of measuring the presence of the
common gist trace in both operations. Which also implies—
as Brainerd points out—the cases in which a gist trace would
be lacking will not produce subadditivity. Similarly, we could
remark that in the dual trace approach ns? is a subspace of
o?. Therefore, the operator for the disjunctive probe coincides
with the operator for the o? probe. As a consequence the EOD
disjunction fallacy is not due to an interference dynamics in
the QEM model, but follows from “double counting” the gist

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component in the outcome of disjunctive probe. Also the EOD
subadditivity is due to this same double counting of gist. Both
subadditivity and the disjunction fallacy are therefore considered
‘’parameter-free” features of the QEM model [1]. In Section 2.3,
we will cover the origin of the EOD effect more extensively and
show how in our Hamiltonian approach of QEM one is not
restricted to subadditive nor fallacious disjunctive scenarios.
The initial state. A short discussion on the initial state
vector in the QEM model is needed since it plays an important
role, both in Brainerd et al.’s development of QEM and our
Hamiltonian driven version of the model. At the start of the
experiment the participants of the experiment are informed
about the equal probability by which each type of cue o, ns,
or nd, will be presented [1]. It can be easily seen however
that this is not possible to implement exactly without forcing
this initial perceived memory state to be voided of all of its
verbatim trace5 . We claim a more appropriate representation of
this initial state is done by addressing this uncertitude on the level
of the probability amplitudes, not the probabilities themselves.
More specifically, we implement each√component probability
amplitude is attributed equal weight 1/ N in the initial state
1
|ψ0 i = √

N



|oi + |nsi + |ndi , with k|ψ0 ik = 1.

(8)

5 Brainerd

et al. [1, p. 239], mentions participants would have roughly
p(o) = p(ns) = p(nd) = 1/3 as baseline probabilities prior to study of {o}. Let
the initial state be represented in VGN space as (α, β, γ )τ . Using the appropriate
projection operators, Equation(6), we find p(o) = |α|2 + |β|2 , p(ns) = |β|2 and
p(nd) = |γ |2 . Equating them all to 1/3 requires α = 0, reducing the perceived
related verbatim trace to nought.

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A Hamiltonian Quantum Like Model for EOD

where N is the vector’s normalization factor. This initial state
can be expressed in terms of perceived verbatim, gist, and
unrelated components. The o-state is composed of components
of verbatim and gist in the perceived memory according a
superposition
of both; |oi = α|Vi+β|Gi or explicitly normalized
p
2
( 1 − |β| , β, 0)τ , where β ∈ C6 . Thus both aspects V and G
contribute with a variable amplitude to the targeted cue o—which
a priori should have been expected since the relative magnitude
of both traces seem variably dependent on the particular instance
of the o-type cue. The memory states for ns and nd on the other
hand do not decompose over multiple traces, and coincide with
the unambiguous eigenvectors of the respective operators My,ns?
and My,nd? , i.e., |nsi = |Gi and |ndi = |Ni. The initial state prior
to cue and probe presentation can thus be expressed in terms of
orthogonal states for V, G, and N 7 :
1
|ψ0 i = √

N



α|Vi + (1 + β)|Gi + |Ni , with |α|2 + |β|2 = 1
(9)

from which we find an expression of the normalization factor
N =

p

|α|2 + |1 + β|2 + 1 =

p

3 + 2ℜ(β)

of the initial state. Most importantly, we have a variable β in
our description, which stands for the average amplitude of gist
trace in the true target set {o} of the experimental paradigm. We
assume that o cues with little relevance to the participants will
correspond to low β, while o cues common to the participants
will increase β.
Given the participant is informed she will be exposed to an
equal amount of o, ns, and nd cues, overall she will expect an
excess of gist in comparison to verbatim or unrelated features.
A “constructive interference” in 1 + β with β > 0 would be
expected (when β ∈ R). In the present experimental paradigm
the cues are semantically forward related words (to its target
word) of the DRM lists, therefore we would expect low or
moderate associated gist traces here, certainly not really intense
gist traces as for instance the Madeleine-cue provoked in Marcel
Proust.

2.2. The Feynman Path Model for EOD by
Busemeyer et al.
The Feynman path model by Busemeyer and Trueblood [23] and
Busemeyer and Bruza [4, Ch. 6] introduces a four dimensional
Hilbert space to encompass the two orderings of the types of
process; verbatim before gist on o cues, and gist before verbatim
on ns and nd cues. This model thus provides a cue dependent
construction of the memory state.
6 An explicit

eigenvector |o(α, β)i of My,o? is given by My,o? |o(α, β)i = |o(α, β)i =
[α, β, 0]τ = α|Vi + β|Gi, with |α|2 + |β|2 = 1. Evidently there is a possible
denomination issue caused by the relative weight of both components, since
diminishing α will eventually turn an o state
p indiscernibly into an ns state.
7 The equally weighed initial state 1/N ( 1 − |β|2 , 1 + β, 1)τ was obtained by
giving each type of cue’s vector |oi, |nsi and |ndi equal weight at start. Our
implementation however does neither reflect equal baseline probability of o, ns,
and nd in the participants memory state as aimed for by Brainerd et al. [1], also
here one cannot have p(o) = p(ns) = p(nd) at the start.
√ For real-valued β, the
initial probabilities come at the closest for β = −2 + 3 at po = 0.59, pns = 0.22,
pnd = 0.41.
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As in the QEM model, the Feynman path approach does not
concatenate reflection time periods. The exposure of the cue or
the probe to the participant does not engender a unitary time
evolution of the memory state. Notably this model provides cuedependence of evolution by ordering verbatim and gist stages
in the process of recollection and depends on interference of
probability amplitudes to form the acceptance probability in the
disjunctive b? probe. Busemeyer and Bruza [4] model requires
only 6 parameters for a satisfactory prediction of the 9 data
points of the disjunctive EOD paradigm. The predictions of
the Feynman path based model by Busemeyer et al. have been
included in the data (Table 1). A short comparative discussion of
the model’s prediction capacity is given at the end of Section 2.
We summarize the Feynman paths in this model and have
adapted the notation of Busemeyer and Bruza [4, Ch. 6] to
conform with the present context8 . We inserted the question
mark to distinguish a probe—o?—from a cue o. The negation of a
probe is indicated by the tilde sign—e.g., õ?—and corresponds to
the negation of the instruction “Is this not an o cue?” This allows
to express the complementarity of the cases o? and õ? according:
|oiho| + |õihõ| = I

(10)

For o cues, verbatim is treated before gist, which means first o?
operates on the initial state |So i for o cues, then followed by the
operation of ns?:
p(o?|o) = |ho|So i|2 ,

p(ns?|o) = |hns|So i|2 = |hns|oiho|So i + hns|õihõ|So i|2 ,
p(b?|o) = p(o?|o) + p(õ?|o)p(ns?|õ)

= |ho|So i|2 + |hõ|So i|2 |hns|õi|2 ,
2
= 1 − |hns|õi|
˜
|hõ|So i|2 ,

requiring two parameters; hns|oi and hns|õi.
For ns cues gist is treated before verbatim, then first ns? operates
on the initial state |Sns i for ns cues, followed by o?
p(o?|ns) = |ho|Sns i|2 = |ho|nsihns|Sns i + ho|nsih
˜ ns|S
˜ ns i|2 ,

p(ns?|ns) = |hns|Sns i|2 ,

p(b?|ns) = p(ns?|ns) + p(ns?|ns)p(o?|
˜
ns)
˜

= |hns|Sns i|2 + |hns|S
˜ ns i|2 |ho|nsi|
˜ 2,

2
= 1 − |hõ|nsi|
˜ 2 |hns|nsi|
˜
,

requiring two more parameters ho|nsi
˜ and ho|nsi.
Also for nd cues gist is treated before verbatim
p(o?|nd) = |ho|Snd i|2 = |ho|nsihns|Snd i + ho|nsih
˜ ns|S
˜ nd i|2 ,

p(ns?|nd) = |hns|Snd i|2 ,

˜
p(b?|nd) = p(ns?|nd) + p(ns?|nd)p(o?|
˜
nd)
˜ 2,
= |hns|Snd i|2 + |hns|S
˜ nd i|2 |ho|ndi|
˜ 2 |hns|S
= 1 − |hõ|ndi|
˜ nd i|2 ,

8 The original notation V for “verbatim,” R for “related,” and U for “unrelated” cues

are here replaced by o, ns, and nd, respectively. One should be attentive to the fact
that V stood for “is the cue verbatim?” (actually meaning old), it does not stand for
the verbatim trace of QEM.
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A Hamiltonian Quantum Like Model for EOD

without new parameter requirements. The parameters appear as
elements of unitary
p transformations and must satisfy unitarity.
Leaving hns|oi = 1 − |hns|õi|2 = hns|õi
˜
and hns|oi
˜
= −hns|õi⋆ .
The initial state is described in a four dimensional Hilbert space,
in which the initial state depends on the presented cue:
|Si = |oo ihoo |Si + |õo ihõo |Si + |oõ ihoõ |Si + |õõ ihõõ |Si,

= |nso ihnso |Si + |ns
˜ o ihns
˜ o |Si + |nsõ ihnsõ |Si + |ns
˜ õ ihns
˜ õ |Si.

where the index represents the cue type. The first expression is
applicable for target cues from {o}, thus |So i (where the index
õ indicates either ns or nd). And the second expression for
the initial state is applicable when the cue is not a target but
comes from {ns} ∪ {nd}, thus for |Sns i and |Snd i. Therefore,
hoõ |So i = hõo |So i = 0 and hnso |Sõ i = hns
˜ o |Sõ i = 0. A
simplification of the formalism is obtained by chosing the phase
ϕ of hns|õi and the phase θ of ho|nsi
˜ equal to each other. This
choice corresponds to a simplification of the dynamics in the
subspaces of the 4-dimensional Hilbert space, in which the gistbefore-verbatim states and the verbatim-before-gist states differ.
Equating the phases on the transition components is considered
a compromise between reducing parameters and prediction
accuracy.
The final six parameters for the Feynman path
model for EOD of Busemeyer et al. then are
{ho|So i, hns|Sns i, hns|Snd i, p
|hns|õi|, |ho|nsi|,
˜ θ }, where pho|So i =
p
p(o?|o), hns|Sns i =
p(ns?|ns), hns|Snd i =
p(ns?|nd),
while |hns|õi|, |ho|nsi|
˜ and the phase angle θ are used to fit the
remaining six data points.

2.3. The Hamiltonian Driven QEM Model
A Hamiltonian based quantum-like model allows the description
of temporal evolution of the perceived memory state of
participants through the stages of the experiment. Although,
the explicit time-dependence of states in this approach would
in principle allow response time predictions, the main goal is
to describe increasing and decreasing tendencies building up
toward the point of decision. We emphasize that while we let
“t” stand for time in our model, it is rather to be considered as
an indicative parameter for temporal ordering than “physical”
time [24].
States and probabilities. Following the Hilbert space
construction of the QEM model, the memory states are
conceived to have one component for accepting o (memory
target, old cues), one component for accepting ns (new
semantically related cues), and one component for accepting nd
(unrelated, new-dissimilar cues). Expressed on the orthogonal
basis vectors for the fuzzy traces the state function following the
VGN ordering of components is denoted as
9probe|cue (t) = [ψp|c V (t), ψp|c G (t), ψp|c N (t)]τ ,

(11)

A state vector is thus defined separately for each combination
of cue in {o, ns, nd} and probe in {o?, ns?, nd?} for partition
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p(o?|o) = |ψo?|o V |2 + |ψo?|o G |2 ,

p(o?|ns) = |ψo?|ns V |2

+ |ψo?|ns G |2 ,

p(ns?|o) = |ψns?|o G |2 ,

p(ns?|ns) = |ψns?|ns G |2 ,

p(nd?|o) = |ψnd?|o N |2 ,

p(nd?|ns) = |ψnd?|ns N |2 ,

p(b?|o) = |ψb?|o V |2 + |ψb?|o G |2 ,

p(b?|ns) = |ψb?|ns V |2

+ |ψb?|ns G |2 ,
(12)

and
p(o?|nd) = |ψo?|nd V |2 + |ψo?|nd G |2

p(ns?|nd) = |ψns?|nd G |2

p(nd?|nd) = |ψnd?|nd N |2

p(b?|nd) = |ψb?|nd V |2 + |ψb?|nd G |2

(13)

where the instruction o or ns? is denoted by shorthand b?—
for “both” o? or ns?. We have noted previously in FTT theory
under b? probe the amplitudes of the V component and
the G component both are in the acceptance subspace. This
leads to formal similarity but not numerical equivalence of the
probabilities p(o?|o) and p(b?|o)—idem for conditionalization on
probes ns and nd—since memory is description-dependent ([1,
3])9 . The quantum model can thus provide explicit expressions
for both the unpacking factor and the subadditivity.
Unpacking factor and subadditivity. First we discuss the
expression for the subadditivity, Equation (1), for some cue c;
p(o?|c) + p(ns?|c) + p(nd?|c) = |ψo?|c V |2 + |ψo?|c G |2

+ |ψns?|c G |2 + |ψnd?|c N |2

= |ψo?|c V |2 + |ψo?|c G |2

+ |ψo?|c N |2 − |ψo?|c N |2




+ |ψns?|c G |2 + |ψnd?|c N |2

= 1 + |ψns?|c G |2 + |ψnd?|c N |2
− |ψo?|c N |2

(14)

We remark that our Hamiltonian driven account of QEM
does not necessarily imply subadditivity of total acceptance
probability10 . Mostly the QEM model will imply subadditivity

with
|ψp|c V |2 + |ψp|c G |2 + |ψp|c N |2 = 1.

subadditivity and in {o?, ns?, o or ns ?} for disjunction EOD.
In contrast with Brainerd et al.’s QEM approach our method
results in nine different state vectors—for each the experimental
paradigms—that are obtained by adapting the Hamiltonian
depending on the choice of probe and the choice of cue. Under
a specific instruction probe and cue, the acceptance probabilities
are obtained by applying the projectors (Equation 6) to the final
state and take the modulus square of the outcome. All acceptance
probabilities for both paradigms are then explicitly given by:

9 Meaning probe-dependence by differing κ and κ in the second stage
o
b
propagation.
10 Brainerd et al. [1, p. 19]) mentions: “[. . . ] a distinct memory state vector is
generated for each of the three types of cues, with corresponding amplitudes vC ,

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

of total acceptance probability as long as a gist component is
present in the memory state [1]. If however in some instance
the gist trace is weak and the unrelated trace N is strongly
description-dependent such that—unexpectedly—the probe o?
engenders stronger response than the probe nd?, it is possible to
have superadditivity in our Hamiltonian driven QEM model.
Next we shortly discuss the expression for the unpacking
factor, Equation (3), for some cue c, which we find by replacing
the respective acceptance probabilities by their modulus squared
amplitude components, Equations (12, 13);
|ψns?|c G |2
p(o?|c) + p(ns?|c)
= 1+
p(b?|c)
|ψb?|c V |2 + |ψb?|c G |2

+

|ψo?|c V |2 − |ψb?|c V |2 + |ψo?|c G |2
−|ψb?|c G |2
|ψb?|c V |2 + |ψb?|c G |2

(15)

We thus remark that our Hamiltonian QEM approach will mostly
show an EOD disjunction fallacy when a gist component is
present in the memory state. However again, when the gist trace
is weak in the ns? condition and the verbatim V and gist trace
G are strongly description-dependent such that the probe b?
engenders stronger response than the probe o?, it is possible
to avoid EOD in the QEM model (rhs will be less than 1 as
the second fraction becomes very small and the final fraction
becomes negative and sufficiently dominant).
Subsequent reflection periods. The experimental paradigm
essentially shows two reflection periods in the participants; the
first period involves processing a cue from {o, ns, nd} after it is
presented, the second period concerns the processing of a probe
from {o?, ns?, nd?} after that one has been presented.
In the first period the participant will do a descriptionindependent effort to evolve the equally weighed initial state
(Equation 8)—a-expressed in VGN base (Equation 9)– as good
as the participant can to one that corresponds to the presented
cue. This type of reflection will be represented by a dedicated cuedependent Hamiltonian Hc , thus requiring three parameters in
total to cover the first stage of the full experimental paradigm.
In the second period the participant receives the probe
instruction and possibly changes her attitude toward the
perception in the first stage, allowing for description-dependent
processing. The input of new information by the probe in the
participant’s mind engenders a change of dynamics (e.g., [25]).
This second type of reflection will thus proceed along a different
Hamiltonian Hp? also requiring three parameters to cover the
experimental paradigm.
First reflection period. We specify now the Hamiltonians
describing the reflection of the first period following the
presentation of the respective cues. This stage will change the
memory state from an undecided equally weighed one to a
gC , and nC .” (see Equations 4, 5). Our present Hamiltonian take of the QEM
structure provides nine memory state vectors. Starting from one single initial state
our Hamiltonian dynamics provides a distinct state vector for each of the nine
configurations of the three cues crossed with the three probes. Therefore we have
nine normalization conditions of the vectors (Equation 11), and can have some
modulation in the unpacking factor and in the subadditivity expression.

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state that reflects the recognition of the cue’s nature by the
participant. Since the Hamiltonian is the generator of change
over infinitesimal time we can model it to cause the required
transitions11 .
Reflection following ns and nd cue. For the reflection
following the presentation of the cue we will construct a
superposition of 2 × 2-dimensional Hadamard gates that transfer
probability amplitude mass toward the targeted components of
the state vector in VGN-space ([4, Ch. 8], [24, 26]). One can
see however that higher matrix powers of such Hamiltonians
will not show the simple closure of transitions we find when
using single parametrized Hadamard gates. Except for shedding
the possibility of simple analytical calculation of the unitary
evolution operator this does not alter the essence of the dynamics.
In the present model we will use parametrized Hadamard gates
with off-diagonal appearance of the parameter12 . We derive
Hamiltonians for the presentation of the ns and nd cue based
on their respective target states (0, 1, 0)τ and (0, 0, 1)τ . On the
presentation of an ns cue to the participant the amplitude mass
has to shift from verbatim trace to gist trace and from the
unrelated trace to the gist trace. In the perceived memory state
vector of the VGN-space this means the Hamiltonian must
transfer amplitude from 1st to 2nd entry and from 3rd to 2nd
entry of the perceived memory state vector:
Hns (γns ) = G12 (γns ) + G32 (γns ),


−1 γns 0
1
⋆ 2 γ .
 γns
= p
ns
2 +1
⋆ −1
γns
0 γns

(16)

Where γns will be the parameter describing the participant’s
ability to recognize an ns cue (γns ∈ R).
Similarly when an nd cue is presented to the participant the
amplitude mass has to shift according the targeted vector from,
from verbatim to unrelated and from gist to unrelated. This
means that the dedicated Hamiltonian must transfer amplitude
from 1st to 3rd entry and from 2nd to 3rd entry of the perceived
memory state in VGN-space.
Hnd (γnd ) = G13 (γnd ) + G23 (γnd ),


−1 0 γnd
1
 0 −1 γnd  .
= q
2 +1
⋆ γ ⋆ 2
γnd
γnd
nd

(17)

11 Applying the Hamiltonian to the initial state gives a first-order approximation of

the change of the state vector for an infinitesimal time interval:
ψδt − ψ0 ≈

iδt
Hψ0
h̄

This allows us to design the Hamiltonian according the needs of the cognitive
process.
12 E.g.,:




1 h 0
1 0 0
1
 h⋆ −1 0  with G21 (h)2 =  0 1 0  .
G21 (h) = p
1 + |h|2
0 0 0
0 0 0
This modification retains the rotation effects of the operator and squares to
the unity operator in VG-space. A main advantage of the present form is the
oscillations of probability over time stop when the parameter is set equal to zero.

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

Where γnd will be the parameter describing the participant’s
ability to recognize an nd cue, (γnd ∈ R).
Reflection following o cue. The Hamiltonian for the dynamics
after o cue presentation
to the participant is again based on its
p
target state vector ( 1 − |β|2 , β, 0)τ . When the o cue is presented
to the participant the amplitude mass has to shift from unrelated
to verbatim and from unrelated to gist. In this case with the o
cue however, both processes must not occur at the same rate. The
dedicated Hamiltonian has to transfer amplitude
p form 3rd to 1st
and from 3rd to 2nd with the respective rates 1 − |β|2 and β in
accordance with the target vector state. Moreover the initial gist
component needs to be redistributed according the target vector
as well, leading
to a complementary transfer from 2nd to 1st entry
p
with rate 1 − |β|2 :
p
p
Ho (γo , β) = G21 (γo 1 − |β|2 ) + G31 (γo 1 − |β|2 )
+ G32 (γo β).

(18)

Where γo will be the parameter describing the participant’s ability
to recognize an o cue, (γo ∈ R, β ∈ C)13 .
Second reflection period. Following the reflection period after
the presentation of the cue, the participant is presented with
a probe stemming from o?, ns?, nd? and matches it with
her recollection memory state post first stage. This comparison
can either lead to an affirmation of the probe or a challenge.
Coincidence of perceived cue and probe may induce to some
degree a tendency to affirm one’s memory state, while contrasting
cue and probe may to some degree induce a challenge or
cognitive dissonance. We remark that affirmation and challenge
are relative; in an o-run a participant with an ns-recollection will
consider the o?-probe as a challenge rather than a confirmation.
The terms affirmation and challenge clearly take their meaning
only for the inter-participant average of acceptance probabilities,
not in general for individual intra-participant occasions (see
Table 3). In the second reflection period the probe thus either
affirms or challenges the recollection effort of the first stage,
dynamically this corresponds to either an amplified continuation
of the first stage dynamics or a reversed evolution with regard to
the probe:
Ho? (κo , γ0 , β) = Ho (κo γo , β),

(19)

Hnd? (κnd , γnd ) = Hnd (κnd γnd ),

(21)

Hns? (κns , γns ) = Hns (κns γns ),

Hb? (κb , γo , γns , β) = Hb (κb γo , κb γns , β).

(20)
(22)

Where κ is the parameter expressing affirmation (κ > 0) or
challenge κ < 0 of the cue by the probe when the parameter
γc > 0, and the other way round when γc < 0. Multiplication
of the driving parameter γc leads to a modified composed
parameter κp γc in the Hamiltonian to affirm or mitigate the
participant’s initial recollection of the cue. We want to emphasize
that the second stage Hamiltonians for the probes are thus
structured exactly in the same way as the Hamiltonians for

TABLE 3 | Affirmation and challenge of cues by probes; + sign indicates
corresponding features, − sign indicates challenge.

o
ns
nd

o?

ns?

nd?

b?

+VGN

+GN −V

−VGN

+VG −N

+GN −V

+VGN

+V −GN

+GN −V

−VGN

+V −GN

+VGN

−VG +N

The subindex indicates the conflicting or affirming feature.

the corresponding cues, except that the driving parameters γ
are modulated by multiplying them with dedicated tweaking
parameters κ.
Reflection following b? probe. While it is not needed in
the first reflection stage, under the disjunctive probe b? in the
second stage a dedicated Hamiltonian, Equation (22), is still
required. Also the Hamiltonian proper to p
the exposure of the b?
probe is based on its target state vector ( 1 − |β|2 , 1 + β, 0)τ .
This consists of the actions of Ho (γo ) and Hns (γns ) where the
parameters in the corresponding gates have been added, or
subtracted if the transport is in opposite direction14 ;
p
p
Hb (γo , γns , β) = G21 (γo 1 − |β|2 − γns ) + G31 (γo 1 − |β|2 )
+ G32 (γns + βγo )

(23)

The Hamiltonian for the b? probe thus uses three parameters
γo , γns and β which it inherits from the Hamiltonians for its
component probes o? or ns?.
Unitary evolution and time of measurement. An issue with
quantum-like models is the typical appearance of oscillations
of probability over time. These oscillations in the evolution are
essentially due to the inherent periodicity of a finite dimensional
and energetically closed quantum system. Simply put, such
systems will always evolve back to their initial state and do over
the exact same itinerary in their Hilbert space—ad infinitum.
Evidently, in the domain of cognition, when quantum-like
modeling of experimental paradigms is done, only within-period
evolution should be given meaningful interpretation [24]. In that
sense a guideline for the time of measurement would be to keep
the reflection times short with respect to the full period. Another
option to arrest the characteristic probability oscillation is to
include a third stage in the experimental paradigm driven by
a ‘grab coat and leave’ Hamiltonian, which would be dedicated
to freeze the perceived memory state (set all driving parameters
γ equal to zero). More elegantly a termination should be
formalized to damp the memory state vector back into its baseline
uninformed state by using Lindblad evolution for an open system
(e.g., [27, 28] Broekaert et al., under review).
A number of alternative criteria could be put forward to
decide this instance of measurement, though at present we keep
to an ad hoc cut to the unitary time propagation as proposed
by Busemeyer et al. [23] and Busemeyer and Bruza [4]15 . With
14 One must take

into account G12 (γ ) = −G21 (−γ ).
t = π2 corresponds to a first extremum when significant parameters in
the Hamiltonian are set equal to zero, i.e., when the actual psychological dynamics
is “turned off ” in the model.
15 Choice

13 With β a complex number one must take care to keep the Hadamard gate
Hermitian.

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June 2017 | Volume 5 | Article 23 | 15

Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

the intent of the possibility of tweaking the observed acceptance
probabilities by description dependence in the second reflection
period, we have taken the ad hoc reflection durations of both
stages somewhat shorter; π3 for each stage.
The first stage ends at t = π3 , the unitary operator of the
second stage picks up there after. The vector of the perceived
memory state at time t after probe presentation is then obtained
by propagating the initial state, Equation (9), by the concatenated
Schrödinger propagators;
π

9p|c (t) = e−iHp? t e−iHc 3 90 , and
p
1
( 1 − |β|2 , (β + 1), 1)τ
90 = p
3 + 2ℜ(β)

(24)

Also the second stage ends at t = π3 , after the first stage. Time
evolution prior to the second stage can be obtained by deleting
the propagator of the second stage and letting the first propagator
have the argument t. The acceptance probabilities p(p?|c) can
then be derived using their expressions, Equations (12, 13),
in terms of state vector components and will be fitted to the
observed data by SSE optimization of the seven free parameters
of our Hamiltonian driven QEM model.

3. FITTING THE MODELS TO THE EOD
DATA
The data fitted post-hoc parameters for Brainerd et al.’s
QEM. QEM provides three amplitudes per cue {vc , gc , ndc }, which
satisfy normalization (Equation 5). Therefore six numbers should
cover the experimental data sets. Our prescription for acceptance
probabilities, Equations (12, 13), coincide with Brainerd et al. [1,
p. 243]:
||MO,y |SC i||2 = |vC |2 + |gC |2 ||Mns,y |SC i||2

= |gC |2 ||MND,y |SC i||2 = |nC |2

and in the same logic we have in QEM, see Equation (7);
||MB,y |SC i||2 = ||MO,y |SC i||2 = |vC |2 + |gC |2 .
We notice that in QEM we will always predict
p(o?|C) ≥ p(ns?|C), which is of course only the case for
the o cue data. Since the modulus square amplitudes are
positive numbers, data with p(o?|C) < p(ns?|C) cannot be
accommodated in the original version of QEM.
Similarly in the disjunction paradigm QEM would always
predict p(o?|C) = p(ns?|C), which is not apparent in
the experimental disjunction data (Table 1) and certainly
not so for ns and nd cues. Without any other means to
fine tune the acceptance probabilities we would expect low
accuracy of prediction for them, while we expect pronounced
total probability and unpacking factor in the subadditivity
paradigm and the disjunction paradigm respectively, Tables 1, 2.
Optimized QEM parameters appear in Tables 4, 5.
The data fitted parameters of the Feynman path based model.
The Feynman path model required six parameters to obtain

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the nine acceptance probabilities of disjunction paradigm ([4],
Ch. 6). The model allows to reproduce very well the general
required pattern of acceptance probabilities at RMSE = 0.0298,
which turn out the precise EOD effects, except for the new
dissimilar cues {nd}, Table 1. In the latter case the unpacking
factor turns out smaller than 1, i.e., the conjunction value turns
out negative.
The Feynman path model was not adapted yet to the
subadditivity paradigm, but since it uses interference of
amplitudes and reversed gist/verbatim processing depending on
the type of cue, the model should be applicable in that paradigm
as well.
The data fitted Hamiltonian driven QEM parameters. With
both experiments reporting different data for similar expressions,
we have fitted the Hamiltonian model to each separately16 . For
the EOD-disjunction paradigm the model obtained closely fitted
parameters to the experimental data, with RMSE = 0.0073 with
β > 0. When β < 0 constraint was imposed a less good RMSE =
0.0341 was obtained. The nine predicted probabilities p(p?|c) by
the parameters of Table 6 are shown in Table 1.
For the EOD-subadditivity paradigm the model obtained a
less efficient fit of parameters to the experimental data, with
RMSE = 0.0565 for β > 0. When β < 0 the parameter fit allowed
an improved RMSE = 0.0191. The Hamiltonian model for the
EOD-paradigm allowed a very good data fit using complex β at
RMSE = 0.0032. We recall that complex numbers consist of a
modulus and a phase, therefore one complex parameter should
actually be counted as two real parameters. We shortly comment
on this issue in the discussion, Section 4. The nine predicted
probabilities p(p?|c) following the parameters of Table 7 for the
three cases are shown in Table 2.
The temporal evolution of acceptance probabilities. With
the optimized values of the driving parameters calculated, the
temporal progression of the acceptance probability can be
graphed (Figures 1–5). The dashed lines represent the first-stage
evolutions when the participant is shown the cue for recognition,
while the full lines represent the second-stage evolutions when
the probes enquire for accepting the type of a cue. The ultimate
instance of measurement happens at the end of the second stage
(t = 2π/3). In all graphs, in the first stage the color indicates
the “probability value” of the traces; red codes for perceiving the
cue’s unrelated features (N), orange codes for perceiving the cue’s
gist (G) and green codes for perceiving the cue’s verbatim and gist
(V + G)—one can quickly check that for the same cue the dashed
red and dashed green values add up to 1 at each moment in the
first stage. Evidently these first stage “probability values” should
not be conflated with the participants acceptance probabilities.
Only after the probe has enquired the participant do these values
evolve as the nine acceptance probabilities. It is worthwhile to
note that the optimalization of parameters has returned initial
states which either contain no gist or no verbatim perception
in the cases with real-valued β (at t = 0 respectively; orange
has almost value zero or, green and orange almost coincide).
Only when β is complex-valued does the initial value show
16 Matlab’s fmincon function on SSE was used with a

grid for the initial vector in the parameter space.

36 21 (β ∈ R) or 36 22 (β ∈ C)

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

TABLE 4 | EOD-disjunction paradigm: Optimized fit of independent QEM
parameters providing RMSE = 0.1112, (vc , gc ∈ R+ ).
vo
0.7810

TABLE 6 | EOD-disjunction paradigm: Optimized fit of Hamiltonian parameters
under β > 0 (RMSE = 0.0073), and β < 0 (RMSE = 0.0341) constraint.

go

vns

gns

vnd

gnd

0.3606

0.0008

0.7506

0.0708

0.4690

0.4528

κns

κb

γo

go

vns

gns

vnd

gnd

0.6557

0.0025

0.7483

0.0012

0.5447

0.19011

1.8354

0.042911

8.5858

0.2701

0.3794

0.99563

0.022764

−1.9232

0.951261

2.5625

0.86085

0.60844

−0.98688

γnd

β

TABLE 7 | EOD-subadditivity paradigm: Optimized fit of Hamiltonian parameters
under constraint β > 0 (RMSE = 0.0565), β < 0 (RMSE = 0.0191) and β ∈ C
(RMSE = 0.0032).
κo

substantial gist and verbatim trace. We have provided two graphs
for the EOD-disjunction paradigm (Figures 1, 2) the first one
was constrained to have β > 0 while the second had to satisfy
β < 0. In this EOD-disjunction paradigm no complex-valued
β offered an optimized fit. For the EOD-subadditivity paradigm
we provide three graphs (Figures 3–5), respectively with the
constraints β > 0, β < 0, and β ∈ C.
One observes that in the first stage the dynamics is mostly
monotonic—except for the one case where β ∈ C (Figure 5). In
the second stage dynamics some intermediary extrema do appear,
which from a cognitive point of view are not to be expected. The
factor of description dependence was expected to be a smaller
modification of the first stage recognition. The second stage
extrema however need to be understood with respect to the ad
hoc instance of measurement at t = π/3 after enquiry, adopting
a shorter measurement time could have mitigated this temporal
behavior. Finally we note that also the outspoken VGN spread
of the initial vector could be related to a too extended period
for evolution. While the fitting of the experimental acceptance
probability data in the Hamiltonian driven QEM has shown
good accuracy, the concomitant intermediate temporal evolution
leaves room for improving the measurement protocol.

4. DISCUSSION
We had set out to develop a Hamiltonian driven model
that would provide temporal evolution of the memory state
of the Quantum Episodic Model of Brainerd et al. [1, 9].
The model uses nine different state vectors for the three
cues crossed with three probe paradigms, and requires six
parameters to drive the Hamiltonians and one parameter to
tweak the gist in the initial state. We provided psychological
interpretation of the parameters fitting the experimental process.
Initially the memory state prior to cue and probe presentation
is an equally weighed mix of o, ns and nd states leading
to an overall amount of the gist component monitored by
the parameter β. In first stage the ability to recognize the
type of the cue c is driven by the cue-specific parameter
γc in the Hamiltonian. In the second stage the instruction
probe p? engenders an amplified or mitigated evolution
driven by the probe-specific parameter κp for descriptiondependence.

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γns

Fitting attempts for a β ∈ C gravitated toward the β < 0 solution, i.e., phase(β) → π .

TABLE 5 | EOD-subadditivity paradigm: Optimized fit of independent QEM
parameters providing RMSE = 0.0963, (vc , gc ∈ R+ ).
vo

κo

κns

0.6461

1.7674

0.5561

0.0075

0.4363

0.1477

κnd
−5.7405
−1.3536
1.180

γo

γns

γnd

β

5.0840

0.3913

0.1000

0.9979

1.5114

94.817

0.4307

0.8451

3.3274

1.1838

−0.9585

0.9323
e2πi0.4143

Our Hamiltonian driven account of QEM shows that the
subadditivity and disjunction fallacy are not a priori guaranteed
or “parameter free” in our model. The occasions in which these
effects would not occur are however very improbable in practice,
Equations (14, 15). This possibility is due to the fact that the
two-staged Hamiltonian evolution produces nine state vectors
9probe|cue (t) instead of regular QEM’s three cue-dependent state
vectors 9cue .
Using two reported experimental data sets showing
subadditivity and over-distribution of the disjunction in
acceptance probabilities for episodic memory recollection, we
were able to provide parameter values with good prediction
capacity in the Hamiltonian model. In practice we provided
values for seven parameters {κo , κns , κnd , γo , γns , γnd , β}
to predict nine acceptance probabilities {po?o , pns?o ,
pnd?o , po?ns , pns?ns , pnd?ns , po?nd , pns?nd , pnd?nd } in the subadditivity
paradigm and did the same for the disjunction paradigm,
Tables 6, 7. Rigorously one should discern the parametrization
case when β ∈ C, which should be counted for two parameters
even if the function of the real and imaginary part of the
parameter take the same position in the model. The present
model thus uses one extra parameter in comparison to the
Feynman path model of Busemeyer and Bruza but provides
better EOD prediction for all type of cues. Moreover the
parameters in the Hamiltonian model do allow psychological
interpretation. The predictions of acceptance probabilities
following the original QEM formulation by Brainerd et al.
showed to be flawed by systematic features. In the disjunction
paradigm QEM’s acceptance probabilities for the both?probe and the old?-probe can only be identical, and in both
experimental paradigms QEM’s acceptance probability for the
old?-probe can only be larger than or equal to the acceptance
probability for the new-similar?-probe, whatever the cue
type.
The issue of “description-dependence” effect seems crucial in
obtaining final acceptance probabilities; the κ factors are rather
large in comparison to the driving parameters γ and cause

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

FIGURE 1 | β > 0 case: temporal evolution of acceptance probabilities for the EOD-disjunction paradigm [3]. Red indicates N probability component, orange
indicates G probability component and green indicates V + G probability component. In the second stage brown indicates the acceptance probability for the b? probe
(V + G). Notice the near absence of verbatim in the initial state.

FIGURE 2 | β < 0 case: temporal evolution of acceptance probabilities for the EOD-disjunction paradigm [3]. Red indicates N probability component, orange
indicates G probability component while green indicates V + G probability component. In the second stage brown indicates the acceptance probability for the b?
probe (V + G). Notice the near absence of gist in the initial state.

outspoken evolution in second stage. This fact is rather counter
intuitive as a priori we had expected small corrective modulation
in second stage evolution (see Tables 6, 7, Figures 1–5).
We found it remarkable that β ≈ ±1 is needed for best fit
in both experimental paradigms when keeping β ∈ R. This
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would suggest that the verbatim trace is almost negligible in
comparison to the gist in the set of true cues {o} in the initial
state, or just the inverse. When β is allowed to be complex a mix
of both traces is present in the best fitting initial state for the
EOD-subadditivity paradigm. In the EOD-disjunction paradigm

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

FIGURE 3 | β > 0 case: temporal evolution of acceptance probabilities for the EOD-subadditivity paradigm [1]. Red indicates N probability component, orange
indicates G probability component while green indicates V + G probability component. Notice the near absence of verbatim in the initial state.

FIGURE 4 | β < 0 case: temporal evolution of acceptance probabilities for the EOD-subadditivity paradigm [1]. Red indicates N probability component, orange
indicates G probability component while green indicates V + G probability component. Notice the near absence of gist in the initial state.

complex β did not provide a best fit (the limit value became
real).
Superposed Hadamard gates with off-diagonal parameters
show to be a viable method in the construction of Hamiltonians.
The “description-dependence” factor κ can indeed mitigate
probability oscillations. The best example can be seen in the

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β < 0 subadditivity Graph 4 where a small κo = 0.022764
acts on an average γo = 2.5625 and gives in the second
stage nearly unmodulated continuation for po?|o (t), po?|ns (t) and
po?|nd (t) (solid green lines).
The ad hoc time π3 avoided most intermediate extrema in the
probabilities in the second reflection stage, except for β ∈ C.
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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

FIGURE 5 | β ∈ C case: temporal evolution of acceptance probabilities for the EOD-subadditivity paradigm [1]. Red indicates N probability component, orange
indicates G probability component while green indicates V + G probability component. Notice both verbatim and gist are present in the initial state.

We remark that lower time of measurement could bring about
the problem of not being able to spread open to a range of
probabilities in time when starting from some pre-defined—
e.g., equal– probability configuration, or just trade of with ever
growing driving parameters γ . A longer time of measurement
would increase the well-known issue of intermediate extrema.
Wep have used the equally weighed initial state
1/N [ 1 − |β|2 , 1 + β, 1]τ in VGN space to give each vector |oi,
|nsi and |ndi equal weight at start which we consider reflected
best the information communicated by the experimenter. The
optimized data fit shows e.g., β ≈ ±1 in both paradigms with
the perceived implicit probabilities at the start at p(o) ≈ 0.8,
p(ns) ≈ 0.8 and p(nd) ≈ 0.2. Which one can observe at t = 0
in both the subadditivity paradigm (Figure 3) and disjunction
paradigm (Figure 1). The precise nature of the initial vector for
the memory state of the participant after studying {o} and having
heard ‘all type of cues will be presented with equal probability’
but prior to cue and probe presentation remains somewhat
puzzling.
In sum we consider to have constructed an acceptable
Hamiltonian driven QEM version, with good prediction capacity
for acceptance probabilities. Future work could include covering
the model fitting of a data set which covers both the subadditivity

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and disjunction paradigm at once –eight parameters for
twelve datapoints– to verify its further prediction capacity,
and to monitor more closely the initial memory state in the
experimental paradigms and the meaurement protocol.

AUTHOR CONTRIBUTIONS
JBB has designed the Hamiltonian model for the EOD paradigm
and provided data fitting and interpretation. JRB is the author of
the Feynman path model for the EOD-paradigm and provided
prior knowledge on the QEM model and Hamiltonian design,
and did critical revision of the work.

ACKNOWLEDGMENTS
JBB gratefully thanks JRB for extensive discussions on
Hamiltonian and Markov dynamical decision models and
their relation to the EOD effect, and also thanks Cole
Rodman for insightful discussions on 3 × 3 categorizationdecision paradigms in quantum-like modeling. This work
was made possible by FWO-Vlaanderen mobility grant
V410016N. Further thanks go to an reviewer for inciting
clarifications.

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Broekaert and Busemeyer

A Hamiltonian Quantum Like Model for EOD

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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Broekaert and Busemeyer. This is an open-access article
distributed under the terms of the Creative Commons Attribution License (CC BY).
The use, distribution or reproduction in other forums is permitted, provided the
original author(s) or licensor are credited and that the original publication in this
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or reproduction is permitted which does not comply with these terms.

June 2017 | Volume 5 | Article 23 | 21

ORIGINAL RESEARCH
published: 06 May 2016
doi: 10.3389/fphy.2016.00017

On the Foundations of the Brussels
Operational-Realistic Approach to
Cognition
Diederik Aerts 1 , Massimiliano Sassoli de Bianchi 2 and Sandro Sozzo 3*
1

Center Leo Apostel for Interdisciplinary Studies, Free University of Brussels, Brussels, Belgium, 2 Laboratorio di Autoricerca
di Base, Lugano, Switzerland, 3 School of Management, Institute for Quantum Social and Cognitive Science, University of
Leicester, Leicester, UK

Edited by:
Andrei Khrennikov,
Linnaeus University, Sweden
Reviewed by:
Jerome Busemeyer,
Indiana University, USA
Irina Basieva,
General Physics Institute, Russia
Giuseppe Sergioli,
University of Cagliari, Italy
*Correspondence:
Sandro Sozzo
ss831@le.ac.uk
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 02 February 2016
Accepted: 19 April 2016
Published: 06 May 2016
Citation:
Aerts D, Sassoli de Bianchi M and
Sozzo S (2016) On the Foundations of
the Brussels Operational-Realistic
Approach to Cognition.
Front. Phys. 4:17.
doi: 10.3389/fphy.2016.00017

Frontiers in Physics | www.frontiersin.org

The scientific community is becoming more and more interested in the research that
applies the mathematical formalism of quantum theory to model human decision-making.
In this paper, we expose the theoretical foundations of the quantum approach to
cognition that we developed in Brussels. These foundations rest on the results of two
decade studies on the axiomatic and operational-realistic approaches to the foundations
of quantum physics. The deep analogies between the foundations of physics and those
of cognition lead us to investigate the validity of quantum theory as a general and
unitary framework for cognitive phenomena, and the empirical success of the Hilbert
space models derived by such investigation provides a strong theoretical confirmation
of this validity. However, two situations in the cognitive realm, “question order effects”
and “response replicability,” indicate that even the Hilbert space framework could be
insufficient to reproduce the expected pattern. This does not mean that the mentioned
operational-realistic approach would be incorrect, but simply that a larger class of
measurements would be in force in human cognition, so that an extended quantum
formalism may be needed to deal with all of them. As we will explain, the recently
derived “extended Bloch representation” of quantum theory (and the associated “general
tension-reduction” model) precisely provides such extended formalism, while remaining
within the same unitary interpretative framework.
Keywords: human cognition, cognitive modeling, quantum structures, foundations of quantum theory,
tension-reduction model

1. INTRODUCTION
A fundamental problem in cognition concerns the identification of the principles guiding human
decision-making. Identifying the mechanisms of decision-making would indeed have manifold
implications, from psychology to economics, finance, politics, philosophy, and computer science.
In this regard, the predominant theoretical paradigm rests on a classical conception of logic and
probability theory. According to this paradigm, people take decisions by following the rules of
Boole’s logic, while the probabilistic aspects of these decisions can be formalized by Kolmogorov’s
probability theory [1]. However, increasing experimental evidence on conceptual categorization,
probability judgments, and behavioral economics confirms that this classical conception is
fundamentally problematical, in the sense that the cognitive models based on these mathematical
structures are not capable of capturing how people concretely take decisions in situations of
uncertainty.

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Aerts et al.

In the last decade, an alternative scientific paradigm has
caught on which applies a different modeling scheme. The
research that uses the mathematical formalism of quantum
theory to model situations and processes in cognitive science is
becoming more and more accepted in the scientific community,
having attracted the interest of renowned scientists, funding
institutions, media, and popular science. And, quantum models
of cognition showed to be more effective than traditional
modeling schemes to describe situations like the “Guppy
effect,” the “combination problem,” the “prisoner’s dilemma,” the
“conjunction and disjunction fallacies,” “similarity judgments,”
the “disjunction effect,” “violations of the Sure-Thing principle,”
“Allais,” “Ellsberg,” and “Machina paradoxes” (see, e.g., [2–22]).
Recently, quantum computational semantics were applied to
natural and musical languages in a novel approach [23, 24].
There is a general acceptance that the use of the term
“quantum” is not directly related to physics, neither this research
in “quantum cognition” aims to unveil the microscopic processes
occurring in the human brain. The term “quantum” rather refers
to the mathematical structures that are applied to cognitive
domains. The scientific community engaged in this research
does not instead have a shared opinion on how and why these
quantum mathematical structures should be employed in human
cognition. Different hypotheses have been put forward in this
respect. Our research team in Brussels has been working in this
domain since early nineties, providing pioneering and substantial
contributions to its growth, and we think it is important to expose
the epistemological foundations of the quantum theoretical
approach to cognition we developed in these years. This is the
main aim of the present paper.
Our approach was inspired by a two decade research on
the mathematical and conceptual foundations of quantum
physics, quantum probability, and the fundamental differences
between classical and quantum structures [25–28]. We followed
an axiomatic and operational-realistic approach to quantum
physics, in which we investigated how the mathematical
formalism of quantum theory in Hilbert space can be derived
from more intuitive and physically justified axioms, directly
connected with empirical situations and facts. This led us
to elaborate a “State Context Property” (SCoP) formalism,
according to which any physical entity is expressed in terms of
the operationally well defined notions of “state,” “context,” and
“property,” and functional relations between these notions [29].
If suitable axioms are imposed to such a SCoP structure, then
one obtains a mathematical representation that is isomorphic to
a Hilbert space over complex numbers (see, e.g., [30]).
Let us shortly explain the “operational-realistic” connotation
characterizing our approach, because doing so we can easily
point out its specific strength, and the reason why it introduces
an essentially new element to the domain of psychology.
“Operational” stands for the fact that all fundamental elements
in the formalism are directly linked to the measurement
settings and operations that are performed in the laboratory
of experimentation. “Realistic” means that we introduce in an
operational way the notion of “state of an entity,” considering
such a “state” as representing an aspect of the reality of the
considered entity at a specific moment or during a specific

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The Brussels Approach to Cognition

time-span. Historically, the notion of “state of a physical entity”
was the “easy” part of the physical theories that were the
predecessors of quantum theory, and it was the birth of quantum
theory that forced physicists to take also seriously the role of
measurement and hence the value of an operational approach.
The reason is that “the reality of a physical entity” was considered
to be a simple and straightforward notion in classical physics and
hence the “different modes of reality of a same physical entity”
were described by its “different states.” That measurements
would intrinsically play a role, also in the description of the reality
of a physical entity, only became clear in quantum physics for the
case of micro-physical entities.
In psychology, things historically evolved in a different way.
Here, one is in fact confronted with what we call “conceptual
entities,” such as “concepts” or “conceptual combinations,” and
more generally with any cognitive situation which is presented
to the different participants in a psychology experiment. Due
to their nature, conceptual entities and cognitive situations are
“much less real than physical entities,” which makes the notion
of “state of a conceptual entity” a highly non-obvious one in
psychology. And, as far as we know, the notion of state is
never explicitly introduced in psychology, although it appears
implicitly within the reasoning that is made about experiments,
their setups and results. Possibly, the notion of “preparation
of the experiment” will be used for what we call “the state
of the considered conceptual entity” in our approach. Often,
however, the notion of state is also associated with the “belief
system” of the participant in the experiment. In our approach
we keep both notions of “state” and “measurement” on equal
footing, whether our description concerns a physical entity or
a conceptual entity. In this way, we can make optimal use of
the characteristic methodological strengths of each one of the
notions. It is in doing so that we observed that there is an
impressive analogy between the operational-realistic description
of a physical entity and the operational-realistic description
of a conceptual entity, in particular for what concerns the
measurement process and the effects of context on the state of
the entity. As a matter of fact, one can give a SCoP description
of a conceptual entity and its dynamics [4, 8, 9]. This justifies
the investigation of quantum theory as a unified, coherent and
general framework to model conceptual entities, as quantum
theory is a natural candidate to model context effects and contextinduced state transformations. Hence, the quantum theoretical
models that we worked out for specific cognitive situations
strictly derive from such investigation of quantum theory as
a scientific paradigm for human cognition. In this respect, we
think that each predictive success of quantum modeling can
be considered as a confirmation of such general validity. It is
however important to observe that, recently, potential deviations
from Hilbert space modeling were discovered in two cognitive
situations, namely, “question order effects” [31] and “response
replicability” [32]. According to some authors, question order
effects can be represented by sequential quantum measurements
of incompatible properties [14, 18, 31]. However, such a
representation seems to be problematical, as it cannot reproduce
the pattern that would be observed in response replicability,
in case the effect were confirmed experimentally [32], nor it

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Aerts et al.

can fit the experimental data, when non-degenerate models are
considered [33, 34], or “exactly” fit the data, when degenerate
models are used, as for instance the quantum identity called the
QQ-equality (see Section 5) is never “perfectly” obeyed by the
data (although it is remarkably almost obeyed by measurements
not including background information [14, 18, 31]).
We put forward an alternative solution for these effects within
a “hidden measurement formalism” elaborated by ourselves (see,
e.g., [26, 35–39] and references therein), which goes beyond
the Hilbert space formulation of quantum theory (probabilities),
though it remains compatible with our operational-realistic
description of conceptual entities [34, 40].
For the sake of completeness, we summarize the content of
this paper in the following.
In Section 2, we present the epistemological foundations
of the quantum theoretical approach to human cognition we
developed in Brussels. We operationally describe a conceptual
entity in terms of concrete experiments that are performed
in psychological laboratories. Specifically, the conceptual entity
is the reality of the situation which every participant in an
experiment is confronted with, and the different states of this
conceptual entity are the different modes of reality of this
experimental situation. There are contexts influencing the reality
of this experimental situation, and the relevant ones of these
contexts are elements of the SCoP structure, the theory of our
approach, and their influence on the experimental situation is
described as a change of state of the conceptual entity under
consideration. There are also properties of this experimental
situation, the relevant ones being elements of the SCoP structure,
and they can be actual or potential, their “amount of actuality”
(i.e., their “degree of availability in being actualized”) being
described by a probability measure. The operational analogies
between physical and conceptual entities suggest to represent
the latter by means of the mathematical formalism of quantum
theory in Hilbert space. Hence, we assume, in our research,
the validity of quantum theory as a scientific paradigm for
human cognition. On the basis of this assumption, we provide a
unified presentation in Section 3 of the results obtained within
a quantum theoretical modeling in knowledge representation,
decision theory under uncertainty and behavioral economics.
We emphasize that our research allowed us to identify new
unexpected deviations from classical structures [41–43], as well
as new genuine quantum structures in conceptual combinations
[44–46], which could not have been identified at the same
fundamental level as it was possible in our approach if we would
have adopted the more traditional perspective only inquiring into
the observed deviations from classical probabilistic structures.
In Section 4, we analyze question order effects and response
replicability and explain why a quantum theoretical modeling
in Hilbert space of these situations is problematical. Finally, we
present in Section 5 a novel solution we recently elaborated
for these cognitive situations [34, 40]. The solution predicts
a violation of the Hilbert space formalism, more specifically,
the Born rule for probabilities is put at stake. We however
emphasize that this solution remains compatible with the general
operational and realistic description of cognitive entities and
their dynamics given in Section 2. In Section 6, we conclude our

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The Brussels Approach to Cognition

article by offering a few additional remarks, further emphasizing
the coherence and advantage of our theoretical approach. We
stress, to conclude this section, that the deviation above from
Hilbert space modeling should not be considered as an indication
that we should better come back to more traditional classical
approaches. On the contrary, we believe that new mathematical
structures, more general than both pure classical and pure
quantum structures, will be needed in the modeling of cognitive
processes.

2. AN OPERATIONAL-REALISTIC
FOUNDATION OF COGNITIVE
PSYCHOLOGY
Many quantum physicists agree that the phenomenology of
microscopic particles is intriguing, but what is equally curious is
the quantum mathematics that captures the mysterious quantum
phenomena. Since the early days of quantum theory, indeed,
scholars have been amazed by the success of the mathematical
formalism of quantum theory, as it was not clear at all how
it had come about. This has inspired a long-standing research
on the foundations of the Hilbert space formalism of quantum
theory from physically justified axioms, resting on well defined
empirical notions, more directly connected with the operations
that are usually performed in a laboratory. Such an operational
justification would make the formalism of quantum theory more
firmly founded.
One of the well-known approaches to the foundations of
quantum physics and quantum probability is the “GenevaBrussels approach”, initiated by Jauch [47] and Piron [48], and
further developed by our Brussels research team (see, e.g., [25,
28]). This research produced a formal approach, called “State
Context Property” (SCoP) formalism, where any physical entity
can be expressed in terms of the basic notions of “state,” “context,”
and “property,” which arise as a consequence of concrete physical
operations on macroscopic apparatuses, such as preparation and
registration devices, performed in spatio-temporal domains, such
as physical laboratories. Measurements, state transformations,
outcomes of measurements, and probabilities can then be
expressed in terms of these more fundamental notions. If suitable
axioms are imposed on the mathematical structures underlying
the SCoP formalism, then the Hilbert space structure of quantum
theory emerges as a unique mathematical representation, up to
isomorphisms [30].
There are still difficulties connected with the interpretation
of some of these axioms and their physical justification, in
particular for what concerns compound physical entities [25].
But, this research line was a source of inspiration for the
operational approaches applying the quantum formalism outside
the microscopic domain of quantum physics [49, 50]. In
particular, as we already mentioned in Section 1, a very similar
realistic and operational representation of conceptual entities
can be given for the cognitive domain, in the sense that
the SCoP formalism can again be employed to formalize the
more abstract conceptual entities in terms of states, contexts,
properties, measurements, and probabilities of outcomes [4, 8, 9].

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Aerts et al.

Let us first consider the empirical phenomenology of cognitive
psychology. Like in physics, where laboratories define precise
spatio-temporal domains, we can introduce “psychological
laboratories” where cognitive experiments are performed. These
experiments are performed on situations that are specifically
“prepared” for the experiments, including experimental devices,
and, for example, structured questionnaires, human participants
that interact with the questionnaires in written answers, or
each other, e.g., an interviewer and an interviewed. Whenever
empirical data are collected from the responses of several
participants, a statistics of the obtained outcomes arises. Starting
from these empirical facts, we identify in our approach entities,
states, contexts, measurements, outcomes, and probabilities of
outcomes, as follows.
The complex of experimental procedures conceived by the
experimenter, the experimental design and setting and the
cognitive effect that one wants to analyze, define a conceptual
entity A, and are usually associated with a preparation procedure
of a state of A. Hence, like in physics, the preparation procedure
sets the initial state pA of the conceptual entity A under study.
Let us consider, for example, a questionnaire where a participant
is asked to rank on a 7-point scale the membership of a
list of items with respect to the concepts Fruits, Vegetables
and their conjunction Fruits and Vegetables. The questionnaire
defines the states pFruits , pVegetables , and pFruits and Vegetables of the
conceptual entities Fruits, Vegetables, and Fruits and Vegetables,
respectively. It is true that cognitive situations exist where the
preparation procedure of the state of a conceptual entity is hardly
controllable. Notwithstanding this, the state of the conceptual
entity, defined by means of such a preparation procedure, is a
“state of affairs.” It indeed expresses a “reality of the conceptual
entity,” in the sense that, once prepared in a given state,
such condition is independent of any measurement procedure,
and can be confronted with the different participants in an
experiment, leading to outcome data and their statistics, exactly
like in physics.
A context e is an element that can provoke a change of
state of the conceptual entity. For example, the concept Juicy
can function as a context for the conceptual entity Fruits
leading to Juicy Fruits, which can then be considered as a state
of the conceptual entity Fruits. A special context is the one
introduced by the measurement itself. Indeed, when the cognitive
experiment starts, an interaction of a cognitive nature occurs
between the conceptual entity A under study and a participant
in the experiment, in which the state pA of the conceptual entity
A generally changes, being transformed to another state p. Also
this cognitive interaction is formalized by means of a context e.
For example, if the participant is asked to choose among a list
of items, say, Olive, Almond, Apple, etc., the most typical one
with respect to Fruits, and the answer is Apple, then the initial
state pFruits of the conceptual entity Fruits changes to pApple , i.e.,
the state describing the situation “the fruit is an apple,” as a
consequence of the contextual interaction with the participant.
The change of the state of a conceptual entity due to a context
may be either “deterministic,” hence in principle predictable
under the assumption that the state before the context acts is
known, or “intrinsically probabilistic,” in the sense that only the

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The Brussels Approach to Cognition

probability µ(p, e, pA ) that the state pA of A changes to the state
p is given. In the example above on typicality estimations, the
typicality of the item Apple for the concept Fruits is formalized
by means of the transition probability µ(pApple , e, pFruits ), where
the context e is the context of the typicality measurement.
Like in physics, an important role is played by experiments
with only two outcomes, the so-called “yes-no experiments.”
Suppose that in an opinion poll a participant is asked to
answer the question: “Is Gore honest and trustworthy?” Only
two answers are possible: “yes” and “no.” Suppose that, for a
given participant, the answer is “yes.” Then, the state pHonesty
of the conceptual entity Honesty and Trustworthiness (which
we will denote by Honesty, for the sake of simplicity) changes
to a new state pGy , which is the state describing the situation
“Gore is honest.” Hence, we can distinguish a class of yes-no
measurements on conceptual entities, as we do in physics.
The third step is the mathematical representation. We have
seen that the Hilbert space formalism of quantum theory is
general enough to capture an operational description of any
entity in the micro-physical domain. Then, the strong analogies
between the realistic and operational descriptions of physical
and conceptual entities, in particular for what concerns the
measurement process, suggest us to apply the same Hilbert space
formalism when representing cognitive situations. Hence, each
conceptual entity A is associated with a Hilbert space H, and the
state pA of A is represented by a unit vector |Ai ∈ H. A yesno measurement is represented by a spectral family {M, 1 − M},
where M denotes an orthogonal projection operator over the
Hilbert space H, and 1 denotes the identity operator over H.
The probability that the “yes” outcome is obtained in such a
yes-no measurement when the conceptual entity A is in the
state represented by |Ai is then given by the Born rule µ(A) =
hA|M|Ai. For example, M may represent an item x that can be
chosen in relation to a given concept A, so that its membership
weight is given by µ(A).
The Born rule obviously applies to measurement with more
than two outcomes too. For example, a typicality measurement
involving a list of n different items x1 , . . . , xn with respect
to a concept A can P
be represented as a spectral measure
{M1 , . . . , Mn }, where nk=1 Mk = 1 and Mk Ml = δkl Mk , such
that the typicality µk (A) of the item xk with respect to the concept
A is again given by the Born rule µk (A) = hA|Mk |Ai.
An interesting aspect concerns the final state of a conceptual
entity A after a human judgment. As above, we can assume the
existence of a nonempty class of cognitive measurements that are
ideal first kind measurements in the standard quantum sense, i.e.,
that satisfy the “Lüders postulate.” For example, if the typicality
measurement of a list of items x1 , . . . , xn with respect to a concept
A gave the outcome xk , then the final state of the conceptual
entity after the measurement is represented by the unit vector
Mk |Ai
|Ak i = √hA|M
. This means that the weights µk (A) given by the
k |Ai
Born rule can actually be interpreted as transition probabilities
µ(pk , e, pA ), where e is the context producing the transitions
from the initial state pA of the conceptual entity A, represented
by the unit vector |Ai, to one of the n possible outcome states pk ,
represented by the unit vectors |Ak i.
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Aerts et al.

Thus, how can a Hilbert space model be actually constructed
for a cognitive situation? To answer this question let us consider
again a conceptual entity A, in the state pA , a cognitive
measurement on A described by means of a context e, and
suppose that the measurement has n distinct outcomes, x1 , x2 ,
. . . , xn . A quantum theoretical model for this situation can be
constructed as follows. Let us assume, for the sake of simplicity,
that the measurement outcomes can be considered to be nondegenerate—this is a special situation which does not hold for
a wide class of cognitive measurements, see Section 3. Then,
we associate A with a n-dimensional complex Hilbert space H,
and then consider an orthonormal base {|e1 i, |e2 i, . . . , |en i} in H
(since H is isomorphic to the Hilbert space Cn , the orthonormal
base of H can be the canonical base of Cn ). Next, we represent the
cognitive measurement described by e by means of the spectral
family {M1 , M2 , . . . , Mn }, where Mk = |ek ihek |, k = 1, 2, . . . , n.
Finally, the probability that the measurement e on the conceptual
entity A in the state pA gives the outcome xk is given by µk (A) =
hA|Mk |Ai = hA|ek ihek |Ai = |hek |Ai|2 .
What about the interpretation of the Hilbert space formalism
above? Two major points should now be reminded, namely:
(i) the states of conceptual entities describe the “modes of
being” of these conceptual entities;
(ii) in a cognitive experiment, a participant acts as a
(measurement) context for the conceptual entity, changing
its state.
This means that, as we mentioned already, the state pA of the
conceptual entity A is represented in the Hilbert space formalism
by the unit vector |Ai, the possible outcomes xk of the experiment
by the base vectors |ek i, and the action of a participant (or
the overall action of the ensemble of participants) as the state
transformation |Ai → |ek i induced by the orthogonal projection
operator Mk = |ek ihek |, if the outcome xk is obtained, so that the
probability of occurrence of xk can also be written as µk (A) =
µ(|ek i, e, |Ai), where e is the measurement context associated
with the spectral family {M1 , M2 , . . . , Mn }.
It follows from (i) and (ii) that a state, hence a unit vector in
the Hilbert space representation of states, does not describe the
subjective beliefs of a person, or collection of persons, about a
conceptual entity. Such subjective beliefs are rather incorporated
in the cognitive interaction between the cognitive situation and
the human participants deciding on that cognitive situation.
In this respect, our operational quantum approach to human
cognition is also a realistic one, and thus it departs from other
approaches that apply the mathematical formalism of quantum
theory to model cognitive processes [12, 14, 17, 18, 31, 32]. Of
course, one could say that the difference between interpreting
the quantum state as a “state of belief ” of a participant in
the experiment, or as a “state of a conceptual entity,” i.e.,
a “state of the situation which the participant is confronted
with during an experiment,” is only a question of philosophical
interpretation, but comes to the same when it concerns the
methodological development of the approach. Although this is
definitely partly true, we do not fully agree with it. Interpretation
and methodology are never completely separated. A certain
interpretation, hence giving rise to a specific view on the matter,

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will give rise to other ideas of how to further develop the
approach, how to elaborate the method, etc., than another
interpretation, with another view, will do. We believe that an
operational-realistic approach, being balanced between attention
for idealist as well as realist philosophical interpretations, carries
in this sense a particular strength, precisely due to this balance.
A good example of this is how we were inspired to use the
superposition principle of quantum theory in our modeling of
concepts as conceptual entities. We represented the combination
of two concepts by a state that is the linear superposition
of the states describing the component concepts. This way of
representing combined conceptual entities captures the nature of
emergence, exactly like in physics. It would not be obvious to put
forward this description when state of beliefs are the focus of what
can be predicted.
We stress a third point that is important, in our opinion.
For most situations, we interpret the effect of the cognitive
context on a conceptual entity in a decision-making process as an
“actualization of pure potentiality.” Like in quantum physics, the
(measurement) context does not reveal pre-existing properties of
the entity but, rather, it makes actual properties that were only
potential in the initial state of the entity (unless the initial state
is already an eigenstate of the measurement in question, like in
physics) [4, 8, 9].
It follows from the previous discussion that our research
investigates the validity of quantum theory as a general, unitary
and coherent theory for human cognition. Our quantum
theoretical models, elaborated for specific cognitive situations
and data, derive from quantum theory as a consequence of the
assumptions about this general validity. As such, these models
are subject to the technical and epistemological constraints of
quantum theory. In other terms, our quantum modeling rests on
a “theory based approach,” and should be distinguished from an
“ad hoc modeling based approach,” only devised to fit data. In
this respect, one should be suspicious of models in which free
parameters are added on an “ad hoc” basis to fit the data more
closely in specific experimental situations. In our opinion, the
fact that our “theory derived model” reproduces different sets of
experimental data constitutes in itself a convincing argument to
support its advantage over traditional modeling approaches and
to extend its use to more complex cognitive situations (in that
respect, see also our final remarks in Section 6).
We present in Section 3 the results obtained in our
quantum theoretical approach in the light of the epistemological
perspective of this section.

3. ON THE MODELING EFFECTIVENESS
OF HILBERT SPACE
The quantum approach to cognition described in Section 2
produced concrete models in Hilbert space, which faithfully
matched different sets of experimental data collected to reveal
“decision-making errors” and “probability judgment errors.”
This allowed us to identify genuine quantum structures in the
cognitive realm. We present a reconstruction of the attained
results in the following.

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The first set of results concerns knowledge representation
and conceptual categorization and combination. James Hampton
collected data on how people rate membership of items with
respect to pairs of concepts and their combinations, conjunction
[51], disjunction [52], and negation [53]. By using the data in
Hampton [52], we reconstructed the typicality estimations of 24
items with respect to the concepts Fruits and Vegetables and their
disjunction Fruits or Vegetables. We showed that the concepts
Fruits and Vegetables interfere when they combine to form Fruits
or Vegetables, and the state of the latter can be represented by the
linear superposition of the states of the former. This behavior is
analogous to that of quantum particles interfering in the doubleslit experiment when both slits are open. The data are faithfully
represented in a 25-dimensional Hilbert space over complex
numbers [15, 16].
In the data collected on the membership estimations of items
with respect to pairs (A, B) of concepts and their conjunction
“A and B” and disjunction “A or B,” Hampton found systematic
violations of the rules of classical (fuzzy set) logic and probability
theory. For example, the membership weight of the item Mint
with respect to the conjunction Food and Plant is higher
than the membership weight of Mint with respect to both
Food and Plant (“overextension”). Similarly, the membership
weight of the item Ashtray with respect to the disjunction
Home Furnishing or Furniture is lower than the membership
weight of Ashtray with respect to both Home Furnishing and
Furniture (“underextension”). We showed that overextension
and underextension are natural expressions of “conceptual
emergence” [10, 16]. Namely, whenever a person estimates the
membership of an item x with respect to the pair (A, B) of
concepts and their combination C(A, B), two processes act in the
person’s mind. The first process is guided by “emergence,” that
is, the person estimates the membership of x with respect to the
new emergent concept C(A, B). The second process is guided by
“logic,” that is, the person separately estimates the membership
of x with respect to A and B and applies a probabilistic logical
calculus to estimate the membership of x with respect to C(A, B)
[54]. More important, the new concept C(A, B) emerges from
the concepts A and B, exactly as the linear superposition of
two quantum states emerges from the component states. A twosector Fock space faithfully models Hampton’s data, and was later
successfully applied to the modeling of more complex situations
involving concept combinations (see e.g., [54, 55]).
It is interesting to note that the size of deviation of classical
probabilistic rules due to overextension and underextension
generally depends on the item x and the specific combination
C(A, B) of the concepts A and B that are investigated. However,
we recently performed a more general experiment in which we
asked the participants to rank the membership of items with
respect to the concepts A, B, their negations “not A,” “not B,”
and the conjunctions “A and B,” “A and not B,” “not A and B,”
and “not A and not B.” We surprisingly found that the size of
deviation from classicality in this experiment does not depend on
either the item or the pair of concepts or the specific combination,
but shows to be a numerical constant. Even more surprisingly,
our two-sector Fock space model correctly predicts the value
of this constant, capturing in this way a deep non-classical

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mechanism connected in a fundamental way with the mechanism
of conceptual formation itself rather than only specifically with
the mechanism of conceptual combination [42, 43].
Different concepts entangle when they combine, where
“entanglement” is meant in the standard quantum sense. We
proved this feature of concepts in two experiments. In the first
experiment, we asked the participants to choose the best example
for the conceptual combination The Animal Acts in a list of
four examples, e.g., The Horse Growls, The Bear Whinnies, The
Horse Whinnies, and The Bear Growls. By suitably combining
exemplars of Animal and exemplars of Acts, we performed four
joint measurements on the combination The Animal Acts. The
expectation values violated the “Clauser-Horne-Shimony-Holt”
version of Bell inequalities [56, 57]. The violation was such that,
not only the state of The Animal Acts was entangled, but also
the four joint measurements were entangled, in the sense that
they could not be represented in the Hilbert space C4 as the
(tensor) product of a measurement performed on the concept
Animal and a measurement performed on the concept Acts
[44]. In the second experiment, performed on the conceptual
combination Two Different Wind Directions, we confirmed the
presence of quantum entanglement, but we were also able to
prove that the empirical violation of the marginal law in this type
of experiments is due to a bias of the participants in picking wind
directions. If this bias is removed, which is what we did in an
ensuing experiment on Two Different Space Directions, one can
show that people pick amongst different space directions exactly
as coincidence spin measurement apparatuses pick amongst
different spin directions of a compound system in the singlet spin
state. In other words, entanglement in concepts can be proved
from only the statistics of the correlations of joint measurements
on combined concepts, exactly as in quantum physics [45].
Since concepts exhibit genuine quantum features when they
combine pairwise, it is reasonable to expect that these features
should be reflected in the statistical behavior of the combination
of several identical concepts. Indeed, we detected quantum-type
indistinguishability in an experiment on the combination of
identical concepts, such as the combination 11 Animals. More
specifically, we found significant evidence of deviation from
the predictions of classical statistical theories, i.e., “MaxwellBoltzmann distribution.” This deviation has clear analogies with
the deviation of quantum mechanical from classical mechanical
statistics, due to indistinguishability of microscopic quantum
particles, that is, we found convincing evidence of the presence of
“Bose-Einstein distribution.” In the experiment, indeed, people
do not seem to distinguish two identical concepts in the
combination of N identical concepts, which is more evident in
more abstract than in more concrete concepts, as expected [46].
The second set of results concern “decision-making errors
under uncertainty.” In the “disjunction effect” people prefer
action x over action y if they know that an event A occurs, and
also if they know that A does not occur, but they prefer y over
x if they do not know whether A occurs or not. The disjunction
effect violates a fundamental principle of rational decision theory,
Savage’s “Sure-Thing principle” and, more generally, the total
probability rule of classical probability [58]. This preference of
sure over unsure choices violating the Sure-Thing principle was

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experimentally detected in the “two-stage gamble” and in the
“Hawaii problem” [59]. In the experiment on a gamble that can
be played twice, the majority of participants prefer to bet again
when they know they won in the first gamble, and also when they
know they lost in the first gamble, but they generally prefer not
to play when they do not know whether they won or lost. In
the Hawaii problem, most students decide to buy the vacation
package when they know they passed the exam, and also when
they know they did not pass the exam, but they generally decide
not to buy the vacation package when they do not know whether
they passed or not passed the exam. We recently showed that, in
both experimental situations, this “uncertainty aversion” can be
explained as an effect of underextension of the conceptual entities
A and “not A” with respect to the conceptual disjunction “A or
not A,” where the latter describes the situation of not knowing
which event, A or “not A,” will occur. The concepts A and “not
A” interfere in the disjunction “A or not A,” which determines
its underextension. A Hilbert space model in C3 allowed us to
reproduce the data in both experiments on the disjunction effect
[55].
Ellsberg’s thought experiments, much before the disjunction
effect, revealed that the Sure-Thing principle is violated in
concrete decision-making under uncertainty, as people generally
prefer known over unknown probabilities, instead of maximizing
their expected utilities. In the famous “Ellsberg three-color
example,” an urn contains 30 red balls and 60 balls that are
either yellow or black, in unknown proportion. One ball will
be drawn at random from the urn. The participant is firstly
asked to choose between betting on “red” and betting on “black.”
Then, the same participant is asked to choose between betting
on “red or yellow” and betting on “black or yellow.” In each
case, the “right” choice will be awarded with $100. As the events
“betting on red” and “betting on black or yellow” are associated
with known probabilities, while their counterparts are not, the
participants will prefer betting on the former than betting on the
latter, thus revealing what Ellsberg called “ambiguity aversion,”
and violating the Sure-Thing principle [60]. This pattern of
choice has been confirmed by several experiments in the last 30
years [61]. Recently, Machina identified in a couple of thought
experiments, the “50/51 example” and the “reflection example,”
a similar mechanism guiding human preferences in specific
ambiguous situations, namely, “information symmetry” [62, 63],
which was experimentally confirmed in L’Haridon and Placido
[64]. In our quantum theoretical approach, ambiguity aversion
and information symmetry are two possible cognitive contexts
influencing human preferences in uncertainty situations and
changing the states of the “Ellsberg and Machina conceptual
entities,” respectively. Hence, an ambiguity aversion context will
change the state of the Ellsberg conceptual entity in such a way
that “betting on red” and “betting on black or yellow” are finally
preferred. In other terms, the novel element of this approach is
that the initial state of the conceptual entity, in its Hilbert space
representation, can also change because of the pondering of the
participants in relation to certain choices, before being collapsed
into a given outcome. This opens the way to a generalization
of rational decision theory with quantum, rather than classical,
probabilities [65].

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The Brussels Approach to Cognition

The results above provide a strong confirmation of the
quantum theoretical approach presented in Section 2, and we
expect that further evidence will be given in this direction in the
years to come. In the next section we instead intend to analyze
some situations where deviations from Hilbert space modeling
of human cognition apparently occur. We will see in Section 5
that these deviations are however compatible with the general
operational-realistic framework portrayed in Section 2.

4. DEVIATING FROM HILBERT SPACE
As mentioned in Section 2, if suitable axioms are imposed on
the SCoP formalism, the Hilbert space structure of quantum
theory can be shown to emerge uniquely, up to isomorphisms
[30]. However, we also know that certain experimental situations
can violate some of these axioms. This is the case for instance
when we consider entities formed by experimentally separated
sub-entities, a situation that cannot be described by the standard
quantum formalism [25].Similarly, one may expect that the
structural shortcomings of the standard quantum formalism can
also manifest in the ambit of psychological measurements, in
the form of data that may not be exactly modelable (or jointly
modelable) by means of the specific Hilbert space geometry
and the associated Born rule. The purpose of this section is
to describe two paradigmatic examples of situations of this
kind: “question order effects” and “response replicability.” In the
following section, we then show how the quantum formalism
can be naturally completed to also faithfully model these data,
in a way that remains consistent with our operational-realistic
approach.
Let us first remark that the mere situation of having to deal
with a set of data for which we do not yet have a faithful
Hilbert space model should not make one necessarily search
for an alternative more general quantum-like mathematical
structure as a modeling framework. Indeed, it is very well
possible that the adequate Hilbert space model has not yet been
found. Recently, however, a specific situation was identified and
analyzed indicating that the standard quantum formalism in
Hilbert space would not be able to be used to model it [32]. This
situation combines two phenomena: “question order effects” and
“response replicability.” We start by explaining “question order
effects” and how the cognitive situation in which they appear can
be represented in Hilbert space.
For this we come back to the yes-no experiment of Section 2,
where participants are asked: “Is Gore honest and trustworthy?”
This experiment gives rise to a two-outcome measurement
performed on the conceptual entity Honesty in the initial state
pH , represented by the unit vector |Hi ∈ H, where H is a
two-dimensional Hilbert space if we assume the measurement to
be non-degenerate, or more generally a n-dimensional Hilbert
space if we also admit the possibility of sub-measurements.
Denoting {MG , M̄G = 1 − MG } the spectral family associated
with this measurement, the probability of the “yes” outcome
(i.e., to answer “yes” to the question about Gore’s honesty and
trustworthiness) is then given by the Born rule µGy (H) =
hH|MG |Hi, and of course µGn (H) = hH|M̄G |Hi = 1 − µGy (H) is
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the probability for the “no” outcome. We then consider a second
measurement performed on the conceptual entity Honesty, but
this time associated with the question: “Is Clinton honest and
trustworthy?” We denote {MC , M̄C = 1 − MC } the spectral
family associated with this second measurement, so that the
probabilities for the “yes” and “no” outcomes are again given by
µCy (H) = hH|MC |Hi and µCn (H) = hH|M̄C |Hi, respectively.
Starting from these two measurements, it is possible to
conceive sequential measurements, corresponding to situations
where the respondents are subject to the Gore and Clinton
questions in a succession, one after the other, in different orders.
Statistical data about “Clinton/Gore” sequential measurements
were reported in a seminal article on question order effects
[66] and further analyzed in Busemeyer and Bruza [14,
67]. More precisely, after fixing a rounding error in Wang
and Busemeyer [67], we have the following sequential (or
conditional) probabilities [34]:
µCyGy (H) = 0.4899 µCyGn (H) = 0.0447 µCnGy (H)
= 0.1767 µCnGn (H) = 0.2887

(1)

= 0.0255 µGnCn (H) = 0.2129

(2)

µGyCy (H) = 0.5625 µGyCn (H) = 0.1991 µGnCy (H)

where Equation (1) corresponds to the sequence where first the
Clinton and then the Gore measurements are performed, whereas
Equation (2) corresponds to the reversed order sequence for the
measurements. Considering that the probabilities in each of the
four columns above are sensibly different, these data describe
typical “question order effects.”
Quantum theory is equipped with a very natural tool to
model question order effects: “incompatible measurements,” as
expressed by the fact that two self-adjoint operators, and the
associated spectral families, in general do not commute. More
precisely, the Hilbert space expression for the probability that,
say, we obtain the answer CyGn when we perform first the
Clinton measurement and then the Gore one, is [14, 67]:
µCyGn (H) = hH|MC M̄G MC |Hi. Similarly, the probability to
obtain the outcome GnCy, for the sequential measurement
in reversed order, is: µGnCy (H) = hH|M̄G MC M̄G |Hi. Since
we have the operatorial identity M̄G MC M̄G − MC M̄G MC =
(MG − MC )[MG , MC ], the difference µGnCy (H) − µCyGn (H) will
generally be non-zero if [MG , MC ] 6= 0, i.e., if the spectral
families associated with the two measurements do not commute.
In the following we will analyze whether non-compatibility
within a standard quantum approach can cope in a satisfying way
with these question order effects, and show that a simple “yes”
to this question is not possible. Indeed, a deep problem already
comes to the surface in relation to the phenomenon of “response
replicability.”
Consider again the Gore/Clinton measurements: if a
respondent says “yes” to the Gore question, then is asked the
Clinton question, then again is asked the Gore question, the
answer given to the latter is expected to be “yes,” independently
of the answer given to the intermediary Clinton question. This
conjectured phenomenon, still necessitating a clear experimental

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confirmation, is called “response replicability1 .” If, in addition,
to question order effects also response replicability is jointly
modeled in Hilbert space quantum mechanics, a contradiction
can be detected, as shown in Khrennikov et al. [32]. Let us
indicate what are the elements that produce this contradiction.
In standard quantum mechanics only if a state is an eigenstate
of the considered measurement the outcome “yes” will be
certain in advance. Also, measurements that can transform an
arbitrary initial state into an eigenstate are ideal measurements
called of the first kind. According to response replicability,
outcomes that once have been obtained for a measurement will
have to become certain in advance if this same measurement
is performed a second time. This means that the associated
measurements should be ideal and of the first kind. For the
case of the Gore/Clinton measurements, and the situation of
response replicability mentioned above, this means that the
Gore measurement should be ideal and of the first kind. But
one can also consider the situation where first the Clinton
measurement is performed, then the Gore measurement and
afterwards the Clinton measurement again. A similar analysis
leads then to the Clinton measurement needing to be ideal
and of the first kind. This means however that after more
than three measurements that alternate between Clinton and
Gore, the state needs to have become an eigenstate of both
measurements. As a consequence, both measurements can
be shown to be represented by commuting operators. The
proof of the contradiction between “response replicability” and
“non-commutativity” worked out in Khrennikov et al. [32]
is formal and also more general than the intuitive reasoning
presented above—for example, the contradiction is also proven
when measurements are represented by positive-operator valued
measures instead of projection valued measures, which is what
we have considered here—and hence indicates that the noncommutativity of the self-adjoint operators needed to account
for the question order effects cannot be realized together with
the “ideal and first kind” properties needed to account for the
response replicability within a standard quantum Hilbert space
setting.
Although refined experiments would be needed to reveal the
possible reasons for response replicability, it is worth to put
forward some intuitive ideas, as we have been developing a
quantum-like but more general than Hilbert space formalism
within our Brussels approach to quantum cognition [35–37], and
we believe that we can cope with the above contradiction within
this more general quantum-like setting in a very natural way. It
seems to be a plausible hypothesis that response replicability is,
at least partly, due to a multiplicity of effects, that however take
place during the experiment itself, such as desire of coherence,
learning, fear of being judged when changing opinion, etc. And
a crucial aspect for both question order effects and response
replicability appearing in the Gore/Clinton situation is that the
sequential measurements need to be carried out with the same
participant, who has to be tested again and again. This is different
1 We

stress here that such a conjecture does require that “all’ psychological
measurements should satisfy “response replicability.” It rather claims that the latter
should hold for a non-empty class of these measurements.

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than the situation in quantum physics, where order effects
appear for non-commuting observables also when sequential
measurements are performed with different apparatuses. Hence,
both question order effects and response replicability seem to
be the consequence of “changes taking place in the way each
subject responds probabilistically to the situation—described by
the state of the conceptual entity in our approach—he or she is
confronted with during a measurement.” Since the structure of
the probabilistic response to a specific state is fixed in quantum
mechanics, being determined by the Born rule, it is clear that such
a change of the probabilistic response to a given measurement,
when it is repeated in a sequence of measurements, cannot
be accounted for by the standard quantum formalism. And
it is exactly such structure of the probabilistic response to a
same measurement with respect to a given state that can be
varied in the generalized quantum-like theory that we have
been developing [35–37]. This is the reason that, when we
became aware of the contradiction identified in Khrennikov
et al. [32], we were tempted to investigate whether in our
generalized quantum-like theory the contradiction would vanish,
and response replicability would be jointly modelizable with
question order effects. And indeed, we could obtain a positive
result with respect to this issue [34], which we will now sketch
in the next section.

5. BEYOND-QUANTUM MODELS
We presented in Section 4 two paradigmatic situations in human
cognition that cannot be modeled together using the standard
quantum formalism. We want now to explain how the latter
can be naturally extended to also deal with these situations, still
remaining in the ambit of a unitary and coherent framework for
cognitive processes.
For this, we introduce a formalism where the probabilistic
response with respect to a specific experimental situation, i.e.,
a state of the conceptual entity under consideration, can vary,
and hence can be different than the one compatible with the
Born rule of standard quantum theory. This formalism, called
the “extended Bloch representation” of quantum mechanics
[35], exploits in its most recent formulation the fact that
the states of a quantum entity (described as ray-states or
density matrix-states) can be uniquely mapped into a convex
portion of a generalized unit Bloch sphere, in which also
measurements can be represented in a natural way, by means of
appropriate simplexes having the eigenstates as vertex vectors.
A measurement can then be described as a process during
which an abstract point particle (representing the initial state of
the quantum entity) enters into contact with the measurement
simplex, which then, as if it was an elastic and disintegrable
hyper-membrane, can collapse to one of its vertex points
(representing the outcomes states) or to a point of one of its
sub-simplexes (in case the measurement would be degenerate).
We do not enter here into the details of this remarkable
process, and refer the reader to the detailed descriptions in Aerts
and Sassoli de Bianchi [34–37]. For our present purposes, it will
be sufficient to observe that a measurement simplex, considered

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as an abstract membrane that can collapse as a result of some
uncontrollable environmental fluctuations, can precisely model
that aspect of a measurement that in the quantum jargon is
called “wave function collapse.” More precisely, when the abstract
point particle enters into contact with the “potentiality region”
represented by such membrane, it creates some “tension lines”
partitioning the latter into different subregions, one for each
possible outcome. The collapse of the membrane toward one of
the vertex points (see Figure 1) then depends on which subregion
disintegrates first, so that the different outcome probabilities
can be expressed as the relative Lebesgue measures of these
subregions (the larger a subregion, the higher the associated
probability). In other terms, this membrane’s mechanism, with
the tension lines generated by the abstract point particle, is a
mathematical representation of a sort of “weighted symmetry
breaking” process. Now, thanks to the remarkable geometry of
simplexes, it can be proven that if the membrane is chosen to
be uniform, thus having the same probability of disintegrating in
any of its points (describing the different possible measurementinteractions), the collapse probabilities are exactly given by the
Born rule. In other terms, the latter can be derived, and explained,
as being the result of a process of actualization of potential
hidden-measurement interactions, so that the extended Bloch
representation constitutes a possible solution to the measurement
problem.
Thus, when the membrane is uniform, the “way of
choosing” an outcome is precisely the “Born way.” However,
a uniform membrane is a very special situation, and it is
natural to also consider membranes whose points do not all
have the same probability of disintegrating, i.e., membranes
whose disintegrative processes are described by non-uniform
probability densities ρ, which we simply call ρ-membranes.
Non-uniform ρ-membranes can produce outcome probabilities
different from the standard quantum ones and give rise to
probability models different from the Hilbertian one (even
though the state space is a generalized Bloch sphere derived from
the Hilbert space geometry2 ). But this is exactly what one needs
in order to account, in a unified framework, for the situation
we encounter when combining the phenomena of “response
replicability” and “question order effects,” as previously described
and analyzed in Khrennikov et al. [32].
We thus see that it is possible to naturally complete the
quantum formalism to obtain a finer grained description of
psychological experiments in which the probabilistic response
of a measurement with respect to a state can be different
to the one described by the Born rule. Additionally, our
generalized quantum-like theory also explains why, despite the
fact that individual measurements are possibly associated with
different non-Born probabilities, the Born rule nevertheless
appears to be a very good approximation to describe numerous
experimental situations. This is related to the notion of “universal
measurement,” firstly introduced by one of us in Aerts [38] and
further analyzed in Aerts and Sassoli de Bianchi [35–37, 68]. In
a nutshell, a universal measurement is a measurement whose
2 More

general state spaces can also be considered, in what has been called the
“general tension-reduction” (GTR) model [36, 37, 40].

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FIGURE 1 | A 2-dimensional measurement simplex, considered as an abstract membrane stretched between the three vertex points x1 , a x2 , and x3 ,
with the abstract point particle attached to it, at point x, giving rise to three disjoint convex regions A1 , A2 , and A3 . The vector λ, here assumed to belong
to region A2 , indicates the initial point of disintegration of the membrane, which by collapsing brings the point particle to point x2 , corresponding to the final outcome
of the measurement.

probabilities are obtained by averaging over the probabilities of
all possible quantum-like measurements sharing a same set of
outcomes, in a same state space. In other terms, a universal
measurement corresponds to an average over all possible nonuniform ρ-membranes, associated with a given measurement
simplex. Following a strategy similar to that used in the definition
of the “Wiener measure,” it is then possible to show that if the
state space is Hilbertian (more precisely, a convex set of states
inscribed in a generalized Bloch sphere, inherited from a Hilbert
space), then the probabilities of a universal measurement are
precisely those predicted by the Born rule.
In Aerts and Sassoli de Bianchi [34] we could show that the
joint situation of question order effects and response replicability
for the data collected with respect to the Gore/Clinton
measurements, and others, is modelizable within our generalized
quantum theory by introducing non-Born type measurements.
However, we were also able to provide a better modeling of
the question order effects data as such. Indeed, using standard
Born-probability quantum theory it was only possible to model
approximately these data in earlier studies [67]. This is due to
the existence of a general algebraic equality about sequential
measurements in Hilbert space quantum theory which is the
following [34, 67, 69]:
Q ≡ MG MC MG − MC MG MC + M̄G M̄C M̄G
−M̄C M̄G M̄C = 0

(3)

where {MG , M̄G = 1 − MG } and {MC , M̄C = 1 − MC } are the
spectral families associated with the Hilbert model of the Gore

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and Clinton measurements introduced in Section 4. Taking the
average q = hH|Q|Hi, one thus obtains, more specifically:
q ≡ µGyCy (H) − µCyGy (H) + µGnCn (H) − µCnGn (H) = 0. (4)
This equality has been called the “QQ-equality,” and can be used
as a test for the quantumness of the probability model, but only
in the sense that a quantum model, necessarily, has to obey it,
although the fact that it does so is not a guarantee that the model
will be Hilbertian. Inserting the experimental values Equations
(1) and (2) into Equation (4), one finds q = 0.0032 6= 0. This
value is small (being only 0.32% of the maximum value q can
take, which is 1), which is the reason that approximate modeling
can be obtained within Hilbert space quantum theory [67]. Note
however that Equation(3) does not depend on the dimension
of the Hilbert space considered, which means that even in
higher dimensional Hilbert spaces, if degenerate measurements
are considered, an exact modeling would still be impossible to
obtain. We have reasons to believe that also question order
effects, with the QQ-equality standing in the way of an exact
modeling of the data, contain an indication for the need to turn
to a more general quantum-like theory, such as the one we used
to cope with the joint phenomenon of question order effects and
response replicability. We present some arguments in this regard
in the following of this section.
First, we note that in case one chooses a two-dimensional
Hilbert space, additional equalities can be written which are
strongly violated by the data this time. As an example, consider

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Aerts et al.

The Brussels Approach to Cognition

the quantity [34]:
q′ ≡ µCyGn (H)µCnGn (H) − µCnGy (H)µCyGy (H)

(5)

−hH|M̄C MG M̄C |HihH|MC MG MC |Hi

(6)

= hH|MC M̄G MC |HihH|M̄C M̄G M̄C |Hi

If the Hilbert space is two-dimensional, one can write MG =
|GihG|, M̄G = |ḠihḠ|, as well as MC = |CihC|, M̄C =
|C̄ihC̄|. Replacing these expressions into Equation (6) one finds,
after some easy algebra, that q′ = 0. However, inserting the
experimental values Equations (1) and (2) into Equation (6), one
finds q′ = −0.073 6= 0, which not only is not zero, but also 29.2%
of the maximum value that q′ can take (which is 0.25).
Second, let us repeat our intuitive reasoning as to why
measurements in the situation of response replicability carry
non-Bornian probabilities. Due to the local contexts of the
collection of sequential measurements, Gore, Clinton, and then
Gore again, the third measurement internally changes into a nonBornian one, and more specifically a deterministic one for the
considered state, since response replicability means that for all
subsequent Gore measurements the same outcome is assured. It
might well be the case, although an intuitive argument would
be more complex to give in this case, that also for the situation
of question order effects, precisely because they only appear if
a same human mind is sequentially interrogated, non-Bornian
probabilities would be required. An even stronger hypothesis,
which we plan to investigate in the future, is that most individual
human minds, and perhaps even all, would carry in general
non-Bornian probabilities, so that the success of Hilbert space
quantum theory and Bornian probabilities would be mainly
an effect of averaging over a sufficiently large set of different
human minds, which effectively is what happens in a standard
psychological experiment. If this last hypothesis is true, the
violation of the Born rule for question order effects and response
replicability would be quite natural, since the same human
mind is needed to provoke these effects. Indeed, our analysis
in Aerts and Sassoli de Bianchi [36, 37] shows that standard
quantum probabilities in the modeling of human cognition can
be explained by considering that in numerous experimental
situations the average over the different participants will be quite
close to that of a universal measurement, which as we observed
is exactly given by the Born rule. In other terms, even if the
probability model of an individual psychological measurement
could be non-Hilbertian, it will generally admit a first order
approximation, and when the states of the conceptual entity
under investigation can be described by means of a Hilbert space
structure, this first order approximation will precisely correspond
to the quantum mechanical Born rule.
If the above considerations provide an interesting piece of
explanation as to why the Born rule is generally successful
also beyond the micro-physical domain, at the same time it
also contains a plausible reason of why it will possibly be not
successful in all experimental situations, i.e., when the average is
either not large enough, or when the experiment is so conceived
that it doesn’t apply as such. This could be the typical situation
of question order effects and response replicability, since in this
case we do not consider an average over single measurements, but

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over sequential (conditional) measurements. And this could be
an explanation of why Hilbertian symmetries like those described
above can be easily violated and that it will not be possible, by
means of the Born rule, to always obtain an exact fit of the data
[34, 40].
Additionally, as we said, it allowed us to precisely fit the
data by using the extended Bloch representation, and more
specifically simple one-dimensional locally uniform membranes
inscribed in a 3-dimensional Bloch sphere that can disintegrate
(i.e., break) only inside a connected internal region [34]. Thanks
to this modeling, we could also understand that the reason the
Clinton/Gore and similar data appear to almost obey the QQequality (Equation 4) is quite different from the reason the
equality is obeyed by pure quantum probabilities. Indeed, in a
pure quantum model two specific contributions to the q-value
(Equation 4), called the “relative indeterminism” and “relative
asymmetry” contributions, are necessarily both identically zero,
whereas we could show, using our extended model, that for the
data (Equation 2), and similar data, these two contributions are
both very different from zero, but happen to almost cancel each
other, thus explaining why the q = 0 equality is almost obeyed,
although the probabilities are manifestly non-Bornian [34].

6. FINAL CONSIDERATIONS
In this article we explained the essence of the operationalrealistic approach to cognition developed in Brussels, which in
turn originated from the foundational approach to quantum
physics elaborated initially in Geneva and then in Brussels (in
what has become known as the “Geneva-Brussels school”). Our
emphasis was that this approach is sufficiently general, and
fundamental, to provide a unitary framework that can be used to
coherently describe, and realistically interpret, not only quantum
theory, but also its natural extensions, like the extended Bloch
model and the GTR-model. In this final section we offer some
additional comments on our approach to cognition, taking into
consideration the confusion that sometimes exists between “ad
hoc (phenomenological) models’ and “theoretical (first principle)
models,” as well as the critique that a Hilbertian model (and a
fortiori its possible extensions) is suspicious because it allows
“too many free parameters’ to obtain an exact fit (and not just
an approximate fit) for all the experimental data.
In that respect, it is worth emphasizing that the principal focus
of our “theory of human cognition” is not to model as precisely
as possible the data gathered in psychological measurements. A
faithful modeling of the data is of course an essential part of
it, but our aim is actually more ambitious. In putting forward
our methodology, consisting in looking at instances of decisionmaking as resulting from an interaction of a decision-maker
with a conceptual entity, we look first of all for a theory
truly describing “the reality of the cognitive realm to which a
conceptual entity belongs,” and additionally also “how human
minds can interact with the latter so that decision-making can
occur.”
In this sense, each time we have put forward a model for some
specific experimental data, it has always been our preoccupation

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Aerts et al.

to also make sure that (i) the model was extracted following
the logic that governs our theory of human cognition, and
(ii) that whatever other experiments would be performed by a
human mind interacting with that same cognitive-conceptual
entity under consideration, also the data of these hypothetical
additional experiments could have been modeled exactly in
the same way. Clearly, this requirement—that “all possible
experiments and data” have to be modeled in an equivalent
way—poses severe constraints to our approach, and it is not
a priori evident that this would always be possible. However,
we are convinced that the fundamental idea underlying our
methodology, namely that of looking upon a decision as an
interaction of a human mind with a conceptual entity in a specific
state (with such state being independent of the human minds
possibly interacting with it), equips the theory of exactly those
degrees of freedom that are needed to model “all possible data
from all possible experiments.”
As we already explained in the foregoing, in all this we have
been guided by how physical theories deal with data coming
from the physical domain. They indeed satisfy this criterion and
are able to model all data from all possible experiments that
can be executed on a given physical entity. What we have called
“conceptual entity” is what in physics corresponds to the notion
of “physical entity.” Now, in our approach we might be classified
as adhering to an idealistic philosophy, i.e., believing that the
conceptual entities “really exist,” and are not mere creations of
our human culture. Our answer to this objection is the following:
to profit of the strength of the approach it is not mandatory
to take a philosophical stance in the above mentioned way, in
the sense that we are not obliged to attribute more existence
to what we call a conceptual entity than that attributed, for
example, to “human culture” in its entirety. The importance
of the approach lies in considering such a conceptual entity as
independently existing from any interaction with a human mind,
and describe the continuously existing interactions with human
minds as processes of the “change of state of the conceptual
entity,” and whenever applicable also as processes of the “change
of context.” And again, let us emphasize that this “hiddeninteraction” methodology is inspired by its relevance to physical
theories. Our working hypothesis is that in this way it will be
possible to advantageously model, and better understand, all of
human cognition experimental situations.
Having said this, we observe that the interpretation of the
quantum formalism that is commonly used in cognitive domains
is a subjectivist one, very similar to that interpretation of
quantum theory known as “quantum Bayesianism,” or “QBism”
[70]. In a sense, this interpretation is the polar opposite of
our realistic (non-subjectivistic) operational approach. Indeed,
QBism originates from a strong critique [71] of the famous
Einstein-Podolsky-Rosen reality criterion [72], whereas at the
foundation of the Geneva-Brussels approach there is the idea
of taking such criterion not only extremely seriously, but also
of using it more thoroughly, as a powerful demarcating tool
separating “actually existing properties” from “properties that are
only available to be brought into actual existence,” and therefore
exist in a potential sense [73]. In other terms, a quantum state is
not considered in QBism as a description of the actual properties

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The Brussels Approach to Cognition

of a physical entity, but of the beliefs of the experimenter
about it. Similarly, for the majority of authors in quantum
cognition, a quantum state is a description of the state of belief
of a participant, and not of the actual state of the conceptual
entity that interacts with the participants. In ultimate analysis,
this difference of perspectives is about taking a clear position
regarding the key notion of “certainty”: is certainty (probability 1
assignments) just telling us something about the very firm belief
of a subject, or also about some objective properties of the world
(be it physical or cultural)? In the same way, are probabilities only
shared personal beliefs, based on habit, or also elements of reality
(considering that in principle their values can be predicted with
certainty)? Although we certainly agree that it is not necessary
to take a final stance on these issues to advantageously exploit
the quantum mathematics in the modeling of many experimental
situations, both in physics and cognition, we also think that the
explicative power of a pure subjectivist view rapidly diminishes
when we have to address the most remarkable properties of
the physical and conceptual entities, like non-locality (nonspatiality) and the non-compositional way with which they can
combine.
It is important to emphasize that the subjectivist view is also
a consequence of the absence, in the usual quantum formalism,
of a meaningful description of what goes on “behind the
scenes” during a measurement. On the other hand, the hiddenmeasurement paradigm, as implemented in the extended Bloch
representation [35], or even more generally in the GTR-model
[36, 37, 40], offers a credible description of the dynamics of a
measurement process, in terms of a process of actualization of
potential interactions, thus explaining a possible origin of the
quantum indeterminism. This certainly allows understanding
the so-called “collapse of the state vector” as an objective
process, either produced by a macroscopic apparatus in a physics
laboratory, or by a mind-brain apparatus in a psychological
laboratory. As we tried to motivate in the second part of
this article, this completed version of the quantum formalism
also allowed us to describe those aspects of a psychological
measurements—the possible different ways participants can
choose an outcome—that would be impossible to model by
remaining within the narrow confines, not only of the quantum
formalism, but also of a strict subjectivistic interpretation
of it.
To conclude, a final remark is in order. Quantum cognition is
undoubtedly a fascinating field of investigation also for physicists,
as it offers the opportunity to take a new look at certain
aspects of the quantum formalism and use them to possibly
make discoveries also in the physical domain. We already
mentioned the example of “entangled measurements,” that were
necessary to exactly model certain correlations. Entangled (nonseparable) measurements are usually not considered in the
physics of Bell inequalities, while they are widely explored in
quantum cryptography, teleportation and information. However,
it is very possible that this stronger form of entanglement
will prove to be useful for the interpretation of certain nonlocality tests and the explanation of “anomalies” that were
identified in EPR-Bell experiments [44]. Also, for what concerns
the notion of “universal measurement,” which is quite natural

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Aerts et al.

in psychological measurements, since data are obtained from
a collection of different minds, could it be that “universal
averages” also happen in the physical domain? In other terms,
could it be that a single measurement apparatus is actually
more like “a collection of different minds” than “a single
Born-like mind”? Considering that the origin of the observed
deviations from the Born rule, in situations of sequential
measurements, can be understood as the ineffectiveness of
the averaging process in producing the Born prescription,
is it possible to imagine, in the physics laboratory, similar
experimental situations where these deviations would be
equally observed, thus confirming that the hypothesis of
“hidden measurement-interactions” would be a pertinent one
also beyond the psychological domain? Whatever the verdict

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application to the locality of quantum mechanics. Am J Phys. (2014)
82:749–54. doi: 10.1119/1.4874855
72. Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of
physical reality be considered complete? Phys Rev. (1935) 47:777–80.
73. Sassoli de Bianchi M. Ephemeral properties and the illusion of microscopic
particles. Found Sci. (2011) 16:393–409. doi: 10.1007/s10699-0119227-x
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
The Handling Editor declared recent co-publications, though no other
collaboration, with the reviewers (IB) and (JB) and states that the process
nevertheless met the standards of a fair and objective review.
Copyright © 2016 Aerts, Sassoli de Bianchi and Sozzo. This is an open-access article
distributed under the terms of the Creative Commons Attribution License (CC BY).
The use, distribution or reproduction in other forums is permitted, provided the
original author(s) or licensor are credited and that the original publication in this
journal is cited, in accordance with accepted academic practice. No use, distribution
or reproduction is permitted which does not comply with these terms.

May 2016 | Volume 4 | Article 17 | 35

ORIGINAL RESEARCH
published: 12 September 2016
doi: 10.3389/fphy.2016.00040

Information and Temporality
Christian Flender *
Faculty of Economics and Behavioral Sciences, University of Freiburg, Freiburg, Germany

Being able to give reasons for what the world is and how it works is one of the defining
characteristics of modernity. Mathematical reason and empirical observation brought
science and engineering to unprecedented success. However, modernity has reached
a post-state where an instrumental view of technology needs revision with reasonable
arguments and evidence, i.e., without falling back to superstition and mysticism.
Instrumentally, technology bears the potential to ease and to harm. Easing and harming
can’t be controlled like the initial development of technology is a controlled exercise
for a specific, mostly easing purpose. Therefore, a revised understanding of information
technology is proposed based upon mathematical concepts and intuitions as developed
in quantum mechanics. Quantum mechanics offers unequaled opportunities because it
raises foundational questions in a precise form. Beyond instrumentalism it enables to
raise the question of essences as that what remains through time what it is. The essence
of information technology is acausality. The time of acausality is temporality. Temporality
is not a concept or a category. It is not epistemological. As an existential and thus more
comprehensive and fundamental than a concept or a category temporality is ontological;
it does not simply have ontic properties. Rather it exhibits general essences. Datability,
significance, spannedness and openness are general essences of equiprimordial time
(temporality).
Edited by:
Emmanuel E. Haven,
University of Leicester, UK
Reviewed by:
Ignazio Licata,
ISEM - Institute for Scientific
Methodology, Italy
Paavo Pylkkanen,
University of Helsinki, Finland
*Correspondence:
Christian Flender
mail@christian-flender.de
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 31 May 2016
Accepted: 22 August 2016
Published: 12 September 2016
Citation:
Flender C (2016) Information and
Temporality. Front. Phys. 4:40.
doi: 10.3389/fphy.2016.00040

Frontiers in Physics | www.frontiersin.org

Keywords: information, technology, temporality, acausality, quantum mechanics

1. INTRODUCTION
In Plato’s famous allegory of the cave chained prisoners only see shadows of things projected on the
wall they are forced to look at. As one of their fellows is freed from the cave, he comes to see reality
and returns to inform about what he experienced. Nobody believes his report. Plato’s idealism
stems from the presupposition that there are pure ideas apart from humanity (the cave) which
only sees instances and appearances (shadows) of perfect shapes. The truth is judged according to
perceptions and conceptions matching or corresponding to a perfect idea (eidos) which may never
be attainable.
With the advent of modern science in the Sixteenth and Seventeenth century correspondence
started to bear fruits again. Descartes was the first who assured himself of what things really are
by claiming cogito ergo sum (I think, therefore I am) [1]. He pulled Plato’s ideas to his cognitive
faculty and made thinking and reasoning the ultimate means for determining being of the self1 .
His thoughts eluded doubt and became the subject-pole (res cogitans) as opposed to objects in the
external world (res extensa). Correspondence was redefined as the relation between propositions
uttered by the thinking ego and properties of things out there in the external world. The truth of
the res extensa depended on its matching the res cogitans. Modern dualism was born.
With the rise of commercial information technology and the Internet in the second half of the
last century dualism has been a fruitful engine for business innovation and economic prosperity.
The template for digital information processing is social phenomena in the analog world.

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Communication, coordination, cooperation and competition
are metaphors employed for building information systems.
Calculating machines, digital storages, and information highways
facilitate and support human activities from the viewpoint
of input-output relations and state transitions. Information is
coded, transformed, stored, and transmitted at high speed over
large distances. Symbol representation and manipulation are at
the heart of computation and information exchange exercised
by digital machines and human minds. Typical artifacts are
algorithms and data structures. Brain-like sub-symbolic networks
are trained to represent and simulate symbolic information
and problem-solving abilities at a higher human-like level
of reasoning. A huge amount of tools and services emerged
to support social activities like manufacturing, information
search, or relationship management. Such informational artifacts
have become pervasive and ubiquitous and alter entrenched
norms of social activities at an increasing level of speed and
sophistication. Their utility and usability can be determined
anthropologically. The way they are developed and used within
a given cultural context partly determines their significance.
With the advent of commercial online social networks in the
late 1990s managing contacts was a major utility. Today, they
serve millions of businesses to advertise their products and
services. Essentially, informational artifacts are instrumental.
Their inner-causality serves humanity as means to an end. Both
human causation and inner-causal-functioning follow design
principles: purposive ideas described in (in-) formal terms
for the sake of computational and material instantiation and
support.
Engineering as problem-solving reduces technology to a
means; technology is instrumental. In contrast, science is
dedicated to find out about what nature and technology
is—its essence (Wesen)—including the essence of utility and
problem-solving. The essence of technology is neither simply
anthropological nor merely instrumental [5, 6]. The essence of
something is its enduring as presence2 . It is temporal and goes
1 Descartes’ ontological mind-matter dualism stems from his understanding of God
as ens perfectissimum. This being is substantial, i.e., self-sustaining; it needs nothing
else than itself. He took this understanding of being as substance and applied it to
thinking and the world, i.e., ens creatum. There is an infinite difference between
the creator and its creations; however, human beings are as self-sustaining as
their creator. More precisely, the res cogitans (ego) and the res extensa (world)
are substances. Ego and world are ontological in the sense of substantial. Like
Kant at a later point in time he acknowledged that these self-sustaining things
are not knowable how they are in-themselves. Therefore, value predicaments are
necessary. However, he like the scholastics and the ancient Greeks presupposed
substances as self-sustaining things nevertheless. Heidegger was the first who
questioned this presupposition of being as substance and came up with a new
ontology (I call it Twenty-first century ontology [2] because it will be our current
millennium and century that Heidegger’s ontology will be understood properly)
for which the essence of being is not infinite like God or finite like ego and world.
The essence of being is temporality. Temporality temporalizing itself is not selfsustaining though remaining and resting-in-itself. There is no infinite difference
between temporality and human beings like Descartes’ ontology presupposed. If
you want God is through and with us. See §20 (The Fundaments of the Ontological
Definition of the “World”) in Heidegger [3, 4].
2 Gumbrecht [7] uses the term presence to signify effects fusing with meaning. Noë
[8] refers to presence in the context of sensor-motor activity. Both ground the
intellect in the physical and socio-cultural world, the place where essences are to
be found [2].

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Information and Temporality

beyond meaning in the sense of a correspondence between an
idea or formal description and its material instantiation and
computational enactment. Engineering builds upon science and
science makes use of technological artifacts. Science wants to
know what things really are. It wants to know essences as that
what remains through time what it is.
From a foundational and scientific point of view it is
reasonable to question instrumental conceptions of information
technology. However, nothing is gained if we play off utility
against foundation. Both applied research and basic science
are legitimate. In the history of science the latter was often a
precursor of the former. Who expected that after the discovery
of the quantum in 1900 transistors and micro-electronics would
enable mobile access to global and personalized services as we
find them today?
In the early Twenty-first century we stand at the brink of
a fourth industrial revolution [9]. After mechanical production
with power from steam and water in the late Eighteenth
century, electricity and mass production a 100 years later, and
production automation through information technology in the
second half of last century, today, cyber-physical systems such
as augmented reality appliances, Industry 4.0, autonomous cars,
and the Internet of Things mark the cornerstone of a next
revolution. Interpreting data truthfully is a key competence in
this context. They call it the cognitive era. When it comes to
explain how the cognitive and the embodied, the mechanical
and the enlivened, humans and machines, actually correlate and
interact with each other interdisciplinary approaches involving
disciplines such as engineering and philosophy appear as firstclass candidates to clarify the very nature of what it means to
make sense of the world both from an applied and foundational
point of view.
However, up to this day we still lack a sound and coherent
understanding of what it means to be a conscious, autonomous,
freedom-loving, situated and culturally-embedded individual.
There are many debates about whether a computing machine one
day will be able to turn into a conscious being like a human [10].
Of course, this depends on our definition of consciousness. For
a panpsychist even a dead stone or a river is somehow enlivened.
Another extreme demarcates certain pathological observations of
people having lost control of their autonomy. Is it possible for
a human to turn into a deterministic machine totally controlled
from outside? These and many other questions will increasingly
pop up the more we advance and extend our industry and culture
with information technology.
What is information? Many answers to this question are
spatial. They refer to a location. For instance, a dialectic approach
may distinguish information from matter and energy and locate
it in the human mind or a storage device such as the front
page of a newspaper or the magnetic tape of a hard disk drive.
But even for matter and energy it is far from clear and settled
if and where they are located. Think of non-local correlations
in quantum physics. For two classically correlated observables
usually a change of property A (e.g., acceleration) causes property
B (e.g., position) to change3 . The time it takes for A to have
3 This

is not to say that A is necessary and sufficient for B to change.

September 2016 | Volume 4 | Article 40 | 37

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an effect on B is constrained by the speed of light. A and B are
spatially localized whereas a change of A exerts a force leading to
a change of B. Non-local correlations between observables (e.g.,
spin of photons) are faster than light and thus instantaneous. A
and B change at two (even far distant) places at the same time
without a local force between them. In other words, A and B are
at two spatially separated locations simultaneously. Their relation
is acausal.
Again what is information? Some scientists claim information
is matter and energy. All information about matter and energy
is encoded in their respective wave function. But where is the
wave function of my information about the latest stock market
news? In my head or in the weekly magazine I read to gather
information about the stock exchange? If information is nonlocal the relevant news about the shareholder value of a particular
company may be distributed among both physical devices, my
brain and the magazine, and thus it may be localized at two places
simultaneously.
Besides the many problems a reductive view on information
raises, it is hard to deny the experience of information being
something extra-physical. It resides over, above or beyond
the material. The development of information technology
starts with an abstract idea—let’s say a diagram of the main
classes and their relationships of an object-oriented software
application to be developed—and ends with the implementation
of a prototype ready to run and be presented at the
customer’s in-house hardware infrastructure. The software and
its design are essentially separated from its implementation
and hardware. But what about the users, how do they relate
and interact with the software and its interfaces? While using
a smart phone, can we clearly separate the device from
its user? Do we transfer information from our minds into
the database of a software application and vice versa? Is
there a correspondence between information in the mind and
information stored on a physical device? Is the essence of
information a correspondence between thinking and the external
world (adaequatio intellectus et rei)? What is the essence of
information technology: software, hardware, interface design, or
usability as experienced?
In this article I’ll argue that the essence of information
technology is temporality. Temporality is the time of acausality.
Acausality is introduced by means of the mathematical apparatus
of quantum mechanics (QM) [11] and takes into account the
current state of what natural science revealed to be form and
matter and how humans actually come to know what form and
matter is.
The paper proceeds as follows. In the next section
anthropology and instrumentalism demarcate the starting
point for a discussion of information technology. Causality is
revealed as a unity of four causes including an anthropological
dimension which philosophy has taught for centuries [5, 6]. In
Section 3 quantum concepts are presented in light of causality as
a precise acausal means for revealing the essence of information
technology. Section 4 argues that temporality is the time of
acausality and temporalizes information technology ecstatically
and horizontally [3, 4]. Finally, Section 5 concludes the
paper.

Frontiers in Physics | www.frontiersin.org

Information and Temporality

2. ANTHROPOLOGY AND
INSTRUMENTALISM
We use information technology in manifold ways. Take browsers
as an example. Through browsers we access web pages, fill out
forms, view statistics, retrieve search results, or leave traces. Our
active engagement with browsers partly determines utility and
results we get out of the web. With our decisions and actions,
clicks and hand movements, we cause browsers to perform
a variety of tasks serving our purposes. In antiquity thinkers
already knew about causation and causality for which effects
were partly determined by a performer (causa efficiens). The
browser is a window through which we trigger calculations and
visualizations. The results we retrieve are not fully determined
by this triggering. A search algorithm implemented on a server
we are connected with takes our queries as input, interprets
our request, and processes results according to its causalfunctioning. This causal-functioning is formally described by an
engineer (causa efficiens) in terms of a counting and calculating
procedure instructed to determine a ranked list of web pages most
relevant to our query (causa formalis). However, a formal search
procedure like the famous page-rank algorithm is not sufficient
for the essence of searching the web. Its materialization and
instantiation on a physical machine is required (causa materialis).
Like the calculating human mind is indebted to its physical
realization—body, arms, hands, fingers, pen, and paper—a search
procedure is caused by its material correlate. Moreover, the
instantiated and materialized algorithm follows certain rules and
these rules were designed to guarantee an end (telos)—the search
result—with respect to degrees of freedom (causa finalis). Also an
end is a cause for which means were developed and implemented.
Together, these four causes make up the essence of causality.
Anthropologically, this essence encompasses the causa efficiens
in terms of an engineer who designed and implemented browser
and search algorithm and an end-user who formulates and puts
queries in order to retrieve results. The former is the original
performer who adopts the perspective of the latter. All four
causes make up instrumentalism. Together with performers who
trigger design, implementation and usage an instrumental and
anthropological conception of information technology stands.
Philosophy has taught these four causes for centuries [5, 6].
It becomes clear that the essence of searching the web is neither
a general idea or form (eidos) of a search algorithm and the
data structures it operates upon formalized as means to an
end. Nor is it its physical implementation and readiness to be
used. Essentially, at the heart of instrumentalism causality is
anthropological too with the performers (engineer and user)
being an integral part of technology.
In antiquity techne was not simply a technological artifact
like a browser or a search procedure. Techne was a way of
revealing truth (aletheia). Revealing was more than a craft. It
also meant knowing (episteme)—the working of the mind—and
artistic work like poetry. Poetic work stems from poiesis and
means revealing in the sense of bringing-forth or disclosing. The
essence of technology is revealing as it shows itself in the world.
This self-revealing encompasses but stands in sharp contrast
to a correspondence theory of truth (adaequatio intellectus

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Flender

et rei). Correspondence starts with a proposition—a linguistic
expression—which is either true or false. Truth and falsity
is decided by referring to an object which either fulfills the
proposition or fails to do so. For instance, the proposition
Plato was a genius refers to the philosopher Plato who
either was a genius or not. Plato himself would have reduced
this proposition to a form or general essence—the proposition
as eidos—from which truth or falsity would have emanated.
He would have reduced his uniqueness and situatedness to
an abstract idea. In contrast techne as revealing and poetic
production brings-forth possibilities for action and affordances
to act as remaining and resting-in-themselves. Possibilities and
affordances of material (hyle), form (eidos) and purpose (telos)
reveal themselves into unconcealment (aletheia). This revealing
bears a concealing (letheia), i.e., a revealing that hides itself, for
instance by means of context-annihilating propositions or ideas.
The key of techne as revealing is to re-contextualize the hidden or
concealed toward the essence of technology. Quantum mechanics
provides the acausal means to do so.

3. QUANTUM MECHANICS AND
INFORMATION TECHNOLOGY
Quantum mechanics is increasingly applied to areas outside of
physics [11]. This has made it possible to investigate quantumlike effects in domains such as computer science, economics,
and psychology. Since the discovery of the quantum in the
beginning of the last century, physics has raised questions
far beyond what has been traditionally conceived as physical.
Determinism, reductionism and physical realism are usually
concepts in philosophy. With the advent of quantum theory
they became entangled not only with physics but a lot of other
disciplines and even popular science. Today, with the success
and economic significance of information technology, a large
number of disciplines related to information exist side by side.
Many of them claim to be an applied science. Institutions offering
information-related research and education may wish to clarify
their subject matter with respect to current scientific progress and
questions related to techne.
This section gives credit to the current state of what natural
science revealed to be form and matter and it takes into account
how humans actually come to know what form and matter is.
It shows that there are physical forms or shapes for which there
is no cause in the sense of causality discussed in the previous
section. Acausality (technology) reveals the essence of physical
forms (information). Acausality has its own time, a primoridal
time (temporality) that temporalizes itself.
What makes formalisms of quantum mechanics interesting
is that they can’t fully abstract from the material world. Pure
mathematics is neither required to put its formal statements to
empirical test4 . Nor does it derive necessarily its formalisms from
empirical data. Symbolic descriptions are supposed to stand on
their own feet5 . Their application to engineering and the natural
4 Ontologically,

the empirical is first and foremost phenomenological.
does not deny the necessity of embodied cognitive skills. If symbolic
representation and manipulation are agnostic toward syntax, i.e., physical form
or shape, it is bad phenomenology.
5 This

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Information and Temporality

and social sciences is of secondary importance. In quantum
mechanics, however, the notion of wave function collapse or
state reduction enforces empirical context. Phenomenological
observation or measurement creates or constructs real states,
which beforehand were indeterminate or didn’t exist.
The so called quantum enigma [12]—also known as the
measurement problem—is one of the outstanding mysteries in
physics and the sciences as a whole. Last but not least, its
explosiveness stems from the fact that causality breaks at the
most fundamental level of objective, third-personal and contextfree descriptions of nature. The way a human experimenter
sets up a measurement device—a decision made by human
consciousness—determines whether he will find matter and
energy behaving like waves spread out and extended in space
or discrete particles whose real existence is determined with the
actual measurement performed. It appears that human decision
making is inseparably connected with the perspective taken upon
one or the other experimental setting and its outcomes. Last but
not least, inseparability of mind and matter is reasonable since
humans have a body and sensing organs built out of atoms and
forces guiding them.

3.1. Inseparability and Acausality
A scientifically and philosophically informed means for revealing
the essence of information technology does not simply take
presuppositions about causality for granted. Therefore, a first step
toward developing such a means is questioning if there are causes
beyond causa efficiens, causa formalis, causa materialis, and causa
finalis. In quantum mechanical systems there are inseparable
states. Such states occur within combined systems composed
of two or more individual systems. Inseparable states can’t be
reduced causally to states of individual systems. They seem to
have no cause; they are acausal.
Mathematically, a combined system is described in
multidimensional vector space. Vectors and linear operators
in combined vector spaces represent states, properties, and
measurements of systems [13]. For instance, suppose one
operator represents two alternating decisions of a human (the
anthropological part)—let’s say observe a (Plato was a
genius) or observe b (Plato was not a genius)—and
another one (the instrumental part) represents outcome a
(Plato was a genius) or outcome b (Plato was not
a genius). These two operators interact in such a way that
alternating decisions and alternating outcomes mix up, entangle,
and evolve toward inseparable states. Such inseparable states
of combined operators can’t be factorized into the states of the
individual systems they emerged from or were a part of all the
way long.
ABGeneral =




p q
r s

pl
 pn
=
 rl
rn





l m
n o

ql
qn
sl
sn


qm
qo 

sm 
so

⊗

pm
po
rm
ro



The 4-dimensional matrix above shows a combined operator
representing decisions and decision outcomes in a general form.
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Information and Temporality

For instance, if A is represented as an operator in 2-dimensional
vector space with two decisions a = (1, 0) and b = (0, 1) and
B is represented as an operator in 2-dimensional vector space
with two decision outcomes a = (1, 0) and b = (0, 1), then
the following 4-dimensional operator represents the state of a
combined system that is separable.

ABSeparable =



1
0

0
1



⊗



1
0

0
1





1
 0
=
 0
0

0
1
0
0

0
0
1
0


0
0 

0 
1

However, the combined state space of decisions A and outcome
alternatives B (ABGeneral ) embraces states which are not separable
into the operators of the individual spaces. Take the following
example:

ABInseparable =

?






⊗

1

 0

? =
0
0

0
0
0
0

0
0
0
0


0
0 

0 
−1

pl = 1 and so = −1 and po = 0 and sl = 0. If pl = 1, then p 6= 0.
If po = 0 and p 6= 0, then o = 0. But so = −1 and therefore
o 6= 0. It is no surprise that some states in AB are inseparable with
regard to A and B. This is a purely structural consideration. It
accounts for the fact that there are (higher) combined states and
properties which are not reducible to (lower) individual states
and properties. ABInseparable is causally not reducible to operators
of the individual systems as it is the case for ABSeparable . Using
the words of a correspondence theorist, the separation between
the proposition that Plato was either a genius or not (A) and
its verification or falsification by referring to Plato as a putative
genius (B) is not tenable anymore.
So far, from a dynamic point of view, there is nothing
said about how such an inseparable state came up in the
first place. How did A and B interact over time in order to
end up in ABInseparable ? Perhaps ABInseparable is presupposed
all the way long? In several previous contributions it was
argued that inseparability is an indicator for phenomena where
presuppositions are at work [2, 14–17]. We are always already
situated in the world and many skills are not propositional
in nature. For instance, we notice that it is raining not by
formulating a proposition and verifying this proposition by
observation. When we are walking on the street, the sky is cloudy,
the air is wet, and our skin is sensing water drops, we understand
that it is raining. Phenomenologically there is no experience
of—and therefore no empirical evidence for—a matchmaking
(adaequatio) between an intellectual understanding of what it
is like to walk in the rain (intellectus) and the ontic fact that
it is raining outside (rei). Nevertheless, in many situations we
separate an idea or proposition from its referential object. This
ontic perspective facilitates a separation between cause and effect.
However, it is not primordial.
It turns out that ABSeparable is an ontic case of ABInseparable .
The former requires an attitude to describe things, in this
case decisions and decision outcomes, as existing independent

Frontiers in Physics | www.frontiersin.org

of observation. Here techne is a revealing that hides itself. A
leveling down or crossing over (letheia) annihilates context
and reveals propositions or ideas and their referential objects
stripped off their situatedness in the world. This is the positivist
viewpoint in epistemology and realism in ontology both of
which are ontic views in a presence culture [2]. A presence
culture acknowledges that decisions are always already situated
within decision situations; they are always already connected
with potential outcomes determined with the actual decision
made. In order to acknowledge causality and thus things in
sequential time, i.e., cause and effect as separate entities, the
ontic viewpoint separates or disentangles inseparable states if
certain structural aspects hold. In the next subsection it will
be demonstrated that these structural aspects bind together
exponents of exponential functions to describe things as separate
entities and things in sequential time. Here there are no absolute
zero points and no derivatives with respect to time. The only
derivative is temporality itself where things appear as being open
to others and even to everyone else.

3.2. Acausality in Time
In a presence culture things are given as available or readyto-hand within a horizon [2]. Things are open and accessible
to others and even to everyone else [15]. They don’t stand in
isolated opposition like the proposition about Plato stands over
and against the historical person Plato, or a glass of wine stands
in direct opposition to a bottle on its right hand side. A piece of
paper and the symbols written on it are real in the sense of being
open to be read by any other reader. While reading letters, words
and sentences on the paper, however, the meaning of a text shows
up as an unbroken reading experience. Therefore, it doesn’t show
up as independent of me reading them, but as meaningful in
alignment with my background knowledge. The meaning of the
text is present within a chronotope spanning across the gap or
separation between me—the reader—and the symbols on the
paper. I do not count time steps while enjoying a poem. Reading
and grasping a poem come with their rhythm and tone, perhaps
even their smell. But temporality is not structured like a causal
chain of arguments where a premise is clearly antecedent to a
conclusion. Time is not spatially located on a horizontal line
with points indicating what was before and what will be after.
Temporality embraces sequential time separated into discrete
steps or continuous events. However, it will be shown that this
is a special case.
A continuous time line can be read from the exponential
function6 . It describes growth and decay in space without
absolute zero points. Its derivative is the function itself. There
is no absolute beginning and no absolute end. The exponential
function is transcendental in the sense of inexhaustible (Euler’s
number is an inexhaustible number). There is no absolute
benchmark for discrete time steps and therefore there is no
absolute causal relation where an antecedent event causes a
subordinate event. Time is acausal, or better, the time of
6 From

a temporal point of view, the dynamics of combined quantum systems
are prescribed by evolution equations, which, in their general form, consist of
exponentiations [13].

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Information and Temporality

acausality is temporality. Temporality temporalizes or befalls
itself (cf. Section 4).
Acausality spans or broadens the present. Presence is
broadening [7]. In a presence culture the future is increasingly
inaccessible and the past increasingly difficult to let behind. In a
meaning culture7 going far back in time is equally transcendental
and inaccessible as future predictions. The very far past and
the far future remain highly speculative, difficult to reproduce,
and impossible to anticipate. Therefore, they are not definite
or determinate. One way to cope with this uncertainty is to
admit that the past and the future are simply inexhaustible or
infinite8 . Transcendentalism embraces uncertainty and openness
to interpretation. It eludes certainty. Causality gives certainty.
However, it is a special case; an ontic viewpoint that establishes
a clear antecedent event and a clear subordinate event, i.e.,
sequential time. It modifies primordial time (temporality) when
exponents P, S, Q, and R relate to each other in such a way that
inseparable states of combined spaces evolve toward separability
as the result of a forgetting, leveling down, or crossing over
(letheia).
 Pt
  Rt

e
0
e
0
InTime
ABSeparable =
⊗
0 eQt
0 eSt
 (P+R)t

e
0
0
0
(P+S)t
 0
e
0
0 

=
 0
0 e(Q+R)t
0 
0

0

0

e(Q+S)t

ABInTime is separable if P + S = Q + R at each instant of time. P,
S, Q, and R are linear operators and can be thought of as a matrix
counterpart of a real number. That P + S = Q + R must not hold
in general and is rather a special ontic case.




ABInTime
Inseparable = ? ⊗ ?

 (P+R)t
e
0
0
0
 0
e(P+S)t
0
0 

=
(Q+R)t
 0
0 e
0 
0
0
0
e(Q+S)t
ABInTime is inseparable if P + S 6= Q + R at each instant
of time. ABInTime
Inseparable is primordial. Up to this date, ontological
emergence of mental causation from material causal laws
has been witnessed nowhere [18]. There are no ontological
InTime
causes leading from ABInTime
Separable to ABInseparable . Therefore, it is
reasonable to assume that the former is a special ontic case of the
latter. If structural aspects hold together exponents distributed
among both individual spaces at each instant of time, a leveling
down or crossing over (letheia) of primordial time (temporality)
separates A and B and provides the condition of the possibility for
experiencing vulgar time9 as a succession of present moments (cf.
7 In a meaning culture the meaning of concepts (e.g., privacy) stands for or
represents something (e.g., a right). Meaning is attributed, predicaments are made.
In contrast a presence culture takes linguistic expressions as a medium that
overcomes the separation between subject and object, mind and matter, physics
and metaphysics.
8 Primordial time (temporality) is finite and the boundary, end or frontier of this
finiteness is authentic future or indeterminacy (cf. Section 4.1).
9 The term “vulgar” is not meant to be a value judgment.

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Section 4). However, in primordial time subjective decision (A)
and objective outcome (B) are always already combined. Instead
of being an aggregate or a unity over time, A and B are combined
acausally and thus equiprimordially. The dynamic viewpoint of
A and B provides a higher degree of inseparability and therefore
a stronger evidence for acausality as each component of A refers
to a component of B at each single moment simultaneously and
thus equiprimordially [13].
At this point, an acausal means for revealing the essence of
information technology stands. ABInTime
Inseparable adds to the essence
of causality as presented in Section 2. Causa formalis and
causa materialis are too sides of the same coin. The formal
representation of an inseparable evolution is not supreme.
Quantum concepts can’t fully abstract from causa materialis.
Vice versa, body, matter and syntax alone are not sufficient for
causing form or even the essence of information technology.
Performer is the information engineer. He or she is the
anthropological component and causa efficiens as part of
an acausal means for revealing the essence of information
technology. Eventually, causa finalis is the essence itself. It
brings itself into unconcealment by resting and remaining-initself. This telos is not simply an end but an end-in-itself. It is
neither subjective (a preference, desire, or value) nor objective
(a common good or value) in the sense of opposed to a subject.
It comes into being out of temporality. Temporality will be
discussed in the next section in more detail. So far it was
introduced as the time of acausality.
Acausality is associated with synchronicity, a term introduced
by Jung and Pauli who searched for correlated events with no
causal link [19]. In physics such events are known under the
label of entanglement and activation at a distance. In the life and
psychological sciences, there are phenomena like social mirroring
or contagious yawning offering an acausal interpretation [15].
They can’t be proved or disproved by means of statistical
methods. Statistics may kill acausal events. Synchronistic events
or acausal means require a non-willing or releasement [2],
whereas correspondence tests enforce separability, a leveling
down or crossing over (letheia) of primordial time (temporality).
The next section introduces temporality as the time of
acausality and the essence of information technology. Primordial
time is neither a subjective stream of present moments in
the observer’s mind nor is it an objective though relative
flow of events in the external world. Temporality temporalizes
ecstatically and horizontally [3, 4]. Temporality is not a concept
or a category. It is not epistemological. As an existential and
thus more comprehensive and fundamental than a concept or
category temporality is ontological; it does not simply have
ontic properties. Rather it exhibits general essences. Datability,
significance, spannedness, and openness are general essences of
equiprimordial time (temporality).

4. TEMPORALITY AND INFORMATION
Science has a natural inclination to strive from epistemology to
ontology. It does not only want to know how we as scientists,
consumers, citizens, etc. come to know; it wants to know
how things really are. A statement as simple as “it is gold” is
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ontological. Being is at stake. Epistemology is concerned with
the ways we come to know that “it is gold,” e.g., by way of
understanding how sensory stimulation from golden surface
material changes as a function of movement. Cognition is at
stake.
In the introduction (cf. Section 1) information technology
was introduced in terms of causality. In the previous section
it was argued that besides traditional causes (instrumentalism
and anthropology) acausality adds to the essence of information
technology. Our implicit understanding of information is often
purely instrumental. We are less interested in what information
thematically is—its essence—and more concerned about its
utility. Information is for processing, education, entertainment,
notification, reporting, etc. Semiotics agrees with such an
instrumental view. There is a pragmatic aspect to information
besides syntax and semantics. Syntax is simply the physical form
of information. Think of the linear operators in Section 3.1 with
components 1 and 0. On a semantic level operators and their
components have a meaning. Operators represent two alternating
decisions or observations: Plato was a genius (1, 0)
or Plato was not a genius (0, 1) and two alternating
outcomes: Plato was a genius (1, 0) or Plato was
not a genius (0, 1). This information turns pragmatic once
it is used to explain acausality.
However, there is more to information than its
meaningfulness and usefulness. Meaning attributions are
arbitrary. Conventionalized meaning, however, often conceals
arbitrariness. Attributing a trait of Plato’s intellect to 1 and 0 is
arbitrary. Certainly genius is not reducible to bits. Acausality
reveals this non-reducible character of traits and information
in general. In many situations of circumspect taking care and
skillful coping we do not attribute meaning and usefulness to
physical forms. Rather meaning and usefulness are made present
[2]. Information encountered shows itself as what it is in a
meaningful and pragmatic way. Semantics and pragmatics are
not something extra to syntax. They are to be found and made
present within the physical form itself. Unless a conspicuous
encounter with information makes me wonder what it really
means—for instance, I may find Chinese letters underneath
a painting without the slightest understanding of Chinese
language—I do not start grappling with meaning and pragmatics
in an explicit and thematic way.
In summary, the essence of information technology is far more
than a (causal) processing of information on different layers of
abstraction (syntax, semantics, and pragmatics). Information is
temporal. Temporality is the time of acausality. This time is
not chronological. Causality requires chronological time. Cause
and effect are separate entities in time. Effect comes after cause
and, vice versa, cause is prior to effect. Chronological time
and causality derive from temporality. They are released by an
awaiting that retains.

4.1. Making Present, Awaiting, and
Retaining
What makes QM particularly apt for modeling and
understanding decision making and other cognitive phenomena

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Information and Temporality

is indeterminacy10 . In contrast to classical uncertainty,
indeterminacy in QM does not presuppose a particle—its
properties like angular momentum or position—to be predetermined though not yet known. Like in the Plato example in
Section 3.1 states and properties represented as vectors or linear
operators may be superposed. Before a decision is made about
Plato’s intellect two even mutually exclusive options constitute
one state of potentiality11 . In quantum physics properties like
position may be superposed and have contradicting values or
values violating the law of total probability. A wave function
is distributed or spread out though the particle it represents
can only exist at one discrete position. The probability that a
particle exists at one particular position A and not at another
position B may not add up to the total probability of 1. It looks
like that in a wave scenario a particle can be at several positions
simultaneously. Unless a measurement determines position
with the measurement made it is undetermined. Grasping this
outmost uncertainty is at the heart of temporality.
Temporality temporalizes out of an authentic future.
Authentic future is not something outstanding. It is not
something missing or lacking. There are no information deficits
in primordial time. For instance, if I want to buy a new car and
my savings already cover 3/4 of the full price, then 1/4 is still
outstanding and expected to add to my savings within the coming
months. My expectation of the remaining amount of money to
be saved is always already in foresight of the full price for the
car to be saved. Future savings are outstanding. Indeterminacy
and authentic future, however, are not outstanding because
uncertainty (position of a particle, Plato’s genius, etc.) is not
epistemological (due to a lack of knowledge) but ontological. In
the car savings case uncertainty is epistemological. I do not yet
know if the remaining amount will add up to my savings within
the coming months. However, I do know what the remaining
amount is: 1/4 of the full price. The full price is pre-determined
and my uncertainty is relative to it.
Authentic future is not chronological. But future
chronologically conceived is founded in indeterminacy or
authentic future. Savings of 3/4 of the full price of a car is prior
to savings of the full price I will or will not have in my account in
the future. For indeterminacy or authentic future there is no full
price. It is not the case that a full price is not known. It doesn’t
exist. Indeterminacy is an end or a future that is not outstanding.
It is always already given though most of time hidden, concealed,
or forgotten12 . Being-toward-indeterminacy is presupposed but
leveled down or crossed over when time is experienced as a
succession of present moments. Such a flow of events or stream
of experience finds its formal expression in separable entities
or ABInTime
Separable (cf. Section 3.1), a requirement for chronological
10 cf. Flender and Müller [16] for an application of QM to privacy decision making.
11 If

Plato really was a genius or not, is, of course, a matter of debate. It is not predetermined. Therefore, such historical examples lend themselves for illustrations
of effects as found in QM.
12 For Heidegger this outmost uncertainty is death or being-toward-the-end [3, 4].
He acknowledges that a common understanding of death is demise. I prefer not
to use the term death as a synonym for indeterminacy. The reason is that the
common or vulgar connotation of death as demise is most difficult to shake off,
a requirement for its transformation into authentic future.

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being. However, ABInTime
Inseparable is primordial. Temporalized
components refer to each other instantaneously, simultaneously,
or equiprimordially. Primordial time (temporality) is finite and
the boundary, end or frontier of this finiteness is indeterminacy.
Having-been, presence and authentic future are equiprimordial
in temporality. The present is released in an awaiting that
retains. Once P, Q, R, S relate to each other (P + S = Q + R)
equiprimordiality is modified in such a way that a succession
of present moments arises. The immediate future is constantly
anticipated and the immediate past is constantly slipping
away. A condition of the possibility for transcendentalism,
inexhaustibility or infinity is that the equiprimordial awaiting
that retains is annihilated or de-contextualized. The making
present of “now and now and now” is predominant; the awaiting
that retains fades into the background. A constant making
present is released without conceiving its origin in an awaiting
(authentic future) that retains (having-been). A stream of present
moments conceals horizontal ecstasies (awaiting, retaining,
and making present) of temporality. This concealment (letheia)
constitutes the modus operandi of everydayness.
As scientists, managers, consumers, citizens, etc. we are
always already in-the-world. This “always already” refers to
presuppositions which are not necessarily resolute, grasped or
conceived but leveled down or crossed over due to one’s being
within a common factual world. In everyday taking care—our
business as usual as scientists, managers, consumers, citizens,
etc.—we observe, manage, consume and participate as “one” does
it. The time of the “one” is a making present that forgets. It forgets
an awaiting that retains as the condition of the possibility for
its release. A good example is taking care of time itself as one
coordinates one’s behavior with other people.
Suppose you have booked a one week meditation retreat
together with a friend. In the evening of the first day of your stay
you make an appointment for the next day. You agree with your
friend on having a first meditative exercise at sunrise. Both of you
and possibly most of the population on earth know what a sunrise
is. In our shared and common world the sun as a natural clock is
always already discovered. Before sundials as well as mechanized,
electrified and digitized clockwork were invented, the sun was a
thing encountered at hand ready to be used. In circumventive
taking care it was used as a natural pointer to sunrise, noon and
sunset according to which everyday activities were coordinated.
The next morning you and your friend wake up at sunrise. Both
of you look into the sky and you see the sun at the horizon.
“Now it is time to have a meditative exercise” is what both of
you understand and share publicly in measuring time with the
oldest clock on earth, the sun. Usually and most of the time we
take care of things and time itself as “one” does it. Implicitly and
unthematically we understand what time it is and what we have
to do. Although primordial time is leveled down or crossed over
we understand temporality temporalizing itself ecstatically and
horizontally. With every “now, that it is time to have a meditative
exercise” (sunrise), an “on that former occasion” (earlier when
the sun rose, yesterday, the days before, etc.) and a “then, when
the sun will have reached its peak or will set” (later on at noon
or sunset) are presupposed and equiprimordially understood
though not explicitly articulated.

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Information and Temporality

Saying “now it is time to have a meditative exercise” is a
discoursing articulation of a making present that temporalizes
itself in unity with an awaiting that retains [3, 4]. In measuring
time, the sun gets made public in such a way that it is encountered
for you, your friend and perhaps other practitioners joining you
as “now” and not later, earlier, tomorrow, or yesterday. Time is
a stream of present moments. Chronology, whether discrete or
continuous, requires a sense of what was before and what will
be after. The “now it is time to have a meditative exercise” is a
present moment within a flow of time, an inner duration or a
continuous time experience whereby the equipromordial making
present of an awaiting that retains is leveled down, crossed over,
or forgotten. This vulgar understanding of time levels down or
forgets the having-been and the awaiting and just reveals at
sunrise “now it is time to have a meditative exercise.”

4.2. Essences of Temporalized Information
Temporality is not a concept or a category. It is not ontic. As an
existential and thus more comprehensive and fundamental than a
concept or category temporality is ontological; it does not simply
have ontic properties. Rather it exhibits general essences13 .

4.2.1. Datability
Datability is a general essence of equiprimordial time
(temporality). In taking care of time itself (time measurement)
every making present or saying “now” is accompanied by a
“then, when” and “on that former occasion, when.” Every
ontic statement like “it is gold” implies a “now, it is gold” and,
equiprimordially, an awaiting (“then, it will still be gold”) and
a retaining (“on that former occasion, it already was gold”). An
athlete who is always already in the flow of what he is doing
(e.g., the running, jumping, or dribbling of a basketball player)
is making present by awaitingly retaining. In taking care of the
game he is within time as a succession of an immediate past (the
not-anymore), the present now, and an immediate future (the
not-yet). However, this flow is derivative or vulgar if datability is
hidden, concealed, crossed over, or leveled down. “If circumspect
taking care were simply a succession of experiences occurring in
time, and even if these experiences were associated with each other
as intimately as possible, letting a conspicuous, unusable tool be
encountered would be ontologically impossible” [3, 4].

4.2.2. Significance
Time is likewise derivative or vulgar if significance is nullified.
In average everydayness, if I wake up in the morning and have an
appointment at sunrise, I do not ponder or reason why I have this
appointment, what it is good for, or for the sake of which desire
or preference I made it. I just have it. Like datability significance
is crossed over or leveled down in circumventive taking care of a
situation. In primordial time, however, temporality temporalizes
“in-order-to” take care of a situation. Its significance tells that
it is time for what shows itself or is given, which may either be
appropriate or inappropriate. For instance, it is appropriate to
catch up with my friend for having our first meditative exercise
and it is inappropriate to go back to bed and have a couple of
hours extra sleep. For a basketball player it is appropriate to
13 An apt

German word for general essences is “Wesensmomente”.

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take a three-point shot when it’s time for taking the lead, or, it’s
inappropriate, when there is a bad defense in the zone depending
on the situation and his circumspect taking care of it.

4.2.3. Spannedness
Temporality broadens the presence. Authentic making present is
broadening14 [20, 21]. Temporal ecstasies broaden the presence
in the sense that they span across future, past, and present.
Making present, awaiting and retaining are equiprimordial
ecstasies. The present is released from an awaiting that retains.
This horizontal spannedness of future, past and present has its
primary moment in an anticipation of indeterminacy (authentic
future). Through temporality temporalizing itself ecstatically
and horizontally out of authentic future information comes
into existence. The meaning or significance of information
in general is temporality. To say that information is this
or that is to let-it-come-toward-itself (awaiting), let-it-beas-it-already-was (retaining), and let-it-be-encountered-as-it-is
(making present). To say that information “is” admits the
existence of information. Existence means being-ahead-ofitself. Being-ahead-of-itself temporalizes out of authentic future.
Indeterminacy shines into what it is not: information. This
spanning or broadening of temporal ecstasies finds it formal
expression in ABInTime
Inseparable (cf. Section 3.1). Here temporalized
components refer to each other instantaneously, simultaneously,
or equiprimordially. Past, present and future are equiprimordial
as long as P, Q, R, S do not relate to each other (P + S 6= Q + R).

4.2.4. Openness
Last but not least, primordial time is open to others and
even to everyone else. It is public or shared. Openness is
a condition of the possibility for coordinating our behavior
with others. Like things and others encountered in circumspect
taking care and scientific investigation we are always already
together with others no matter if they are physically present
or not. Time shows itself as open or public and thus it
is shared like things and others encountered in everyday
taking care. Being together with things and others encountered
is always already being-in-time. For instance, we share an
astronomical calendar and use it for coordinating our behaviors,
from the planning of our careers to weekly meetings. Perhaps
in the natural sciences and historiography such shared and
agreed upon conceptions of time are shaken more than in
any other realm of human life. Temporality is prior to any
specialized discipline and prior to any distinction. It is the
condition of the possibility for information to come into
being out of indeterminacy (authentic future) by temporalizing
itself. It is the time of acausality, a techne for revealing the
essence of information as that what remains and rests-initself.
14 Heidegger calls this making present “Moment” (Augenblick). See §65
(Temporality as the Ontological Meaning of Care) in [3, 4]. I prefer speaking of
broadening because it captures the other ecstasies (awaiting and retaining) more
elegantly.

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Information and Temporality

5. CONCLUSION
In his 1946 foreword of Brave New World, in retrospective,
Aldous Huxley speculates about how he would have had
rewritten his dystopian novel 15 years earlier [22]. He reasons
about a third alternative between an insane life in Utopia
where genetic engineering, brainwashing and recreational use
of drugs produce happy consumers who appear to be plugged
into a universal happiness machine and a lunatic world of
primitive people who resisted any economic and technological
progress.
“Religion would be the conscious and intelligent pursuit of
man’s Final End, the unitive knowledge of the immanent Tao or
Logos, the transcendent Godhead or Brahman. And the prevailing
philosophy of life would be a kind of High Utilitarianism, in
which the Greatest Happiness principle would be secondary to the
Final End principle—the first question to be asked and assured in
every contingency of life being: ‘How will this thought or action
contribute to, or interfere with, the achievement, by me and the
greatest possible number of other individuals, of man’s final end?”’
Huxley’s dream of a society composed of freely co-operating
individuals devoted to the pursuit of sanity has a Final End,
a causa finalis, in mind. Today, 70 years after he wrote his
foreword, science may be in a position to enter the middle way,
a third alternative between naive technological enthusiasm and
nostalgic, ultra-conservative, or even total rejection of progress.
Perhaps it is an irony of fate that science—the prestigious
and success-laden project of modernity and representative
of an enlightened, reasonable and secularized world—offers
reconciliation with the spiritual, enchanted, and numinous.
For a long time the presupposition of knowledge being
freed from value has been responsible for scientific progress.
Objective, third-personal, and context-free knowledge—the fruit
of science—is rid of subjective ends, motives, desires, interests,
and feelings, all of which can be subsumed as being valuable.
Value-free knowledge, however, is a fallacy. There is no
science without presuppositions. There is only science whose
foundations so far have remained unexamined. This is not to
deny that even a traditional (meta-) discipline like philosophy
presupposes conditions upon which its interpretations rest.
However, it makes a difference whether presuppositions
are simply taken for granted or if they are well-founded
by means of reasonable arguments and phenomenological
evidence.
Arguments and evidence employed in this contribution draw
from QM [11]. QM offers unequaled opportunities because it
raises foundational questions in a precise form. So far, applying
QM to phenomena and problems outside of physics has been
highly successful and, last but not least, its explanatory power
for concepts (i.e., existentials to be precise) as general and
specific as information and temporality has been substantiated
in this article. There is growing evidence that effects and laws
of QM also hold for macroscopic phenomena. However, far
more revolutionary is the fact that applying QM to cognition
is not equivalent but the same as altering and refining the
cognitive apparatus of the scientist as an acausal measuring
instrument.

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Now we live in an age with unprecedented possibilities for
extending our capacities and abilities to reveal. Information
technology and the Internet are extensions. They challenge us
and we challenge them. For a long time we thought about
technological apparatus being something purely instrumental,
i.e., causality as traditionally conceived (cf. Section 2). Galilei
was the first who employed an apparatus—a telescope—to verify
a scientific hypothesis. He wanted to observe and verify if
earth is indeed orbiting the sun. Today, our apparatus still
extends into the material world but acausality and temporality
as developed in this article challenge us to become part of
observation.
In this article I tried to be as objective and value-free as
possible. The method of doing so may appear unconventional.
Questioning the very nature of causes, means, ends and values—
the presumptive opposition of subjectivity and objectivity—
seems like a deconstruction. Perhaps it is a deconstruction
with the supposition of a value-free connotation—a branch of
consciousness studies—which bears the potential to bring science
and technology forward and guide us through the cognitive era
that just started.
Investigating the relationship of first-person experience and
third-person facts has been at the center of consciousness studies
for quite a while. Unfortunately, we are still in the dark when
it comes to give causally necessary and sufficient conditions for
consciousness to arise. We are able to package reasons and causal
chains of arguments into narratives explaining how causality,
means, ends and values may have evolved. However, explanations
after the fact still lack causally necessary and sufficient conditions
as desired for a full-blown materialist theory of consciousness.
For instance, retinal cells and the visual cortex may be necessary
for seeing shape and color. However, they will never be sufficient
for explaining visual consciousness. Repeatable and reproducible
observations of a particular constellation of firing cells in the

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Information and Temporality

visual cortex prior to or “simultaneous with” a visual experience
of a particular object is not sufficient evidence for visual
consciousness of that particular object (post hoc ergo propter hoc).
Non-materialist accounts are equally problematic as nobody can
convincingly deny having a body and living in a material world.
Abstaining from causal material explanations leaves at least
two other options for making sense of consciousness [23]. First,
including one or many intentional beings into narratives is
subject of various religious traditions, in particular monotheism,
e.g., the Judeo-Christian tradition, or polytheism of antiquity.
Second, there are (pantheist) approaches neither claiming
mechanistic laws and necessity underlying consciousness nor
referring to a higher being or eternal creator as the missing link
to the mystery.
A vision like Huxley’s third alternative supposes a final end:
a deity or end-in-itself. Such an approach abandons valuefree explanations in the traditional sense. It acknowledges
individualism and relativities of preferences, ends, and values.
However, relativities are not merely subjective as opposed to
objective. They are objective in the sense of information and
temporality as developed in this contribution.

AUTHOR CONTRIBUTIONS
The author confirms being the contributor of this work who
wrote and conducted what is laid-out and approved it for
publication.

ACKNOWLEDGMENTS
For their support the author would like to thank all individuals
and institutions who know they made an essential contribution
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11. Khrennikov A. Quantum-like modeling of cognition. Front Phys. (2015) 3:77.
doi: 10.3389/fphy.2015.00077
12. Rosenblum B, Kuttner F. Quantum Enigma: Physics Encounters Consciousness.
New York, NY: Oxford University Press (2011).
13. Kronz FM, Tiehen JT. Emergence and quantum mechanics. Philos Sci. (2002)
69:324–47. doi: 10.1086/341056
14. Flender C, Kitto K, Bruza P. Beyond ontology in information systems.
In: Bruza P, editor. Proceedings of the 3rd Quantum Interaction
Symposium, DFKI, No. 5494 in LNAI. Saarbrücken: Springer (2009).
p. 276–88.
15. Flender C, Kitto K, Bruza P. Nonseparability of shared intentionality.
In: Bruza P, editor. Proceedings of the 3rd Quantum Interaction
Symposium, DFKI, No. 5494 in LNAI. Saarbrücken: Springer (2009).
p. 211–24.
16. Flender C, Müller G. Type indeterminacy in privacy decisions: the privacy
paradox revisited. In: Busemeyer J, editor. Proceedings of the 6th International
Symposium on Quantum Interaction (QI 2012), No. 4620 in LNCS. Paris:
Springer (2012). p. 148–160.
17. Flender C. Quantum phenomenology and dynamic co-emergence. In: Song D,
editor. Proceedings of the 5th Quantum Interaction Symposium (QI 2011), No.
7052 in LNCS. Aberdeen: Springer (2011). p. 205–210.
18. Bitbol M. Ontology, matter and emergence. Phenomenol Cogn Sci. (2007)
6:293–307. doi: 10.1007/s11097-006-9041-z

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19. Atmanspacher H, Primas H. Recasting Reality - Wolfgang Pauli’s
Philosophical Ideas and Contemporary Science. Berlin; Heidelberg: Springer
(2009).
20. Gumbrecht HU. Unsere breite Gegenwart. Berlin: Suhrkamp (2010).
21. Gumbrecht HU. Our Broad Present: Time and Contemporary Culture.
New York, NY: Columbia University Press (2014).
22. Huxley A. Brave New World. London: Vintage Classics (2004).
23. Nagel T. Mind & Cosmos - Why the Materialist Neo-Darwinian Conception
of Nature is Almost Certainly False. New York, NY: Oxford University Press
(2012).

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Information and Temporality

Conflict of Interest Statement: The author declares that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2016 Flender. This is an open-access article distributed under the terms
of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) or licensor
are credited and that the original publication in this journal is cited, in accordance
with accepted academic practice. No use, distribution or reproduction is permitted
which does not comply with these terms.

September 2016 | Volume 4 | Article 40 | 46

ORIGINAL RESEARCH
published: 26 January 2017
doi: 10.3389/fphy.2016.00053

Toward a Quantum Theory of Humor
Liane Gabora 1* and Kirsty Kitto 2
1
Department of Psychology, University of British Columbia, Kelowna, BC, Canada, 2 Department of Mathematical Sciences,
Queensland University of Technology, Brisbane, QLD, Australia

This paper proposes that cognitive humor can be modeled using the mathematical
framework of quantum theory. We begin with brief overviews of both research on humor,
and the generalized quantum framework. We show how the bisociation of incongruous
frames or word meanings in jokes can be modeled as a linear superposition of a set
of basis states, or possible interpretations, in a complex Hilbert space. The choice of
possible interpretations depends on the context provided by the set-up vs. the punchline
of a joke. We apply the approach to a verbal pun, and consider how it might be extended
to frame blending. An initial study of that made use of the Law of Total Probability, involving
85 participant responses to 35 jokes (as well as variants), suggests that the Quantum
Theory of Humor (QTH) proposed here provides a viable new approach to modeling
humor.
Edited by:
Andrei Khrennikov,
Linnaeus University, Sweden
Reviewed by:
Haroldo Valentin Ribeiro,
Universidade Estadual de Maringá,
Brazil
Raimundo Nogueira Costa Filho,
Federal University of Ceará, Brazil
Irina Basieva,
Graduate School for the Creation of
New Photonics Industries, Russia
*Correspondence:
Liane Gabora
liane.gabora@ubc.ca
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 01 September 2016
Accepted: 21 December 2016
Published: 26 January 2017
Citation:
Gabora L and Kitto K (2017) Toward a
Quantum Theory of Humor.
Front. Phys. 4:53.
doi: 10.3389/fphy.2016.00053

Frontiers in Physics | www.frontiersin.org

Keywords: bisociation, context, humor, incongruity, law of total probability, pun, quantum cognition, quantum
interaction

1. INTRODUCTION
Humor has been called the “killer app” of language [1]; it showcases the speed, playfulness, and
flexibility of human cognition, and can instantaneously put people in a positive mood. For over a
100 years scholars have attempted to make sense of the seemingly nonsensical cognitive processes
that underlie humor. Despite considerable progress with respect to categorizing different forms of
humor (e.g., irony, jokes, cartoons, and slapstick) and understanding what people find funny, there
has been little investigation of the question: What kind of formal theory do we need to model the
cognitive representation of a joke as it is being understood?
This paper attempts to answer this question with a new model of humor that uses a
generalization of the quantum formalism. The last two decades have witnessed an explosion
of applications of quantum models to psychological phenomena that feature ambiguity and/or
contextuality [2–4]. Many psychological phenomena have been studied using quantum models,
including the combination of words and concepts [5–10], similarity and memory [11, 12],
information retrieval [13, 14], decision making and probability judgment errors [15–19], vision
[20, 21], sensation–perception [22], social science [23, 24], cultural evolution [25, 26], and creativity
[27, 28]. These quantum inspired approaches make no assumption that phenomena at the quantum
level affect the brain, but rather, draw solely on abstract formal structures that, as it happens,
found their first application in quantum mechanics. They utilize the structurally different nature
of quantum probability. While in classical probability theory events are drawn from a common
sample space, quantum models define states and variables with reference to a context, represented
using a basis in a Hilbert space. This results in phenomena such as interference, superposition and
entanglement, and ambiguity with respect to the outcome is resolved with a quantum measurement
and a collapse to a definite state.

January 2017 | Volume 4 | Article 53 | 47

Gabora and Kitto

This makes the quantum inspired approach an interesting
new candidate for a theory of humor. Humor often involves
ambiguity due to the presence of incongruous schemas: internally
coherent but mutually incompatible ways of interpreting or
understanding a statement or situation. As a simple example,
consider the following pun:
“Time flies like an arrow. Fruit flies like a banana.”
This joke hangs on the ambiguity of the phrase FRUIT FLIES,
where the word FLIES can be either a verb or a noun. As a verb,
FLIES means “to travel through the air.” However, as a noun,
FRUIT FLIES are “insects that eat fruit.” Quantum formalisms
are highly useful for describing cognitive states that entail this
form of ambiguity. This paper will propose that the quantum
approach enables us to naturally represent the process of “getting
a joke.”
We start by providing a brief overview of the relevant research
on humor.

2. BRIEF BACKGROUND IN HUMOR
RESEARCH
Even within psychology, humor is approached from multiple
directions. Social psychologists investigate the role of humor in
establishing, maintaining, and disrupting social cohesion and
social status, developmental psychologists investigate how the
ability to understand and generate humor changes over a lifetime,
and health psychologists investigate possible therapeutic aspects
of humor. This paper deals solely with the cognitive aspect of
humor. Much cognitive theorizing about humor assumes that it
is driven by the simultaneous perception [29, 30] or “bisociation”
[31] of incongruent schemas. Schemas can be either static frames,
as in a cartoon, or dynamically unfolding scripts, as in a pun.
For example, in the “time flies” joke above, interpreting the
phrase FRUIT FLIES as referring to the insect is incompatible
with interpreting it as food traveling through the air. Incongruity
is generally accompanied by the violation of expectations and
feelings of surprise. While earlier approaches posited that humor
comprehension involves the resolution of incongruous frames
or scripts [32, 33], the notion of resolution often plays a
minor role in contemporary theories, which tend to view the
punchline as activating multiple schemas simultaneously and
thereby underscoring ambiguity (e.g., 34, 35).
There are computational models of humor detection and
understanding (e.g., 36), in which the interpretation of an
ambiguous word or phrase changes as new surrounding
contextual information is parsed. For example, in the “time flies”
joke, this kind of model would shift from interpreting FLIES as
a verb to interpreting it as a noun. There are also computational
models of humor that generate jokes through lexical replacement;
for example, by replacing a “taboo” word with a similar-sounding
innocent word (e.g., [37, 38]). These computational approaches
to humor are interesting, and occasionally generate jokes that
are laugh-worthy. However, while they tell us something about
humor, we claim that they do not provide an accurate model of
the cognitive state of a human mind at the instant of perceiving

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Quantum Theory of Humor

a joke. As mentioned above, humor psychologists believe that
humor often involves not just shifting from one interpretation of
an ambiguous stimulus to another, but simultaneously holding
in mind the interpretation that was perceived to be relevant
during the set-up and the interpretation that is perceived to be
relevant during the punchline. For this reason, we turned to
the generalized quantum formalism as a possible approach for
modeling the cognitive state of holding two schemas in mind
simultaneously.

3. BRIEF BACKGROUND IN GENERALIZED
QUANTUM MODELING
Classical probability describes events by considering subsets of
a common sample space [39]. That is, considering a set of
elementary events, we find that some event e occurred with
probability pe . Classical probability arises due to a lack of
knowledge on the part of the modeler. The act of measurement
merely reveals an existing state of affairs; it does not interfere with
the results.
In contrast, quantum models use variables and spaces that
are defined with respect to a particular context (although this is
often done implicitly). Thus, in specifying that an electron has
spin “up” or “down,” we are referring to experimental scenarios
(e.g., Stern-Gerlach arrangements and polarizers) that denote the
context in which a measurement occurred. This is an important
subtlety, as many experiments have shown that it is impossible
to attribute a pre-existing reality to the state that is measured;
measurement necessarily involves an interaction between a state
and the context in which it is measured, and this is traditionally
modeled in quantum theory using the notion of projection. The
state |9i representing some aspect of interest in our system is
written as a linear superposition of a set of basis states {|φi i} in
a Hilbert space, denoted H, which allows us to define notions
such as distance and inner product. In creating this superposition
we weight each basis state with an amplitude term, denoted ai ,
which is a complex number representing the contribution of
aPcomponent basis state |φi i to the state |9i. Hence |9i =
i ai |φi i. The square of the absolute value of the amplitude
equals the probability that the state changes to that particular
basis state upon measurement. This non-unitary change of state
is called collapse. The choice of basis states is determined by the
observable, Ô, to be measured, and its possible outcomes oi . The
basis states corresponding to an observable are referred to as
eigenstates. Observables are represented by self-adjoint operators
on the Hilbert space. Upon measurement, the state of the entity is
projected onto one of the eigenstates.
It is also possible to describe combinations of two entities
within this framework, and to learn about how they might
influence one another, or not. Consider two entities A and B with
Hilbert spaces HA and HB . We may define a basis |iiA for HA
and a basis |jiB for HB , and denote the amplitudes associated
with the first as aA
i and the amplitudes associated with the second
as aBj . The Hilbert space in which a composite of these entities
exists is given by the tensor product HA ⊗ HB . The most general
state in HA ⊗ HB has the form
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Gabora and Kitto

Quantum Theory of Humor

|9iAB =

X

i,j

aij |iiA ⊗ |jiB

(1)

B
This state is separable if aij = aA
i aj . It is inseparable, and
A
therefore an entangled state, if aij 6= ai aBj .
In some applications, the procedure for describing
entanglement is more complicated than what is described
here. For example, it has been argued that the quantum field
theory procedure, which uses Fock space to describe multiple
entities, gives a kind of internal structure that is superior to the
tensor product for modeling concept combination [5]. Fock
space is the direct sum of tensor products of Hilbert spaces, so
it is also a Hilbert space. For simplicity, this initial application
of the quantum formalsm to modeling humor will omit such
refinements, but such a move may become necessary in further
developments of the model.
Quantum models can be useful for describing situations
involving potentiality, in which change of state is
nondeterministic and contextual. The concept of potentiality
has broad implications across the sciences; for example, every
biological trait not only has direct implications for existing
phenotypic properties such as fitness, but both enables and
constrains potential future evolutionary changes for a given
species. The quantum approach been used to model the
biological phenomenon of exaptation—wherein a trait that
originally evolved for one purpose is co-opted for another
(possibly after some modification) [40]. The term exaptation
was coined by Gould and Vrba [41] to denote what Darwin
referred to as preadaptation1 . Exaptation occurs when selective
pressure causes this potentiality to be exploited. Like other kinds
of evolutionary change, exaptation is observed across all levels of
biological organization, i.e., at the level of genes, tissue, organs,
limbs, and behavior. Quantum models have also been used to
model the cultural analog of exaptation, wherein an idea that was
originally developed to solve one problem is applied to a different
problem [40]. For example, consider the invention of the tire
swing. It came into existence when someone re-conceived of a
tire as an object that could form the part of a swing that one sits
on. This re-purposing of an object designed for one use for use
in another context is referred to as cultural exaptation. Much
as the current structural and material properties of an organ or
appendage constrain possible re-uses of it, the current structural
and material properties of a cultural artifact (or language, or
art form, etc.) constrain possible re-uses of it. We suggest that
incongruity humor constitutes another form of exaptation;
an ambiguous word, phrase, or situation, that was initially
interpreted one way is revealed to have a second, incongruous
interpretation. Thus, it is perhaps unsurprising that, as with
other forms of exaptation, a quantum model is explored.

4. A QUANTUM INSPIRED MODEL OF
HUMOR
A quantum theory of humor (QTH) could potentially inherit
several core concepts from previous cognitive theories of humor
1 The terms exaptation, preadaptation and co-option are often used interchangeably.

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while providing a unified underlying model. Considering the past
work discussed in Section 2, it seems reasonable to focus on the
notion that cognitive humor involves an ambiguity brought on by
the bisociation of internally consistent but mutually incongruous
schemas. Thus, cognitive humor appears to arise from the
double think that is brought about by being forced to reconsider
some currently held interpretation of a joke in light of new
information: a frame shift. Such an insight opens humor upto
quantum-like models, as a frame shift of an ambiguous concept
is well modeled by the notion of a quantum superposition
described using two sets of incompatible basis states within some
underlying Hilbert space structure.
In what follows we sketch out a preliminary quantum inspired
model of humor and discuss what would be required for a
full-fledged formal QTH. Next, we outline a study aimed at
discovering whether humor behaves in a quantum-like manner.
The last section discusses how the QTH opens up avenues for
future investigation in a field that to date has not been well
modeled.

4.1. The Mathematical Structure of QTH
We start our journey toward a QTH by building upon an existing
model of conceptual combination [8]: the State–COntext–
Property (SCOP) model. As per the standard approach used
in most quantum-like models of cognition, |9i represents the
state of an ambiguous element, be it a word, phrase, object,
or something else, and its different possible interpretations are
represented by basis states. Core to the SCOP model is a
treatment of the context in which every measurement of a state
occurred, and the resultant property that was measured. These
three variables are stored as a triple in a lattice.

4.1.1. The State Space
Following Aerts and Gabora [6], the set of all possible
interpretation states for the ambiguous element of a joke is given
by a state space 6. Specific interpretations of a joke are denoted
by |pi, |qi, |ri, · · · ∈ 6 which form a basis in a complex Hilbert
space H. Before the ambiguous element of the joke is resolved,
it is in a state of potentiality, represented by a superposition
state of all possible interpretations. Each of these represents a
possible understanding arising due to activation of a schema
associated with a particular interpretation of an ambiguous word
or situation. The interpretations that are most likely are most
heavily weighted. The amplitude term associated with each basis
state represented by a complex number coefficient ai gives a
measure of how likely an interpretation is given the current
contextual information available to the listener. We assume that
all basis states have
P unit length, are mutually orthogonal, and are
complete, thus i |ai |2 = 1.

4.1.2. The Context
In the context of a traditional verbal joke, the context consists
primarily of the setup, and the setup is the only contextual
element considered in the study in Section 5. However,
it should be kept in mind that several other contextual
factors not considered in our analysis can affect perceived
funniness. Prominent amongst these is the delivery; the way

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Gabora and Kitto

Quantum Theory of Humor

in which a joke is delivered can be everything when it
comes to whether or not it is deemed funny. Other factors
include the surroundings, the person delivering the joke, the
power relationships among different members of the audience,
and so forth.
As a first step, we might represent the set of possible contexts
for a given joke as ci ∈ C . Each possible interpretation of a
joke comes with a set fi ∈ F of properties (i.e., features or
attributes), which may be weighted according to their relevance
with respect to this contextual information. The weight (or
renormalized applicability) of a certain property given a specific
interpretation |pi in a specific context ci ∈ C is given by ν. For
example, ν(p, f1 ) is the weight of feature fi for state |pi, which is
determined by a function from the set 6 × F to the interval [0, 1].
We write:
ν : 6 × F → [0, 1]

(2)

(p, fi ) 7→ ν(p, fi ).

4.1.3. Transition Probabilities
A second function µ describes the transition probability from
one state to another under the influence of a particular context.
For example, µ(q, e, p) is the probability that state |pi under the
influence of context ci changes to state |qi. Mathematically, µ is
a function from the set 6 × C × 6 to the interval [0, 1], where
µ(q, e, p) is the probability that state |pi under the influence of
context |ei changes to state |qi. We write:
µ : 6 × C × 6 → [0, 1]

(3)

(q, e, p) 7→ µ(q, e, p).

Thus, a first step toward a full quantum model of humor consists
of the 3-tuple (6, C , F ), and the functions ν and µ. Next we
address a key question that should be asked of any cognitive
theory of humor: what is the underlying cognitive model of the
funniness of a joke?

4.2. The Humor of a Joke
As the listener hears a joke, more context is provided, and
in our model the listener’s understanding evolves according to
the transition probabilities associated with the cognitive state
and the emerging context. When the listener hear the joke
a bisociation of meaning is percieved; that is, the listener
realizes that a second way of interpreting it is possible.
A projective measurement onto a funniness frame is the
mechanism that we use to model the likelihood that a given joke is
considered funny.
Thus, in our model, funniness plays the role of a measurement
operator, and it is affected by the shift that occurs in the
understanding of a joke with respect to two possible framings:
one created by the setup, and one by the punchline. The
probability of a joke being regarded as funny or not is
proportional to the projection of the individual’s understanding
of the joke (|9i) onto a basis representing funniness. This means
that the probability of a joke being considered as funny, pF is
given by a projection onto the |1i axis in HF2 , a two-dimensional
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Hilbert sub-space where |0i represents “not funny” and |1i
represents “funny.”
pF = ||1ih1|9i|2

(4)

Similarly, the probability of a joke being regarded as not funny is
represented by
pF̄ = ||0ih0|9i|2 .

(5)

Note that |9i evolves as the initial conceptualization of the joke is
reinterpreted with respect to the frame of the punchline. This is a
difficult process to model, and we consider the work in this paper
to be an early first step toward an eventually more comprehensive
theory of humor that includes predictive models.
We now present two examples in which specific instances
of humor are considered within the perspective of this basic
quantum inspired model. First the approach is applied to a pun.
Second, the approach is applied to a cartoon that is a frame blend.
Both scenarios will help to deepen our understanding of the
significant complexity of humor, and the difficulties associated
with creating a mathematical model of this important human
phenomenon.

4.3. Applying QTH to a Pun
Consider the pun: “Why was 6 afraid of 7? Because 789.” The
humor of this pun hinges on the fact that the pronunciation
of the number EIGHT, a noun, is identical to that of the verb
ATE. We refer to this ambiguous word, with its two possible
meanings, as EYT. An individual’s interpretation of the word
EYT is represented by |9i, a vector of length equal to 1. This
is a linear superposition of basis states in the semantic sub2 which represents possible states (meanings) of the
space HM
word EYT: EIGHT or ATE2 . The interpretation of EYT as a
noun, and specifically the number EIGHT, is denoted by the
unit vector |ni. The verb interpretation, ATE, is denoted by the
2 . Thus,
unit vector |vi. The set {|ni, |vi} forms a basis in HM
we have now expanded our original two-dimensional funniness
space with an additional two-dimensional semantic space, where
2 . We note that these two spaces
the full space H4 = HF2 ⊗ HM
should not be considered as mutually orthogonal, but that they
will overlap. If they were orthogonal then the funniness of a
joke would be independent of the interpretation that a person
attributes to it.
With this added mathematical structure, we can represent the
2
interpretation of the joke as a superposition state in HM
|9i = an |ni + av |vi,

(6)

where an and av are amplitudes which, when squared, represent
the probability of a listener interpreting the joke in a noun or
a verb form (|ni and |vi) respectively. This state is depicted in
Figure 1A, which shows a superposition state in the semantic
space. When given no context in the form of the actual
presentation of the joke, these amplitudes represent the prior
2 We acknowledge that other interpretations are possible, and so this is a simplified

model. It is straightforward to extend the model into higher dimensions by adding
further interpretations as basis states.

January 2017 | Volume 4 | Article 53 | 50

Gabora and Kitto

Quantum Theory of Humor

FIGURE 1 | The humor of a joke can be explained as arising from a measurement process that occurs with respect to two incompatible frames. Using
the example of the pun, (A) the meaning of the set-up is reinterpreted with EYT updating toward the interpretations ATE. (B) Funniness is then treated as a
measurement, with the probability of funniness being judged with respect to a projection on the {|0i, |1i} basis. In this case there is a large probability of the joke being
considered funny due to the dominant component of the projection of |9i lying on the |1i axis. (C) The cognitive state of the subject then collapses to the observed
state (i.e., funny or not).

likelihood of a listener interpreting the uncontextualized word
(i.e., EYT) in either of the noun or verb senses (e.g., a free
association probability; see [12] for a review). However, we would
expect to see these probabilities evolving throughout the course
of the pun as more and more context is provided (in the form
of additional sentence structure). Throughout the course of the
joke, the state vector |9i therefore evolves to a new position
in H4 .
Since the set-up of the joke,“Why was 6 afraid of 7?,” contains
two numbers, it is likely that the numbers interpretation of
EYT is activated (a situation represented in Figure 1A). The
listener is biased toward an interpretation of EYT in this sense,
and so we would expect that an >> av . However, a careful
listener will feel confused upon considering this set-up because
we do not think of numbers as beings that experience fear.
This keeps the interpretation of EYT shifted away from an
equivalence with the eigenvector |ni. As the joke unfolds, the
predator interpretation that was hinted at in the set-up by the
word “afraid,” and reinforced by “789,” activates a more definite
alternative meaning, ATE, represented by |vi. This generates an
alternative interpretation of the punchline: that the number 7
ate the number 9. The cognitive state |9i has evolved to a new
position in H4 , a scenario that is represented in Figure 1B. At
this point a measurement occurs: the individual either considers
the joke as funny or not within the context represented by the
funniness sub space HF2 , and a collapse to the relevant funniness
basis state occurs (see Figure 1C). Note that this final state still
2 —the
contains a superposition within the meaning subspace HM
funniness judgment merely shifts the interpretation of the joke, it
does not eliminate the bisociation. Rather, it depends upon it.
If we consider the set of properties associated with EYT
then we would expect to see two very different prototypical
characteristics associated with each interpretation. For example,

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the EIGHT interpretation is difficult to map into properties such
as “food” denoted f1 , and “not living” denoted f2 (since when
something is eaten it is usually no longer alive). Because “food”
and “not living” are not properties of EIGHT, ν(p, f0 ) << ν(n, f0 ),
and similarly ν(p, f1 ) << ν(n, f1 ). However, “food” and “not
living” are properties of EYT, ν(p, f0 ) << ν(v, f0 ), and similarly
ν(p, f1 ) << ν(v, f1 ).
We can now start to construct a model of humor that could
be correlated with data. If jokes satisfy the law of total probability
(LTP) then their funniness should satisfy the distributive axiom,
which states that the total probability of some observable should
be equal to the sum of the probabilities of it under sets of more
specific conditions. Thus, considering a funniness observable ÔF
(with eigenstates {|1i, |0i} and the semantic observable ÔM (with
a simplified two eigenstate structure {|Mi, |M̄i} representing two
possible meanings that could be attributed to the joke). We can
take the spectral decomposition of ÔM = m|MihM| + m̄|M̄ihM̄|,
where m, m̄ are eigenvalues of the two eigenstates {|Mi, |M̄i}.
Doing this, we should find that if this system satisfies the LTP
then the probability of the joke being judged as funny is equal to
the sum of the probability of it being judged funny given either
semantic interpretation
p(F) = p(|1i) = p(M) · p(F|M) + p(M̄) · p(F|M̄).

(7)

We can manipulate the interpretation that a participant is likely
to attribute to a joke by changing the semantics of the joke itself.
Thus, changing the joke should change the semantics, and so
affect the humor that is attributed to the joke. We shall return
to this idea in Section 5.
This section has demonstrated that a formal approach
to concept interactions that has been previously shown to
be consistent with human data [5] can be adapted to

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Gabora and Kitto

the simultaneous perception of incongruous meanings of an
ambiguous word or phrase in the understanding of a pun.

4.4. Applying QTH to a Frame Blend
Although our first example used a pun for simplicity, we believe
that quantum inspired models may also be useful for more
elaborate forms of humor, such as jokes and cartoons referred
to as frame blends. A frame blend involves the merging of
incongruous frames of reference [42]. A common example of a
frame blend is a cartoon in which animals are engaged in some
kind of human behavior (such as a cartoon of a cow with all
her teats pierced saying “Just gotta be me”). In a frame blend
rather than being led “down the garden path” by the setup
and subsequent re-interpretation in light of the punchline, the
humor results from the simultaneous presentation of seemingly
incompatible frames. Using QTH, the two interpretations of the
incongruous situation would be designated by the unit vectors
{|di, |oi}. The cognitive state of perceiving the blended frames
is represented as a superposition of the two frames. As with
phenomena such as conceptual combination, there are likely to
be constraints on how frames can be successfully blended, and it
will be necessary to consider this when constructing models of
frame blends. We reserve further exploration of this interesting
class of humor for future work.

5. PROBING THE STATE SPACE OF
HUMOR
Returning to the question raised by Equation (7), a QTH should
be justified by considering whether humor does indeed violate
the Law of Total Probability (LTP) [3]. However, the complexity
of language makes it difficult to test how humor might violate
the LTP using a method similar to those followed for decision
making [11]. Past work on humor is unlikely to yield the data
required to perform tests such as this. For example, we currently
have no experimental understanding of how the semantics of a
joke interplays with its perceived funniness. It seems reasonable
to suppose that the two are related, but how? We are not aware
of any data that provide a way to evaluate this relationship. This
is problematic, as there are a number of interdependencies in the
framing of a joke that make it difficult to construct a model (even
before considering factors such as the context in which the joke
is made, and the socio-cultural background of the teller and the
listener). In this section we present results from an exploratory
study designed to start unpacking whether humor should indeed
be considered within the framework of quantum cognition. As an
illustrative example, consider the following joke:
VO : “Time flies like an arrow. Fruit flies like a
banana.”
As with the joke discussed in Section 4.3, the humor arises from
the ambiguity of the words FRUIT and FLIES. The first frame (F1,
the set-up), leads one to interpret FLIES as a verb and LIKE as a
preposition, but the second frame (F2, the punchline), leads one
to interpret FRUIT FLIES as a noun and LIKE as a verb. A QTH
must be able to explain how the funniness of the joke depends

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Quantum Theory of Humor

upon a shift in the semantic understanding of the two frames, F1
and F2.
We now outline a preliminary study that has helped us to
explore the state space of humor.

5.1. Stimuli
We collected a set of 35 jokes and for each joke we developed a
set of joke variants. A VS variant consisted of the set-up only for
the original, VO . Thus, the VS variant of the VO joke is
VS : “Time flies like an arrow.”
A VP variant consists of the original punchline only. Thus, the VP
variant of the VO joke is
VP : “Fruit flies like a banana.”
We then considered the notion of a congruent punchline as one
that does not introduce a new interpretation or context for an
ambiguous element of the set-up (or punchline). Congruence was
achieved by modifying the set-up to make it congruent with the
punchline, or by modifying the punchline to make it congruent
with the set-up. Thus, if the original set-up makes use of a noun,
then so does a congruent modification (and similarly for the
punchline).
A CP variant consists of the original set-up followed a
congruent version of the punchline. Thus, a CP variant of the O
joke is:
CP: “Time flies like an arrow; time flies like a bird.”
A CS variant consists of the original punchline preceded by a
congruent version of the set-up. Thus, a CS variant of the O’
joke is
CS: “Horses like carrots; fruit flies like a banana.”
For some jokes we had a fifth kind of variant. A IS variant consists
of the original set-up followed an incongruent version of the
punchline that we believed was comparable in funniness to the
original. Thus, considering the joke discussed in Section 4.3:
O: “Why was 6 afraid of 7? Because 789.”
A IS variant of this joke is:
IS: “Why was 6 afraid of 7? Because 7 was a six
offender.”
Thus the stimuli consisted of a questionnaire containing original
jokes, and the above variants presented in randomized order. The
complete collection of jokes and their variants is presented in the
Appendix (Supplementary Material).

5.2. Participants
The participants in this study were 85 first year undergraduate
students enrolled in an introductory psychology course at
the University of British Columbia (Okanagan campus). They
received partial course credit for their participation.

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Quantum Theory of Humor

5.3. Procedure
Participants signed up for the study using the SONA recruitment
system, and subsequently responded at their convenience to
an online questionnaire hosted by FluidSurveys. They were
informed that the study was completely voluntary, and that they
were free to withdraw at any point in time. They were also
informed that the researcher would not have any knowledge of
who participated in the study, and that their participation would
not affect their standing in the psychology class or relationship
with the university. Participants were told that the purpose of
the study was to investigate humor, and to help contribute to a
better understanding the cognitive process of “getting” a joke.
Participants were asked to fill out consent forms. If they agreed
to participate, they were provided a questionnaire consisting of a
series of jokes and joke variants (as described above) and asked
to rate the funniness of each using a Likert scale, from 1 (not
funny) to 5 (hilarious). The questionnaire took approximately
25 min to complete. They received partial course credit for their
participation.

5.4. Results
The mean funniness ratings across all participants for the entire
collection of jokes and their variants (as well as the jokes and
variants themselves) is provided in the Appendix (Supplementary
Material). Table 1 provides a summary of this information (the
mean funniness rating of each kind of joke variant across
all participants) aggregated across all joke sets. As expected,
the original joke (O) was funniest (mean funniness = 2.70),
followed by those jokes that had been intentionally modified to
be funny: Incongruent Setup (IS) (mean funniness = 2.37) and
Incongruent Punchline (IP) (mean funniness = 2.12). Next in
funniness were the jokes that had been modified to eradicate
the incongruency and thus the source of the humor: Congruent
Setup (CS) (mean funniness = 1.41) and Congruent Punchline
(CP) (mean funniness = 1.47). The joke fragments without a
counterpart–i.e., either Setup (S) or Punchline (P) alone–were
considered least funny of all (the mean funniness of both was
1.22). The dataset is entirely consistent with the view that the
humor derives from incongruence due to bisociation.

5.5. Toward a Test of the QTH
Recall that the Law of Total Probability (LTP) as represented
by Equation (7) suggests that the mean funniness of a joke
should be equal to the sum of its funniness as judged under
all possible semantic interpretations. This is not an equality
that we can directly test given our current understanding of
language and how it might interplay with humor. However, the

TABLE 1 | The mean funniness ratings across all participants and all joke
sets for each kind of joke variant.
Joke variant

O

Mean funniness

2.70

S

P

CS

CP

IS

IP

1.22

1.22

1.41

1.47

2.37

2.12

O, Original; S, Set-up only; P, Punchline only; CS, Congruent Set-up; CP, Congruent
Punchline; IS, Incongruent Set-up; IP, Incongruent Punchline.

Frontiers in Physics | www.frontiersin.org

dataset reported here gives us some initial ways to address this.
With a methodology for converting the Likert scale ratings into
projective measurements of a joke being funny or not, we can
start to consider the relative frequency that an original joke is
judged as funny and compare this result with the individual
components.
We start by translating the Likert scale responses into a
simplified measurement of funniness, by mapping the funniness
ratings into a designation of funny or not. In order to run a quick
comparison between the relative frequencies that participants
decided the full joke (VO ) was funny when compared to the
components of the joke (VS and VP ), we took the mean value
of the components for each subject. Given that puns are not
generally considered particularly funny (a result backed up by
our participant ratings) we used a fairly low threshold value of
2.5 (i.e., if the mean was less than 2.5 then the components
were judged as unfunny, and vice versa). Exploring the results
of this mapping gives us the data reported in Figure 2 for the
VO , VS and VP variants of the jokes, listing the frequency at
which participants judged the joke and subcomponents funny. A
mean value for the joke fragments is also presented. All data uses
confidence intervals at the 95% level.
We see a significant discrepancy between the funniness of the
original and the combined funniness of its components. This is
not a terribly surprising result; jokes are not funny when the
set-up is not followed by the punchline, and participants usually
rated VS and VP variants as unfunny (i.e., scoring them at 1).
Table 2 in the Appendix (Supplementary Material) shows that in
the participant pool of 85, the set-up and punchline variants of
the joke rarely had a mean funniness rating above 1.5. However,
to extract a violation of the LTP for this scenario, we would need
to construct expressions such as the following
p(F) = p(EIGHT).p(F|EIGHT) + p(ATE).p(F|ATE).

(8)

How precisely could such a relationship be tested? Two forms
of data are required to test whether the simple puns used in our
experiment actually violate the LTP:
1. Funniness ratings: These are the probabilities regarding
the probability that different components of the joke are
considered funny (the whole joke (p(F)); just the setup
(p(F|EIGHT)); and just the punchline (p(F|ATE)); and
2. Semantic probabilities: These list the probability of EYT
being interpreted as EIGHT: p(EIGHT), or ATE: p(ATE),
within the context of the specific joke fragment.
We have demonstrated a method for extracting the funniness
ratings above. How might we obtain data for the semantic
probabilities? We must consider the precise interpretation of
what these probabilities might actually be. Firstly, we note that it
seems likely participants will interpret just a set-up or a punchline
in the sense that the fragment represents. The bisociation that
humor relies upon is not present for a fragment, and so a person
hearing a fragment will be primed by its surrounding context
toward interpreting an ambiguous word in precisely the sense
intended for that fragment. Indeed, the incongruity that results
from having to readjust the interpretation of the joke, and the

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Gabora and Kitto

Quantum Theory of Humor

FIGURE 2 | A comparison of the frequency with which a specific joke and its fragments are considered funny for participants in the pilot trial (using a
threshold value of 2.5, n = 85). A mean of the set-up and the punchline variants (VS and VP ) is also given. Confidence intervals are set at 95%.

resulting bisociation, lies at the very base of the humor that arises.
Free association probabilities will not give these values. To test
the LTP, it would be necessary to extract information about how
a participant is interpreting core terms in the joke as it progresses;
some form of nondestructive measurement is required, and a new
experimental protocol will have to be defined. We reserve this for
future work.
However, the significant difference between the rated
funniness of the fragments and that of the original joke allows
us to formulate an alternative mechanism for testing equations
of the form (7) and (8). We can do this by asking whether there
is any way in which the semantic probabilities could have values
that would satisfice the LTP? An examination of Figure 2 for the
setup and punchline variants of the jokes suggests that there is
no way in which to chose semantic probabilities that will satisfy
the LTP. Thus, we have preliminary evidence that humor should
perhaps be treated using a quantum inspired model.

6. DISCUSSION
It would appear that there is some support for the hypothesis
that the humor arising from bisociation can be modeled by
a quantum inspired approach. Furthermore, the experimental
results presented in section 5 suggest that this model might more
appropriate than one grounded in classical probability. However,
much work remains to be completed before we can consider these
findings anything but preliminary.
Firstly, the model presented in Section 4 is simple, and
will need to be extended. While an extension to more senses
for an ambiguous element of a joke is straightforward with a
move to higher dimensions, the model is currently not well
suited to the set of variants discussed in Section 5.3. A model
that can show how they interrelate, and how their underlying
semantics affects the perceived humor in a joke is desirable.
Furthermore, the funniness of the joke was simplistically
represented by a projection onto the “funny”/“not funny”
axis. A more theoretically grounded treatment of the Likert
data is desirable. For example, the current threshold value

Frontiers in Physics | www.frontiersin.org

of 2.5 was chosen somewhat arbitrarily [although could be
justified by a consideration of the mean values for funniness
scores reported in the Appendix (Supplementary Material)—see
Table 2]. A more systematic way of considering the Likert scale
measures to allow for a normalization of funniness ratings at
the level of an individual is also desirable. As a highly subjective
phenomenon, funniness is liable to be judged by different
individuals inconsistently and so it will be important that we
control for this effect in comparing Likert responses among
individuals.
Considering experimental results, the sample size of the data
set is somewhat small (85 participants), although our funniness
ratings appear to be reasonably stable for this cohort. A more
concerning problem revolves around the construction of a LTP
relationship for our simple model. There are many alternative
ways in which a LTP could be constructed for puns, and
more sophisticated models need to be investigated before we
can be confident that our results conclusively demonstrate that
humor must be modeled using a quantum inspired approach.
In particular, we require a more sophisticated method that
facilitates the extraction of data about the semantics attributed
by a participant to a joke. A two stage protocol may be
the answer for obtaining the necessary semantic information
for a more rigorously founded test of the violation of LTP.
It would be useful to construct a systematic study of the
manner in which adjusting the congruence of the set-ups and
punchlines influences perception of the joke. The quantum
inspired semantic space approaches of Van Rijsbergen [13]
and Widdows [43] may prove fruitful in this regard, as
they would facilitate the creation of similarity models such
as those explored by Aerts et al. [44] and Pothos and
Trueblood [45].
In summary, humor is complex, and it will take an ongoing
program of research to understand the interplay between the
semantics of a joke and its perceived funniness. However, at
this point we might pause to consider the broader question of
why humor might be better modeled by a quantum inspired
approach than by one grounded in classical probability? To

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Quantum Theory of Humor

this end we return to the discussion of Section 3. As we saw,
the humor of a pun involves the bisociation of incongruent
frames, i.e., re-viewing a setup frame in light of new contextual
information provided by a punchline frame. Moreover, the
broader contextuality of humor means that even the funniest
of jokes can become markedly unfunny if delivered in the
wrong way (e.g., a monotone voice), or in the wrong situation
(e.g., after receiving very bad news). Funniness is not a preexisting “element of reality” that can be measured; it emerges
from an interaction between the underlying nature of the
joke, the cognitive state of the listener, and other social and
environmental factors. This makes the quantum formalism an
excellent candidate for modeling humor, as this interaction is
well described by the concept of a vector state embedded in
a space which is represented using basis states that can be
reoriented according to the framing of the joke. However, this
paper only provides a preliminary indication that a QTH may
indeed provide a good theoretical underpinning for this complex
process. Much more work remains to be done.

is reason to suppose that a quantum inspired model is indeed
appropriate.
Our QTH is not proposed as an all-encompassing theory of
humor; for example, it cannot explain why laughter is contagious,
or why children tease each other, or why people might find
it funny when someone is hit in the face with a pie (and
laugh even if they know it will happen in advance). It aims to
model the cognitive aspect of humor only. Moreover, despite the
intuitive appeal of the approach, it is still rudimentary, and more
research is needed to determine to what extent it is consistent
with empirical data. Nevertheless, we believe that the approach
promises an exciting step toward a formal theory of humor. It is
hoped that future research will build upon this modest beginning.

7. CONCLUSIONS

AUTHOR CONTRIBUTIONS

This paper has provided a first step toward a quantum theory
humor (QTH). We constructed a model where frame blends
are represented in a Hilbert space spanned by two sets of basis
states, one representing the ambiguous framing of a joke, and
the other representing funniness. The process of “getting a
joke” then consists of a dual stage scenario, where the cognitive
state of a person evolves toward a re-interpretation of the
meaning attributed to the joke, followed by a measurement of
funniness. We conducted a study in which participants rated
the funniness of jokes as well as the funniness of variants of
those jokes consisting of setting or punchline by alone. The
results demonstrate that the funniness of the jokes is significantly
greater than that of their components, which is not particularly
surprising, but does show that there is something cognitive taking
place above and beyond the information content delivered in
the joke. A preliminary test to see whether the humor in a joke
violates the law of total probability appears to suggest that there

LG had the idea for the paper and designed and conducted the
study. Both authors contributed equally to all other aspects of the
research and the writing of the paper.

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ETHICS STATEMENT
This research was approved by the Behavioral Research Ethics
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ACKNOWLEDGMENTS
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Gabora and Kitto. This is an open-access article distributed
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distribution or reproduction in other forums is permitted, provided the original
author(s) or licensor are credited and that the original publication in this journal
is cited, in accordance with accepted academic practice. No use, distribution or
reproduction is permitted which does not comply with these terms.

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REVIEW
published: 22 September 2015
doi: 10.3389/fphy.2015.00077

Quantum-like modeling of cognition
Andrei Khrennikov *
Department of Mathematics, Linnaeus University, Växjö, Sweden

Edited by:
Wei-Xing Zhou,
East China University of Science and
Technology, China
Reviewed by:
Zhi-Qiang Jiang,
East China University of Science and
Technology, China
Qing Yun Wang,
Beihang University, China
*Correspondence:
Andrei Khrennikov,
Department of Mathematics,
Linnaeus University, PJ vägen 1,
Växjö S-35195, Sweden
andrei.Khrennikov@lnu.se
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 16 July 2015
Accepted: 28 August 2015
Published: 22 September 2015
Citation:
Khrennikov A (2015) Quantum-like
modeling of cognition.
Front. Phys. 3:77.
doi: 10.3389/fphy.2015.00077

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This paper begins with a historical review of the mutual influence of physics and
psychology, from Freud’s invention of psychic energy inspired by von Boltzmann’
thermodynamics to the enrichment quantum physics gained from the side of psychology
by the notion of complementarity (the invention of Niels Bohr who was inspired by William
James), besides we consider the resonance of the correspondence between Wolfgang
Pauli and Carl Jung in both physics and psychology. Then we turn to the problem of
development of mathematical models for laws of thought starting with Boolean logic
and progressing toward foundations of classical probability theory. Interestingly, the laws
of classical logic and probability are routinely violated not only by quantum statistical
phenomena but by cognitive phenomena as well. This is yet another common feature
between quantum physics and psychology. In particular, cognitive data can exhibit a kind
of the probabilistic interference effect. This similarity with quantum physics convinced a
multi-disciplinary group of scientists (physicists, psychologists, economists, sociologists)
to apply the mathematical apparatus of quantum mechanics to modeling of cognition.
We illustrate this activity by considering a few concrete phenomena: the order and
disjunction effects, recognition of ambiguous figures, categorization-decision making.
In Appendix 1 of Supplementary Material we briefly present essentials of theory of
contextual probability and a method of representations of contextual probabilities by
complex probability amplitudes (solution of the “inverse Born’s problem”) based on a
quantum-like representation algorithm (QLRA).
Keywords: quantum-like models, cognition and psychology, two slit experiment, order and disjunction effects

1. Introduction
Recently, scientists working in various disciplines (physicists, psychologists, economists,
sociologists) started to apply the mathematical apparatus of quantum mechanics (QM), [1, 2]
especially quantum probability calculus [3] (based on Born’s rule), to multi-disciplinary problems
[4–36]. Some physicists regard such an activity as totally “illegal.” They argue that the mathematical
apparatus of QM was designed specifically for description of particular physical phenomena and it
cannot be used in, e.g., psychology. Why? Some elaborate that the apparatus of QM is relevant
to micro phenomena only (though this viewpoint is debatable even in the quantum physics
community). One aim of this paper is to convince physicists, especially those working in the
quantum information theory and quantum probability, that applications of the methods of QM
to cognition can be justified. We show that the present sharp separation of subjects of physics and
psychology/cognition is only a peculiarity of the present moment, that 19th and the first part of
20th century were characterized by mutual influence of physical and psychological theories and
the fruitful exchange of ideas between the brightest representatives from both sides. One of the
best known examples is the impact made by psychology on QM which resulted in borrowing the

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principle of complementarity [37] by Niels Bohr from William
James’ book [38], see also books of Plotnitsky [39–41]. It
may be less known that, in turn, the idea of complementarity
was elaborated by James under the influence of the 19th
century studies in thermodynamics which led him (as well as
later Freud [42, 43]) to the notion of psychic energy; initially,
complementarity in psychology was about complementarity of
different representations of psychic energy [38].
Meanwhile, we point out that quantum-like modeling of
cognition considered here must be distinguished from theories
of physical quantum brain in the spirit of Hameroff [44], Penrose
[45, 46]. We work in the purely operational framework: it was
found that some experimental studies in cognitive psychology,
economics, and social science generate statistical data which
match well quantum description of measurements and the
corresponding probabilistic outputs (see e.g., [4, 5, 9, 13, 14,
18]). Therefore, it is natural to model cognition with the aid of
QM formalism. The quantum cognition project does not try to
explain the physiological origin of quantum rules for information
processing and probability, similarly to Copenhageners in QM
(following Bohr [37]). As in physics, this approach does not
exclude a possibility to go beyond the operational quantum
formalism. However, for the moment, there is no commonly
accepted “prequantum model of cognition,” cf., however, with
Khrennikov [47].
In this paper we also mark the turning points in the
development of mathematical models for laws of thought starting
with the book of Boole [48] and considering the foundations of
classical probability theory as established by Kolmogorov [49] in
1933.
Then, we briefly review the violations of the laws of classical
logic and probability in quantum statistical experiments, in
particular we discuss the probabilistic structure of the two slit
experiment [50] and adress no-go theorems [1, 51, 52] (von
Neumann, Kochen-Specker, Bell), see also [53]. We demonstrate
that such violations (including the interference effect) also occur
in statistics collected in cognitive experiments. This similarity
with effects in quantum physics convinced scientists from physics
and cognitive science to apply the mathematical apparatus of QM
to modeling of cognition. For illustration we use two concrete
applications [12–18]: the order and disjunction effects. The
paper is concluded with a short review of recent research in
quantum(-like) cognition, in particular, cognitive applications of
the theory of open quantum systems [23, 24, 30, 31] and positive
operator valued measures [4, 7, 36].
We remark that the use of the mathematical apparatus of
QM for problems of cognition is motivated not only by the
existence of non-classical statistical data collected in cognitive
psychology, but also by similarities of basic features of (1)
states of a system under study and (2) possible observations
performed on the system, in physics and cognition. First feature
concerns the representation of a state (e.g., a mental state) as a
superposition of other (basis) states. In quantum(-like) modeling
of cognition, superpositions play the crucial role because they
represent states of very deep uncertainty which can not be
modeled by classical probability distributions. Secondly, the
representation of incompatible quantum physical observables by

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Quantum-like modeling of cognition

non-commuting operators also corresponds well to psychological
intuition, since the majority of observables used in psychology,
in particular, in the theory of decision making, exhibit the
order effect. The property of entanglement of the states of two
(or more) different systems is crucial for most peculiar QM
effects (such as quantum teleportation and quantum computing).
Entanglement also plays an important role in cognitive studies
but as an exhibition of contextuality of cognitive phenomena (in
the spirit of Cabello [54]) rather than physical non-locality (see
also [53, 55–58]).
The problem of a proper interpretation of a quantum state
(represented by a wave function) is still one of the most intriguing
problems of quantum foundations [53]. The present situation is
characterized by a huge diversity of interpretations (which can
be considered as a sign of deep foundational crisis). Working
with applications of the QM formalism in new fields of science
one also meets this problem. In QM there are, roughly speaking,
two big classes of interpretations: (a) quantum state is a physical
state of an individual system; (b) quantum state is a special
(probabilistic) representation of information about the results of
possible measurements on an ensemble of (identically prepared)
systems. The first one can be called the physical interpretation
and the second one the information interpretation. Recently, the
latter became very popular in quantum information theory and
led (in its extreme forms) to subjective interpretation of quantum
states, including quantum Bayesianism of Fuchs [59–61] and
the information interpretation of Zeilinger [62, 63], Brukner
[64]. Such interpretations match the ideology of quantum(-like)
cognition. (Though, as we have seen in QM, the problem of
interpretation is very complex, and it would be too risky to try
to fix firmly the interpretation of quantum(-like) states used
in cognitive studies.) Meanwhile, there is one crucial difference
between conventional QM and quantum cognition. In QM, in
accordance with Bohr’s views, there is a system and an observer,
the latter considered as external with respect to the system.
This ideology, although working successfully in experimental
studies of micro-world phenomena, is problematical where the
possibility of separation between a system under observation
and the observer is questionable, e.g., in quantum cosmology.
Trying to solve this problem, the problem of interpretation of
the “wave function of the Universe,” Hugh Everett proposed
the many worlds interpretation of the wave function, probably
the most exotic among all interpretations. In fact, in quantum
cognition we meet the same problem. The brain is a self-observer;
here it is not easy to separate the system under measurement
from the observer. However, it seems that the information
interpretation in the spirit of Zeilinger-Brukner-Fuchs gives a
possibility to resolve it: in the brain, one information subsystem
makes predictions about the result of the observation on another
information subsystem. Still, the problem of interpretation of
the “mental wave function” is complicated. In this paper, we
do not keep to any fixed interpretation, while we are most
sympathetic to the information interpretation. At the same
time we are very cautious (maybe, too cautious) with respect
to the use of the many worlds interpretation for quantum
cognition, in spite of novel possibilities and yet unexplored
ways.

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FIGURE 2 | Sigmund Freud.

FIGURE 1 | Hermann von Helmholtz.

2. From Psychology to Physics and Back
Reviewing a variety of definitions from dictionaries and
encyclopedias, we believe that we can safely state the following.
Physics is the science that deals with the properties of matter.
Psychology is the science that deals with mental processes and
behavior. In accordance with the views of Rene Descartes there
are two basic types of substance, material and mental, and one
is not reduced to the other1 . Although during the last century
physical reductionism captured the headlines in psychology,
Descartes’ ideology still penetrates the body of modern science.
Naturally, physics and psychology are considered as different
fields of science as they can be, each with its specific theoretical
and experimental methodologies. It seems that there is nothing
or very little in common between them. Most physics students
would probably not like to spend their time studying psychology
courses and vice versa. However, developments in physics
and psychology are connected much stronger than one can
imagine. We can point to a few big names who contributed to
establishing a connection between the two most fundamental
sciences (one about nature and the other about psyche):
Hermann von Helmholtz (Figure 1), Sigmund Freud (Figure 2),
Gustav Theodor Fechner, William James (Figure 4), Niels Bohr
(Figure 3), Carl Jung, Wolfgang Pauli, Albert Einstein,....
Freud was strongly influenced by works of von Helmholtz on
thermodynamics and especially on the energy conservation law2 .
He noted similarities between thermodynamics and the human
psyche and developed a kind of mental thermodynamics known
as psycho-dynamics [42, 43]. Freud actively used the notion of
psychic energy (libido) and the law of its conservation. (Primarily
libido represents the sexual energy. However, according to
1 It is also a Buddhist

dogma that life is comprised of mind and matter.
course, when discussing this law we have to mention the works of Germain
Henri Hess, James Prescott Joule, and Rudolf Clausius. But, for Freud, the
influence of von Helmholtz’s ideas was especially strong. He started his research
in physiology under the supervision of Ernst Brucke who previously worked with
Hermann von Helmholtz.
2 Of

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FIGURE 3 | Niels Bohr.

Freud, the sexual energy is one of the forms of the psychic
energy which can be transformed into other forms.) At the
first stage of his psycho-dynamical studies Freud was influenced
by the ideas of Fechner: considering physical facts (related to
human body) and mental facts as sides of one reality. Fechner
concluded that both physical and mental phenomena has to
be described by the same mathematical apparatus [65]. (This
remark is very important for us as foretelling the main idea
of this paper: behavior of both mind and matter nicely fits
the framework of the mathematical formalism of quantum
theory).
The notion of psychic energy played an important role in
theorizing of James [38]. Following physicists (who at that time
were already using the field theory) he started to operate with the
notion of psychic field. This psychic field as well as a physical
field can have different modes. This analogy led James [38]
to the fundamental principle of complementary of information
belonging to different modes of consciousness (the words of
James are italicized):

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of nature in the ordinary space-time system, and in its place to set
up invisible fields of probability in multidimensional spaces.”

FIGURE 4 | William James.

“It must be admitted, therefore that in certain persons, at least, the
total possible consciousness may be split into parts which coexist
but mutually ignore each other, and share the object of knowledge
beteen them. More remarkable still, they are complementary. Give
an object to one of the consciousnesses, and by this very act you
remove it from the other or others. Barring a certain common
fund of information , like the command of language, etc., what
the upper self knows the under self is ignorant of, and vice
versa.”

Above we pointed to the “knowledge transfer” in one direction,
from physics to psychology. However, the opposite also took
place. In particular, the principle of complementarity was invented
in quantum physics by Bohr under the strong influence of James’
“Principles of Psychology” [38] (cf. the above citation with the
principle of complementarity in QM).
Now we point to the famous correspondence between Pauli
and Jung [66] on comparative analysis of foundations of physics
and psychology. These letters were written in a free style
of discussion between friends (and, in part, a patient and a
psychoanalyst)3 . This freedom allowed them to express (in
psychoanalytic manner) many thoughts which would be never
presented in formal scientific discussions and publications. From
the letters it is clear that Jung was deeply influenced by quantum
theory in Pauli’s presentation; e.g., Jung wrote to Pauli:
“As the phenomenal world is an aggregate of the processes of
atomic magnitude, it is naturally of the greatest importance to
find out whether, and if so how, the photons (shall we say)
enable us to gain a definite knowledge of the reality underlying
the mediative energy processes Light and matter both behave
like separate particles and also like waves. This ... obliged us to
abandon, on the plane of atomic magnitudes, a causal description
3 At the beginning Pauli wanted to discuss with Jung his psychical problems
which might be a subject of psychoanalytic treatment. However, Jung smartly
redirected Pauli to a young female psychoanalyst and the most part of Pauli-Jung
correspondence is about psyche-physics inter-relation.

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Inspired by acausal features of quantum mechanics, Jung
developed his famous theory of synchronicity [67]; the theory
about the experiences of two or more events as meaningfully
related, where they are unlikely to be causally related (The
subject sees it as a meaningful coincidence). The use by quantum
physicists of “invisible fields of probability on multidimensional
spaces” strongly supported Jung’s interest in psychic fields,
invisible, probabilistic, and defined not on the physical space time
R4 , but on some kind of “mental space,” cf. [68]. This was a
clue to unification of psychic and quantum physical fields in one
psycho-physical field. The idea was very appealing to both Pauli
and Jung and it was one of the topics of their correspondence.
Jung also discussed field models with Einstein, and Einstein’s
attempts to create a unified pure field model of physical reality
(see e.g., [69]), also supported Jung’s studies on psychical fields.
Finally, however, neither the Einstein dream about a purely field
description of physical reality nor the Jung-Pauli dream about
the unified (quantum) psycho-physical field found a rigorous
mathematical realization.
Our discussion on mutual influence of physics and psychology
can be shortly represented as the following (of course,
incomplete) diagram:
[Hess, Joule, Clausius, and von Helmholtz] → [Freud,
Fechner, James] → [Bohr] ↔ [Pauli] ↔ [Jung] ← [Einstein]...

3. Modeling of Cognition with
Classical-nonclassical Logic vs.
Classical-nonclassical Probability
Now we concentrate on problems in cognition (keeping
in mind our ultimate goal—the quantum modeling in
cognitive psychology). Recall that “cognition” usually treats
psychological functions of an indvidual from the viewpoint
of information processing. (Sometimes “cognition” is treated
more tendentiously as the “science of mind”). We shall use
mathematics as an instrument for linkage of cognition and
physics.

3.1. From Boolean Logic to Kolmogorovian
Probability
In 19th century George Boole wrote the book “An Investigation
of the Laws of Thought on Which are Founded the Mathematical
Theories of Logic and Probabilities” [48], see also [70]. This was
the first mathematical model of the thinking process based on the
laws of reasoning nowadays known as the Boolean logic. The role
of Boolean logic in modern science is impossible to overestimate,
it plays the crucial role in information theory, decision making,
artificial intelligence, digital electronics. Boolean logic is the basic
mathematical model of classical logic.
One of the most important features of Boolean logic is that
it serves as the basis of the modern probability theory [49]:
representation of events by sets, subsets of some set , the
so-called sample space, or space of elementary events. The

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system of sets representing events, say F , allows operations of
Boolean logics; F is the so-called σ -algebra of sets4 . It is closed
with respect to the (Boolean) operations of (countable) union,
intersection, and complement (or in logical terms “and,” “or,”
“no”). Thus, the first lesson for a physics student is that by
applying any theorem of probability theory, e.g., the law of large
numbers, one has to be aware that paradigm of Boolean logic is
being used. The set-theoretic model of probability was presented
by Kolmogorov in 1933 [49]; it is based on the following two
natural (from the Boolean viewpoint) axioms:
• (AK1) events are represented as elements of a σ -algebra and
operations on events are described by Boolean logic;
• (AK2) probability is represented as a probabilistic measure.
We remind that a probabilistic measure p is a (countably)
P∞
additive function on a σ -algebra F : p(∪∞
j = p(Aj )
j = Aj ) =
for Aj ∈ F , Ai ∩ Aj = ∅, i 6= j, which is valued in [0, 1]
and normalized by 1. We also recall the definition of a random
variable as a measurable function, a :  → R5 . In classical
probability theory random variables represent observables.
Thus, the second lesson for a physics student is that probability
is an axiomatic theory, as, e.g., geometry. (My experience of
probabilistic discussions with physicists is that only a few of them
understand this. Majority tries to treat probability heuristically,
e.g., as frequency. This approach may work well in applied
research, e.g., with experimental data. However, it may lead to
paradoxic conclusions in foundational studies, as e.g., in the case
of violation of Bell’s inequality, see [53], for details)6 .

3.2. Formula of Total Probability, Bayesian
Analysis
One of the basic laws of the Kolmogorovian model, the formula
of total probability (FTP), will play very important role in our
further considerations. Before addressing FTP, we point to an
exceptional role which is played by conditional probability in the
Kolmogorov model. This sort of probabilities is not derived in
any way from “usual probability”; conditional probability is per
definition given by the Bayes formula:
p(B|C) = p(B ∩ C)/p(C), p(C) > 0.

(1)

By Kolmogorov’s interpretation it is the probability of an event B
to occur under the condition that an event C has occurred. One can
immediately see that this formula is one of strongest exhibitions
of the Boolean structure of the model; one cannot even assign
conditional probability to an event without using the Boolean
operation of intersection.
4 Here the symbol σ encodes “countable.” In American terminology such systems
of subsets are called σ -fields.
5 Here measurability has the following meaning. The set of real numbers R is
endowed with the Borel σ -algebra B: the minimal σ -algebra containing all open
and closed intervals. Then for any A ∈ B its inverse image a−1 (A) ∈ F . This
gives a possibility to define on B the probability distribution of a random variable,
pa (A) = p(a−1 (A)).
6 I see a big problem in the absence of mathematically advanced courses in
probability theory for physics students. It seems that education in physics suffers
from this problem throughout the world.

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Let us consider a countable family of disjoint sets Ak belonging
to F such that their union is equal to  and p(Ak ) > 0, k = 1, ....
Such a family is called a partition of the space .
Theorem 1 Let {Ak } be a partition. Then, for every set B ∈ F , the
following formula of total probability holds
p(B) =

X

p(Ak )p(B|Ak )

(2)

k

Especially interesting for us is the case where a partition is
induced by a discrete random variable a taking values {αk }. Here,
Ak == {ω ∈  : a(ω) = αk }. Let b be another discrete random
variable. It takes values {βj }. For any βj , we have
p(b = βj ) =

X
k

p(a = αk )p(b = βj |a = αk ).

(3)

This formula plays a crucial role in classical decision theory:
knowing probabilities of the a-variable and the corresponding
conditional probabilities for the b-variable one can obtain the
“total probability” for any value of the latter. We also point
out that FTP is the cornerstone for the Bayesian procedure
for probability updating which is also widely used in decision
making.

3.3. Probability-geometry: Comparison of
Evolutions
To understand better the role of the axiomatic nature of the
modern set-theoretic model of probability it is useful to make
comparison with another axiomatic theory - geometry. We
can learn a lot from history of development of geometry. Of
course, the biggest name in geometry is Euclid. His axiomatics
of geometry was considered as the only possible for about 2000
years. It became so common that people started to identify
Euclidean model of geometry with physical space. In particular,
Immanuel Kant presented deep philosophic arguments [71] that
physical space is Euclidean. The Euclidean dogma was rejected
as the result of internal mathematical activity, the study of a
possibility of derivation of one of axioms from others. This axiom
was the famous fifth postulate: given a line and a point not on
the line, there is precisely one line parallel to the given one and
containing the given point. Nikolay Ivanovich Lobachevsky was
the first to understand that this postulate can be replaced with one
of its negations. This led him to a new geometric axiomatics, the
model which nowadays is known as Lobachevsky geometry (or
hyperbolic geometry). Thus, the Euclidean geometry started to be
treated as just one of possible models of geometry. This discovery
revolutionized, first, mathematics (with contributions of Gauss,
Bolyai, and especially Riemann) and then physics (Minkowski,
Einstein, Hilbert).
This geometry lesson tells us that there is no reason to
expect that the Kolmogorovian model is the only possible
axiomatic model of probability. One can expect that by modifying
the Kolmogorovian axioms in the same spirit as Lobachevsky
modified the Euclidean axiomes, mathematicians could create
non-Kolmogorovian models of probability which may be useful for
various applications, in particular in physics. However, in the case

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of probability the historical pathway of development of geometry
was not repeated. Mathematicians did not have 2000 years to
rethink the Kolmogorovian axiomatics...

3.4. Non-Kolmogorovian Nature of Quantum
Probability; No-go Theorems

3.5. Quantum Logic

New physics, QM, intervened brutally in the mathematical
kingdom. The probabilistic structure of QM did not match
classical probability theory based on the set-theoretic approach
of Kolmogorov. At the first stage of development of QM this
mismatching was not so visible. The first sign can be seen in
Born’s rule:
p(x) = |ψ(x)|2 ,

(4)

where ψ(x) is the wave function and p(x) is the probability
to detect a particle at point x. The wave function is
primary here, not the probability. What is encoded in these
complex amplitudes pre-existing behind probabilities obtained in
quantum measurements? One of the most evident consequences
of Equation (4) is violation of the formula of total probability
(FTP), one of the basic laws of classical probability theory, see
Section 4 for details. In the two slit experiment constructive
and destructive interference of the wave functions corresponding
to passing through different slits is probabilistically represented
as violation of FTP, so to say, interference of probabilities.
(Moreover, in QM only such interference of probabilities can
be observed, nothing closer to probability amplitudes, since
“quantum waves” are not directly approachable).
John von Neumann was the first to pay attention to the
peculiar probabilistic structure of QM as compared to the
probabilistic structure of classical statistical mechanics [1]. In
particular, he generalized Born’s rule to quantum observables
represented by Hermitian operators. For an observable
represented by an operator with purely discrete spectrum, the
probability to obtain the value λ as the result of measurement is
given as
p(λ) = kPλ ψk2 ,

(5)

where Pλ isPthe projector corresponding to the eigenvalue λ.
(Here A = λ λPλ ).
In his seminal book [1] von Neumann pointed out that,
opposite to classical statistical mechanics where randomness of
the results of measurements is a consequence of variability of
physical parameters such as, e.g., the position and momentum
of a classical particle, in QM the assumption about the existence
of such parameters (for the moment, probably, still hidden
and unapproachable by the existing measurement devices)
leads to a contradiction. This statement presented in Von
Neumann [1] is known as von Neumann no-go theorem, theorem
about impossibility to go beyond the description of quantum
phenomena based on quantum states: it is impossible to construct
a theoretical model providing a finer description of those
phenomena than given by QM7 . Thus, von Neumann was sure
that it is impossible to construct a classical probability measure
7 This theorem was criticized for unphysical assumptions used by von Neumann
to approach his no-go conclusion; especially strong critique was from the side of

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on the space of some hidden variables which would reproduce
probabilities obtained in quantum measurements. Later this
statement was confirmed by other no-go theorems, e.g., of
Kochen and Specker [51] and Bell [52].

These “theorems” are consequences of the mathematical
structure of QM. While classical probability theory is based
on the set-theoretical description, QM is founded on the
premise that events are associated with subspaces (or orthogonal
projectors on these subspaces) of a vector space, complex Hilbert
space. The adoption of subspaces as the basis for predicting
events also entails a new logic, the logic of subspaces (projectors),
which relaxes some of the axioms of classical Boolean logic (e.g.,
commutativity and distributivity).
First time this viewpoint that QM is based on a new type of
logic, quantum logic, was expressed in the book of Von Neumann
[1], where he treated projectors corresponding to the eigenvalues
of quantum observables (represented by Hermitian operators)
as propositions (see also [72]). The explicit formulation of logic
of QM as a special quantum logic is based on the lattice [73]
of all orthogonal projectors. For reader’s convenience, below
we present the mathematical structure of quantum logic (see
[74], for details). However, in principle one can jump directly to
Section 3.6.

3.5.1. Logical Operations on for Projectors
For an orthogonal projector P, we set HP = P(H), its image, and
vice versa, for subspace L of H, the corresponding orthogonal
projector is denoted by the symbol PL .
The set of orthogonal projectors is a lattice with the order
structure: P ≤ Q iff HP ⊂ HQ or equivalently, for any ψ ∈
H, hψ|Pψi ≤ hψ|Qψi.
We recall that the lattice of projectors is endowed with
operations “and” (∧) and “or” (∨). For two projectors P1 , P2 ,
the projector R = P1 ∧ P2 is defined as projector onto the
subspace HR = HP1 ∩ HP2 and the projector S = P1 ∨ P2 is
defined as projector onto the subspace HR defined as the minimal
linear subspace containing the set-theoretic union HP1 ∪ HP2 of
subspaces HP1 , HP2 : this is the space of all linear combinations of
vectors belonging these subspaces. The operation of negation is
defined as the orthogonal complement: P⊥ = {y ∈ H : hy|xi =
0 for all x ∈ HP }.
In the language of subspaces the operation “and” coincides
with the usual set-theoretic intersection, but the operations “or”
and “not” are non-trivial deformations of the corresponding
set-theoretic operations. It is natural to expect that such
deformations can induce deviations from classical Boolean logic.
Consider the following simple example. Let H be two
dimensional Hilbert space
with the orthonormal basis (e1 , e2 )
√
and let v = (e1 + e2 )/ 2. Then Pv ∧ Pe1 = 0 and Pv ∧ Pe2 = 0,
but Pv ∧ (Pe1 ∨ Pe2 ) = Pv . Hence, for quantum events, in general
the distributivity law is violated:
Bell [52], the author of another famous no-go theorem; calmer critical arguments
were presented by Ballentine [2]. (We also remark that, although in the modern
literature the von Neumann statement is called “theorem,” in the German edition
it was called an “ansatz”).

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P ∧ (P1 ∨ P2 ) 6= (P ∧ P1 ) ∨ (P ∧ P2 )

(6)

As can be seen from our example, even mutual orthogonality of
the events P1 and P2 does not help to save the Boolean laws8 .
We remark that for commuting projectors quantum logical
operations have the Boolean structure. Thus, non-commutativity
can be considered as algebraic representation of non-classicality
of quantum logic. In particular, for aPsingle observable
(with purely discrete spectrum) A =
λ λPλ , projectors
corresponding to different eigenvalues are orthogonal and, hence,
commutative. Therefore, deviations from classical logic and
probability can be found only through analysis of results of a few
incompatible measurements.
The idea that cognition and quantumness have something in
common has been discussed during last 80 years, starting with
the philosophic studies of Alfred North Whitehead.

3.6. Toward Quantum Modeling of Cognition
As we have seen, quantum logic relaxes some of the axioms
of classical Boolean logic, e.g., commutativity and distributivity.
Human judgments are not always commutative (order effects
are pervasive) and often violate the probabilistic implications
of the distributive axiom. The principles of QM resonate with
deeply rooted psychological intuitions and conceptions about
human cognition and decision. Therefore, it is natural to
try to use the mathematical apparatus, developed to describe
the aforementioned quantum deformations of Boolean logics,
to model cognition and, in particular, to apply quantum
measurement theory to model decision making. Also, the
mathematical apparatus of QM is actively applied to probabilistic
problems of psychology, cognitive science, social science,
economics, and finances (see e.g., the monographs [4–7]).
We remark that non-commutativity of incompatible
observables can be considered as the algebraic representation
of the principle of complementarity. Thus, the loop in the
inter-relation of physics and psychology was finally closed:
complementarity came back to psychology, but in the advanced
mathematical form.
We remark that in QM probabilities can only be expressed
through elements of quantum logics, see Equation (5). Thus,
non-classicality is a statistical effect. In the same way nonclassicality of human reasoning can be observed only as a
statistical effect. In fact, such an effect has been well known
in psychology for long, but it was interpreted as irrational
behavior of people which was statistically exhibited in the form
of various probability fallacies. Their role (both in psychology
and economics) was emphasized in the influential Tversky (over
30,000 citations), Kahneman (Nobel prize in economics) research
8 At first glance, representation of events by projectors/linear subspaces may look
exotic because of the very common use of the set-theoretic representation of
events in the modern classical probability theory. We want to fight this prejudice
and support the view that alternatives are possible and sometimes desirable. The
tradition to represent events by subsets was firmly established by Kolmogorov [49]
only in 1933. We remark that before him the basic classical probabilistic models
were not of the set-theoretic nature. For example, the main competitor of the
Kolmogorov model, the von Mises frequency model [75], was based on the notion
of a collective (see [76], for formulation of QM on the basis of the von Mises
model).

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program [77]: the conjunction and disjunction fallacies, order
effects in decisions, over- and under- extension errors in conceptual
combinations, and ambiguous concepts [78, 79].
In author’s works [4, 10] it was pointed out that violation
of FTP can serve as a statistical test of non-classicality of
data generated by both physical and cognitive phenomena. The
coefficient of interference expressed in the probabilistic terms, see
Equation (9) Section 4 can be interpreted as quantitative measure
of non-classicality (non-Kolmogorovness). These papers deal
with an important case of dichotomous observables of the
inverse Born problem: a complex probability amplitude ψ
is reconstructed with the aid of the interference coefficient,
see Appendix 1 in Supplementary Material for a detailed
presentation. This constructive wave function method is especially
important for cognitive applications. In QM the space geometry
is often used to construct the corresponding wave functions,
e.g., for a free particle with a fixed momentum p, ψ(x) = eixp ;
generally one can use the Schrödinger equation in R3 with a
potential V(x) and initial and boundary conditions. The main
problem of the quantum cognition project is that a proper notion
of mental space has not yet been elaborated (cf. [68]). We cannot
directly use physics methods, such as introducing functions (e.g.,
energy) on physical space. A possibility to construct a “mental
wave function” directly from data is properly justified. The author
designed an algorithm for inversion of Born’s rule, the so-called
Quantum-like representation algorithm (QLRA) [4], see Section 8
for a few applications.
Author’s article [10] served as the theoretical basis for a
series of experiments on contextual effect (of Gestalt type) in
recognition of ambiguous figures by Conte et al. [13, 14, 18],
see Section 8.1 for brief presentation of these results. Analysis of
obtained statistical data showed that classical FTP is violated and
that the “belief state” of students participated in the experiment
can be described by a complex amplitude ψ and observables by
non-commutative Hermitian operators.
Busemeyer et al. performed extended studies [5, 12, 16, 25–
27, 32–34], see also the monograph of Busemeyer and Bruza [5],
on violation of FTP for well-known data on probability fallacies
obtained in experiments by Shafir and Tversky, Hofstader,
Grosson and other cognitive psychologists [80–84]. It was shown
that such data can be modeled with the aid of the mathematical
formalism of QM [5]. Besides, Busemeyer et al. lauched the
project on quantum(-like) decision making; see also the pioneer
work of Aerts and Aerts [8], the paper of Phothos and Busemeyer
[16] and the series of works of Asano et al. [23, 24, 30, 31, 35].

4. Violation of Formula of Total Probability
and Non-Kolmogorov Probability Theory
The two slit experiment is the basic example demonstrating that
QM describes statistical properties of microscopic phenomena, to
which the classical probability theory seems to be not applicable
(see e.g., Feynman and Hibbs [50]). In this section, we consider
the experiment with the symmetric setting: the slits are located
symmetrically with respect to the source of photons, Figure 5.
Consider a pair of random variables a and b. We select a as the

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slit variable, i.e., a = 0 (the photon passes through the upper slit),
a = 1 (the photon passes through the lower slit), see Figure 5,
and b as the position on the photo-sensitive plate, see Figure 5.
Remark that the b-variable has the continuous range of values,
the position x on the photo-sensitive plate.
For the experimental context with both slits open, see
Figure 6, by Born’s rule Equation (4) the probability that a
photon is detected at position x on the photo-sensitive plate is
represented as

2
 1
2

1
1
p(b = x) =  √ ψ0 (x) + √ ψ1 (x) = ψ0 (x)
2
2
2


2 
1 
+ ψ1 (x) + ψ0 (x) ψ1 (x) cos θ,
2

(7)

0 and ψ1 are two wave functions, whose absolute values
whereψ
ψi (x)2 give the distributions of photons passing through the slit
i = 0, 1, respectively, see Figures 7, 8
The term




δ(x) = ψ0 (x) ψ1 (x) cos θ

represents quantitively the interference effect of two wave
2

functions. Let us denote ψi (x) by p(b = x|a = i), then
Equation (7) is represented as
p(b = x) = p(a = 0)p(b = x|a = 0) + p(a = 1)p(b = x|a = 1)
+δ(x),

where the “interference term” δ has the form:
p
2 p(a = 0)p(b = x|a = 0)p(a = 1)
δ(x) =
×p(b = x|a = 1) cos θ .

(9)

Here the values of probabilities p(a = 0) and p(a = 1) are equal
to 1/2, since we consider the symmetric setting. For a general
experimental setting, p(a = 0) and p(a = 1) can be taken as
the arbitrary non-negative values satisfying p(a = 0) + p(a =
1) = 1. In the above form, the classical probability law—FTP, see
Equation (3),
X
p(b = x) =
p(a = i)p(b = x|a = i),
(10)
i

is violated, and the interference term Equation (9) quantifies
the violation. The additional interference term appears not

FIGURE 5 | Two slit experiment.

FIGURE 7 | Context with only slit1 is open.

FIGURE 6 | Context with both slits are open.

FIGURE 8 | Context with only slit0 is open.

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(8)

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only in the two slit experiment, but in any experiment with
arbitrary incompatible quantum observables represented by noncommuting Hermitian operators A, B : [A, B] 6= 0 (see [53], for
details).
Now consider two random variables of any origin, from
physics, cognitive science, biology, sociology. Let FTP be violated.
Of course, for a classical probabilist this is impossible, but
P plenty
of such data exist, see Section 3.6. Here p(b = x) 6= i p(a =
i)p(b = x|a = i), i.e., a kind of (probabilistic) interference term
appears:
X
δ(x) = p(b = x) −
p(a = i)p(b = x|a = i),
(11)
i

The point is that we cannot use the Kolmogorov probability
model. For example, psychologists can look for special
psychological explanations of such strange data, e.g.,
altruism. However, such a psychological “resolution” does
not change the mathematical problem: how to describe such
data mathematically? The previous analysis of quantum
measurements of the interference type (more generally of pairs
of incompatible quantum observables) demonstrated that the
appearance of the interference type term matches the predictions
of quantum probability theory, where probabilities are based
on complex probability amplitudes. Therefore, it is natural to
use this non-classical probability theory to model phenomena
generating data with non-trivial interference terms which
violate FTP. This was one of the starting points for quantum
probability theory to impact mathematical modeling of cognition
[4, 5, 10, 12].
We remark that (Equation 11) can be (tautologically)
rewritten in the form similar to the formula for quantum
interference (Equation 8) and the interference term can be always
represented similarly: Equation (9):
p
δ(x) = 2λ(x) p(a = 0)p(b = x|a = 0)p(a = 1)p(b = x|a = 1).
(12)
The only difference is that for arbitrary data we cannot guarantee
that |λ(x)| ≤ 1. Thus, for arbitrary statistical data, we have FTP
with the interference term:
X
p(b = x) =
p(a = i)p(b = x|a = i)
i

p
+ 2λ(x) p(a = 0)p(b = x|a = 0)p(a = 1)p(b = x|a = 1).

(13)

5. Savage Sure Thing Principle, Disjunction
Effect
STP [85] If you prefer prospect B0 to prospect B1 if a possible
future event A happens, and you prefer prospect B0 still if future
event A does not happen, then you should prefer prospect B0
despite having no knowledge of whether or not event A will
happen.
Savage’s illustration refers to a person deciding whether or not
to buy a certain property shortly before a presidential election,
the outcome of which could radically affect the property market.

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“Seeing that he would buy in either event, he decides that he
should buy, even though he does not know which event will
obtain,” [85], p. 21.
The crucial point is that the decision maker is assumed
to be rational. Thus, the sure thing principle was used as
one of foundations of rational decision making and rationality
in general. It plays an important role in economics in the
framework of Savage’s utility theory. Mathematically Savage’s STP
is a simple consequence of FTP. Thus, this principle, widely
used in economics, is mathematically based on the classical
probability (and Boolean logic). In particular, the Bayes formula
for conditional probabilities (Equation 1) plays the crucial role.
Therefore, rationality determined by this principle is Bayesian
rationality.
Experimentally observed [80, 81] violations of STP were
interpreted by Shafir and Tversky as a new effect, the disjunction
effect (see also Hofstader [82, 83] and Croson [84]). STP was
also confronted by a number of famous (in cognitive psychology,
economics, and decision making) paradoxes, Ellsberg, Allais, and
Simpson paradoxes [6].
As was discovered by professor of cognitive psychology
Jerome Busemeyer, statistical exhibiting the disjunction effect
can be treated as non-classical, violating FTP, and hence
these data has to be described by some non-Kolmogorovian
probability model, e.g., quantum probability. Detailed analysis
of data collected in Shafir and Tversky [81] and Tversky
and Shafir [80] experiments as well as experiments of
other cognitive psychologists was performed by the author
Khrennikov [4]: FTP is violated; the corresponding quantum
representations were constructed. Below we consider the simplest
experiment.
In Section 8.3 we produce the quantum-like representation
for statistical data obtained in one of experiments on disjunction
effect which was performed by Tversky and Shafir [80]. By using
the constructive wave function approach and QLRA, see Section
3.6 and Appendix 1 Supplementary Material, we construct the
representation of data with the aid of a complex probability
amplitude, “belief state,” “mental wave function,” such that
experimental probabilities (frequencies) are given by the Born
rule.

6. The General Scheme of Representation
of Measurements in Quantum Physics and
Cognition
In this section we repeat the discussion [36] on similarity between
the schemes of representation of measurements in quantum
physics and cognition.
On a very general level, QM accounts for the probability
distributions of measurement results using two kinds of entities,
called observables A and states ψ (of the system on which the
measurements are made). Let us assume that measurements are
performed in a series of consecutive trials numbered 1, 2, . . .. In
each trial t the experimenter decides what measurement to make
(e.g., what question to ask), and this amounts to choosing an
observable A. Despite its name, the latter is not an observable per

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se, in the colloquial sense of the word. Still, it is associated with
a certain set of values, which are the possible results one can get
when measuring A. In a psychological experiment these are the
responses that a participant is allowed to give, such as Yes and
No.
The probabilities of these outcomes in trial t (conditioned on
all the previous measurements and their outcomes) are computed
as some function of the observable A and of the state ψ (t) in
which the system (a particle in quantum physics, or a participant
in psychology) is at the beginning of trial t,
p(A = v in trial t | measurements in trials 1, . . . , t − 1) =


F ψ (t) , A, v .
(14)
This measurement changes the state of the system, so that at the
end of trial t the state is ψ (t+1) , generally different from ψ (t) . The
change ψ (t) → ψ (t+1) depends on the observable A, the state
ψ (t) , and the value v = v (A) observed in trial t,


ψ (t+1) = G ψ (t) , A, v .

(15)

On this level of generality, a psychologist will easily recognize in
Equations (14, 15) a probabilistic version of the time-honored
Stimulus-Organism-Response (S-O-R) scheme for explaining
behavior [86]. This scheme involves stimuli (corresponding
to A), responses (corresponding to v), and internal states
(corresponding to ψ). It does not matter whether one simply
identifies A with a stimulus, or interprets A as a kind of
internal representation thereof, while interpreting the stimulus
itself as part of the measurement procedure (together with
the instructions and experimental set-up, that are usually
fixed for the entire sequence of trials). What is important is
that the stimulus determines the observable A uniquely, so
that if the same stimulus is presented in two different trials
t and t ′ , one can assume that A is the same in both of
them.
QM is characterized by linear representation of observables—
by Hermitian operators; pure states are represented by
normalized vectors of complex Hilbert space H. Consider an
observable which is mathematically represented byP
the Hermitian
operator A with purely discrete spectrum: A =
v vPv , where
Pv is the projector onto the eigensubspace corresponding to the
eigenvalue v. Then
p(A = v in trial t | measurements in trials 1, . . . , t − 1) =


F ψ (t) , A, v = kPv ψ (t) k2
(16)
and


Pv ψ (t)
ψ (t+1) = G ψ (t) , A, v =
.
kPv ψ (t) k

(17)

This state transform expresses the von Neumann-Lüders
projection postulate of QM and represents the quantum state
update as a back reaction on measurement.

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Nowadays these transformations are actively used in
psychology; for example, to describe the order effect [32].

7. Short Review on Various Directions of
Research on Quantum Modeling of
Cognition
As was emphasized in Khrennikov [4], some statistical data
from psychology cannot be described by the standard von
Neumann model in which observables are represented by
Hermitian operators and state transformations (resulting from
the back actions of measurements) by the von Neumann-Lüders
projection postulate. As well as in quantum physics, one
have to use generalized quantum observables represented by
positive operator valued measures (POVMs) with corresponding
state transformers [4, 36]. In quantum physics POVM-type
observables naturally arise in the framework of theory of open
quantum systems describing interaction of a quantum system
with an environment; especially useful is the Markovian
approximation in the form of the Gorini-KossakowskiSudarshan-Lindblad equation. This advanced formalism
was widely applied to problems of cognition, in psychology,
social and political sciences [23, 24, 30, 31, 87]. In this framework
the process of decision making is represented as the process of
interaction of a concrete psychological function with a mental
environment: decision making as decoherence. This approach
was used to model irrational behavior of players in games of the
Prisoner’s Dilemma type. In such games the rational behavior is
associated with selection of the Nash equilibrium as the optimal
strategy. However, there were found numerous experimental
evidences that players can select strategies different from the
Nash equilibrium [80, 81]. Such behaviors were modeled with
the aid of theory of open quantum systems in a series of works of
Asano et al. [23, 24, 30, 31].
As was already pointed out, no-go theorems play a crucial role
in distinguishing classical and quantum probabilistic behaviors.
In quantum physics the Bell-type inequalities are explored as
experimental tests. In cognitive science the first experimental
violation of a Bell-type inequality (in the form of the Wigner
inequality) was reported in the article of Conte et al. [14, 18],
see also [5]. In quantum physics the Leggett-Garg inequality
was explored to test compatibility of macroscopic realism
with QM. Harald Atmanspacher and Thomas Filk used this
inequality [28] to study the problem of bistable perception (see
also [88]).
Violations of the Bell-type inequalities can be coupled to the
problem of contextuality, e.g., [53]. The contextual interpretation
of the aforementioned results on violations of these inequalities
in cognitive science and psychology is most natural. Cognition
is irreducibly contextual. The contextual modeling of cognition
was performed on the large scale in the monograph [4] in
which a general contextual theory of probability was developed.
Theory of contextual probability contains quantum probability
as a special case. Recently Ehtibar Dzhafarov initiated extended
studies on contextuality and Bell-type inequalities in psychology
and psychophysics [89, 90].

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8. Examples of Applications of the
Mathematical Formalism of Quantum
Theory
Here we present some examples of application the mathematical
formalism of quantum theory to psychology and decision
making.

8.1. Recognition of Ambiguous Figures
Let us explain our experiment on recognition of ambiguous
figures [13], see also [4], and its connection with Gestalt
psychology.
It is well known that, starting in 1912, Gestalt psychology
moved a devastating attack against the structuralism
formulations of perception in psychology. The classical
structuralism theory of perception was based on a reductionistic
and mechanistic conception that was assumed to regulate the
mechanism of perception. For any perception there exists a set of
elementary defining features that are at the same time necessary
(each of them) and jointly sufficient in order to characterize
perception also in cases of more complex conditions. The
Gestalt approach introduced instead a holistic new approach,
showing that the whole perception behavior of complex images
can never be reduced to the simple identification and sum of
elementary defining features defined in the framework of our
experience.
During the 1920s and 1930s Gestalt psychology dominated
in the study of perception. Its aim was to identify the natural
units of perception, explaining it in a revised picture of the
manner in which the nervous system works. Gestalt psychology’s
main contributions have provided some understanding of the
elements of perception through the systematic investigation of
some fascinating features, such as the causes of optical illusions,
the manner in which the space around an object is involved in the
perception of the object itself, and, finally the manner in which
ambiguity plays a role in the identification of the basic laws of the
perception.
In particular, Gestalt psychology also made important
contributions to the question of how it is that sometimes we see
movements even though the object we are looking at is not really
moving. As we know, when we look at something we never see
just the thing we look at. We see it in relation to its surroundings
(underlying context). An object is seen against its background.
In each case we distinguish between the figure, the object or the
shape, and the space surrounding it, which we call background or
ground, see Figures 9, 10, 11.
The psychologist Rubin was the first to systematically
investigate this phenomenon, and he found that it was possible
to identify any well-marked area of the visual field as the figure,
leaving the rest as the ground.
However, there are cases in which the figure and the ground
may fluctuate and one is forced to consider the dark part as the
figure and the light part as the ground, and vice versa, alternately.
Subjects of the experiment respond (recognize the image)
based on subjective and context-dependent factors, and output
of the experiment is principally probabilistic. The early work

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FIGURE 9 | Ambiguity Figure 1A.

FIGURE 10 | Ambiguity Figure 1B.

of Rubin, which observed the importance of the figure–
ground relationship, marked the starting point from which
Gestalt psychologists began to explain what today is known
as the organizing principles of perception. A number of
organizing or grouping principles emerged from such studies of
ambiguous stimuli. Three identified principles may be expressed

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followed by b appeared on a large screen for a time of only 3 s, and
simultaneously the students were asked to mark on a previously
prepared personal schedule their decision as to whether the
figures were equal or not. Test a after Test b presentation had the
objective of evaluating whether the perception of the first image
(Test a) can alter the perception of the subsequent image (Test b).
All the experiments were computer assisted and in each phase of
the experiment the following probabilities were calculated:
pb (+), pb (−), pa (+), pa (−),
p(b = +|a = +), p(b = −|a = +), p(b = +|a = −),

p(b = −|a = −).

FIGURE 11 | Ambiguity Figure 2.

as similarity, closure and proximity. Gestalt psychologists
attempted to extend their work also at a more physiological
level, postulating the existence of a strong connection between
the sphere of the experience and the physiology of the system,
by admitting the well-known principle of isomorphism. This
principle establishes that the subjective experience of a human
being and the corresponding nervous event have substantially the
same structure.
In our experiment, we examined subjects by Tests a and b in
order to test quantum-like behavior. For Tests a and b we used the
ambiguity figures of Figures 9, 10 as they were widely employed
in Gestalt studies:
• (a) Are these segments equal?
• (b) Are these circles equal?
Thus, the a-test is based on the following cognitive task: look
at Figure 9 and reply to question (a). The b-test is based on
Figure 10: look a this figure and reply to question (b).
The reasons for using such ambiguity tests here for analyzing
quantum-like behavior in perception may be summarized as
it follows. First of all, the Gestalt approach was based on the
fundamental acknowledgment of the importance of the context
in the mechanism of perception. Quantum-like behavior also
postulates this basic importance and role of the context in the
evolution of the considered mechanism, see Section 4. Finally,
we have seen that in ambiguity tests, the figure and the ground
may fluctuate during the perception. Consequently, a nondeterministic (a quantum-like) behavior should be involved.
Ninety-eight medical students of University of Bari (Italy)
were enrolled in this study, with about equal distribution of
females and males, aged between 19 and 22 years, after giving
their informed consent to participate in the experiment. In the
first experiment a group of 53 students was subjected in part
to Test b (presented with Test b only) and in part to Tests a
and b (presented with Test a and soon after presented with Test
b with prefixed time separation of about 2 s between the two
tests). The same procedure was employed in the second and third
experiments for groups of 24 and 21 students, respectively. All
the students of each group were subjected to Test b or to Test
a followed by Test b. The ambiguity figures of Test b or Test a

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Here the role of context, say C, is played by the selection
procedure of a sample for the experiment. All probabilities
depend on C.
A statistical analysis of the results was performed in order to
ascertain whether coefficients of interference λβ are non-zero or
zero in Tests b, a and b|a. The first experiment gave the following
results
Test b : pb (+) = 0.6923; pb (−) = 0.3077,

Test a : pa (+) = 0.9259; pa (−) = 0.0741,

Test b|a : p(b = +|a = +) = 0.68; p(b = −|a = +) = 0.32,
p(b = +|a = −) = 0.5; p(b = −|a = −) = 0.5.

(18)

The calculation of conditional probability gave the following
result with regard to pb (+):
pa (+)p(b = +|a = +)+pa (−)p(b = +|a = −) = 0.6666. (19)
The second experimentation gave the following results:
Test b : pb (+) = 0.5714; pb (−) = 0.4286,

Test a : pa (+) = 1.0000; pa (−) = 0.0000,

Test b|a : p(b = +|a = +) = 0.7000; p(b = −|a = +) =

0.3000,

p(b = +|a = −) = 1.0000; p(b = −|a = −) = 0.0000.

(20)

The calculation of the conditional probability gave the following
result with regard to pb (+):
pa (+)p(b = +|a = +) + pa (−)p(b = +|a = −) = 0.7.

(21)

Finally, the third experimentation gave the following results:
Test b : pb (+) = 0.4545; pb (−) = 0.5455,

Test a : pa (+) = 0.7000; pa (−) = 0.3000,

Test b|a : p(b = +|a = +) = 0.4286; p(b = −|a = +) =

0.5714;

p(b = +|a = −) = 1.0000, p(b = −|a = −) = 0.0000.

(22)

The calculation of the conditional probability with regard to
pb (+) gave the following result:
pa (+)p(b = +|a = +)+pa (−)p(b = +|a = −) = 0.6000. (23)
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The mean value ± SD of pb (+) resulted in pb (+) = 0.5727 ±
0.1189 in Test b and calculated using Equations (18), (20),
and (22), while instead a mean value of 0.6556 ± 0.0509
resulted for pb (+) when calculated in Test b|a and thus using
Equations (19), (21), and (23). The two calculated mean values
are different and thus give evidence of quantum-like behavior of
cognitive mental states as they were measured by testing mental
observables by Tests b, a, and b|a. Student’s t-test showed that
the probability that the obtained differences between the two
estimated values of pb (+) by Test b and by Test b|a are accidental,
does not exceed 0.30. Thus, with probability 0.70 the coefficients
of supplementarity are non-zero and, hence, students behave
(think) in a quantum-like way (with respect to observables based
on the ambiguous figures). We also found that these coefficients
are bound by 1, so behavior is trigonometric, see Appendix 1 in
Supplementary Material.
As the final step, we calculate cos θβ on the basis of
the coefficient of interference λβ given by Equation (13) in
Supplementary Material. In our experiments we obtained
cos θ+ = −0.2285, θ+ = 1.8013
and
cos θ− = 0.0438, θ− = 1.5270,
which are quite satisfactory phase results indicating quantumlike behavior for the investigated mental states.
The above results present a preliminary evidence of the
existence of quantum-like behavior in the dynamics of some
mental states. Luckily, we were able to capture mental conditions
of subjects in which the context influenced decision making
in an essential way. We have established equivalence between
quantum-like entities and corresponding cognitive entities.
As the performed experiment suggests a quantum-like
behavior of cognitive entities, a consequence could be that
cognitive entities as well as quantum entities exhibit a highly
contextual nature. In the same manner as quantum entities are
influenced by the routine physical act of measurement, cognitive
entities are influenced by the act of measurement (decision). In
the case of cognitive entities, the measurement is characterized
by cognitive interaction.
Mathematical modeling of the experiment considered above
was beased on a behavioral similarity between cognitive and
quantum-like entities, so we were able to make direct use of
an abstract quantum-like formalism and apply it to cognitive
entities. Moreover, we were able to account for quantum-like
dynamics of the cognitive entities. The numerical results of
the previous experiment give us an opportunity to delineate
basic features of cognitive entities not known in the past. Let
us outline this approach in more detail. We can introduce a
complex quantum-like amplitude, which represents the state
of our cognitive entity expressed in relation to some selected
mental observables. Let us suppose that we selected the mental
observable b, belonging to a given cognitive entity. Suppose also
that b can assume only two possible values (b = +, −). This
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complex quantum-like amplitude can be produced by QLRA,
Appendix 1 in Supplementary Material. The Born rule holds
|ψ(±)|2 = pb (±).

(24)

The complex quantum-like amplitude can represent the state
of our cognitive entity in relation to the considered mental
observable b.
The experiment indicates a methodological way for quantumlike processing of future experiments. We will briefly reconsider
the case of the experiment we have performed, showing how
to calculate quantum-like complex amplitudes and thus to give
a quantum-like characterization of the state of the cognitive
entity that was employed in the experiment. Let us consider in
detail the model entities of our experiment. As we indicated
previously, we managed to calculate two different values for
cos θ (+) and cos θ (−), whose meaning is now clear. In our case,
as we found above, cos θ+ = −0.2285, θ+ = 1.8013 and cos θ− =
0.0438, θ− = 1.5270, which nicely corresponds to quantum-like
behavior of the investigated cognitive entity. As a final step, we
present a detailed calculation of the quantum-like model of the
mental state of the cognitive entity as characterized during the
course of the experiment.
By using the obtained data, we can write a mental wave
function ψ = ψC of the mental state C of the group of students
who participated in the experiment—corresponding to a mental
context denoted by the same symbol C. QLRA, see Appendix 1 in
Supplementary Material, produces
ψ(β)

=

p

p(a = +)p(b = β|a = +)
p
+ e
p(a = −)p(b = β|a = −).
iθ (β)

(25)

The ψ is a function from the range of values {+, −} of the
mental observable b to the field of complex numbers. Since b
may assume only two values, such a function can be represented
by two-dimensional vectors with complex coordinates. Our
experimental data give
ψ(+) =

√
0.8753 × 0.6029
√
+eiθ (+) 0.1247 × 0.5 ≈ 0.7193 + i0.2431 (26)

ψ(−) =

√
0.8753 × 0.3971
√
+eiθ (−) 0.1247 × 0.5 ≈ 0.5999 + i0.2494. (27)

and

8.2. Quantum Representation of Order Effect in
Psychology
For example, in a typical opinion-polling experiment, a group of
participants is asked one question at a time, e.g., A = “Is Bill
Clinton honest and trustworthy?” and then B = “Is Al Gore honest
and trustworthy?” or in the opposite order, B and then A[91]. The
corresponding probability distributions, p(A = i, B = j) - “first
the B-question with the result j and then the A-question with the
result i” and p(B = j, A = i) - “first the A-question with the result
i and then the B-question with the result j” do not coincide.

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Quantum-like modeling of cognition

For classical probability theory this is a problem. Here the
observables A and B have to be represented by functions A, B :
 → 0, 1 (random variables). Set Ai = {ω ∈  : A(ω) = i}, Bj =
{ω ∈  : B(ω) = j}. Then
p(A = i, B = j) = p(Ai ∩ Bj ) = p(Bj ∩ Ai ) = p(B = j, A = i).
(28)
The order effect is washed out as the result of commutativity
of conjunction. For comparison with the quantum approach,
it is useful to write the previous equality by using conditional
probabilities:
p(A = i, B = j) = p(B = j)p(A = i|B = j)

= p(A = i)p(B = j|A = i) = p(B = j, A = i).

(29)

In the quantum model of the opinion poll,Pobservables are
represented
by Hermitian operators, A =
i = 0,1 iPi , B =
P
j = 0,1 jQj . Here
p(A = i, B = j) ≡ p(B = j)p(A = i|B = j),

p(B = j, A = i) ≡ p(A = i)p(B = j|A = i).

(30)
(31)

Opposite to Equation (29) which is a consequence of Equation
(28), these are the definitions of the “sequential probabilities.”
Here the joint probability distribution is, in general, not well
defined. Quantum conditional probability is defined as the
probability with respect to the state obtained as the update of
the initial state ψ after the first measurement (and crucially
dependent on the first measurement result)
p(A = i|B = j) =

kPi Qj ψk2
kQj Pi ψk2
,
p(B
=
j|A
=
i)
=
.
kQj ψk2
kPi ψk2

The order effect takes place if and only if kPi Qj ψk2 6= kQj Pi ψk2 ,
or h[Pi , Qj ]ψ|ψi 6= 0. If the operators do not commute, then such
a state ψ exists.

8.3. “Hawaii Experiment”
Tversky and Shafir [80] considered the following psychological
test demonstrating the disjunction effect. They showed that
significantly more students report that they would purchase
a non-refundable Hawaii vacation if they knew that they had
passed or failed an important exam than report they would
purchase if they did not know the outcome of the exam (So, a
student is going to travel to Hawaii in any event, whether she
passed exam or not, but only under the condition that she knows
the result).
There can be introduced the following two variables; a = 1
(exam passed), a = 0 (exam failed) and b = 1 (go to Hawaii),
b = 0 (not to go to Hawaii). The data [80] has the form:
p(b = 1) = 0.32 and hence p(b = 0) = 0.68 (these are
the probabilities in the context of uncertainty). Then we also
have p(a = 0) = p(a = 1) = 0.5. In the experiment 50% of
students were informed that they passed/not passed the exam.
The general structure of the experiment was the following. There
were two groups of students; one was used for the unconditional
measurement of the b-variable and generated the probabilities

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p(b = 0), p(b = 1) and the second group was used for the
conditional measurement of b: under the conditions a = 1 or
a = 0. The data collected in the second setting was
p(b = 1|a = 1) = 0.54, p(b = 1|a = 0) = 0.57;
p(b = 0|a = 1) = 0.46, pb = 0|a = 0 = 0.43.

The transition probabilities can
in the form of
 be represented

0.54
0.57
the following matrix: Pb|a =
. These data violate
0.46 0.43
FTP and the degree of violation is given by the coefficients of
interference, see EquationP(11): δ(1) = 0.17, δ(0) = −0.17.
(We remark that always x δ(x) = 0). These coefficients can
be represented in the form Equation (9) (as for interference
of wave functions in the two slit experiment) with θ1 =
1.3, θ0 = 2. For dichotomous variables, the data easily allow
to reconstruct the quantum(-like) state and observables, by
using the constructive wave function approach and QLRA,
see Appendix 1 in Supplementary Material. We present the
formula giving the “belief state” ψ of students in the basis of
eigenvectors of the Hermitian operator B representing the bobservable, i.e., B = diag(0, 1).
p
p It has the form: ψ(x) =
p(a = 0)p(b = x|a = 0) + eiθx p(a = 1)p(b = x|a = 1). By
inserting the values of probabilities and angles into this
expression we obtain the vector with complex coordinates, x =
0, 1. The direct calculation shows that Born’s rule Equation
(4) holds, i.e., p(b = x) = |ψ(x)|2 , x = 0, 1. Thus,
statistical data from this cognitive psychology experiment can
be mathematically represented with the aid of the quantum
formalism.

8.4. Categorization-decision Experiment
One of the most elucidating examples of quantum theory as
applied to psychology is the experiment on interference of
categorization in decision making. Statistical data collected in
such experiments exhibits non-classical feature in the form of
violation of FTP with high statistically significance. In particular,
it is impossible model such data with the aid of classical Markov
dynamics. Therefore, it is natural to proceed with the quantumlike model justifying violation of laws of classical probability
theory. In coming presentation of this model we follow the paper
[92].
Often decision makers need to make categorizations before
choosing an action. For example, a military operator has to
categorize an agent as an enemy before attacking with a drone.
How does this overt report of the category affect the later
decision? This paradigm was originally designed to test a Markov
model of decision making that is popular in psychology [93].
Later it was adapted to investigate quantum-like interference
effects in psychology [17, 92].
We begin by briefly summarizing the methods used in the
experiments (see [92], for details) . On each trial of several
hundred training trials, the participant is first shown a picture
of a face that may belong to a “good guy” category (category G)
or a “bad guy” category (category B), and they have to decide
whether to “attack” (action A) or “withdraw” (action W). The
trial ends with feedback indicating the category and appropriate

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Khrennikov

action that was assigned to the face on that trial. There are
many different faces, and each face is probabilistically assigned
to a category, and the appropriate action is probabilistically
dependent on the category assignment. Some of the faces are
usually assigned to the “good guy” category, while other faces
are usually assigned to the “bad guy” category. The category
is important because participants are usually rewarded (win
points worth money) for “attacking” faces assigned to “bad
guys” and they are usually punished (lose points worth money)
for “attacking” faces assigned to the “good guys;” likewise they
are usually rewarded for “withdrawing” from “good guys” and
punished for “withdrawing” from “bad guys.” Participants are
given ample training during which they learn to first categorize
a face and then decide an action, and feedback is provided on
both the category and the decision. Although the feedback given
at the end of each trial is probabilistic, the optimal decision is to
always “attack” when the face is usually assigned to a “bad guy”
category, and always “withdraw” when the face is usually assigned
to a “good guy” category. The key manipulation occurs during a
transfer test phase which includes the standard “categorization–
decision” (C-D) trials followed by either “category alone” (Calone) trials or “decision alone” (D-alone) trials. For example, on
a “decision alone” trial, the person is shown a face, and simply
decides to “attack” or “withdraw,” and recieves feedback on the
decision. The categorization of the face on the D-alone trials
remains just as important to the decision as it is on C-D trials, and
some implicit inference about the category is necessary before
making the decision, but the person does not overtly report this
implicit inference.
Note that the C-D condition in the psychology experiment
allows the experimenter to observe which “path” the participant
follows before reaching a final decision. This is analogous to
a “double slit” physics experiment in which the experimenter
observes which “path” a particle follows before reaching a final
detector. In contrast, for the D-alone condition in the psychology
experiment, the experimenter does not observe which “path” the
decision maker follows before reaching a final decision. This is
analogous to the “double slit” physics experiment in which the
experimenter does not observe which “path” the particle follows
before reaching a final detector9 .
According to the Markov model proposed in Townsend et al.
[93], for the D-alone condition, the person implicitly performs
the same task as explicitly required by the C-D condition. More
specifically, for the D-alone condition, once a face (denoted f )
is presented, there is a probability that the person implicitly
categorizes the face as a “good” or “bad” guy. From each category
inference state, there is a probability of transiting to the “attack”
or “withdraw” decision state. So the probablity of “attack” in
the D-alone condition (denoted as p(A|f )) should equal the
total probability of “attacking” in the C-D condition (denoted
as pT (A|f )). The latter is defined by the probability that the
person categorizes a face as a “good guy” and then “attacks”
9 We remark that here the picture of path is used only for illustrative purpose;
therefore we placed path in quotation marks. In QM there is no such a concept as a
“path” (trajectory) of a particle. We can only ascertain, and this is only statistically,
a singular event of an electron “passing” through a slit. In fact this way of seeing
the situation provides an even better parallel here.

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Quantum-like modeling of cognition

plus the probability that the person categorizes the face as
a “bad guy” and then “attacks” (pT (A|f ) = p(G ∩ A|f ) +
p(B ∩ A|f )). Using this categorization-decision paradigm, one
can examine how the overt report of the category interferes with
the subsequent decision. An interference effect of categorization
on decision making occurs when the probability of “attacking”
for D-alone trials differs from the total probability pooled
across C-D trials. The Markov model for this task originally
investigated by Townsend et al. [93] predicts that there should
be no interference, and the law of total probability should be
satisfied.
Beginning with our first study [17], we have conducted a
series of four experiments on this paradigm (see 92, for review).
All results of these experiments show similar results, but we
briefly report a summary of findings from the fourth experiment
that included 246 participants (a minimum 34 observations per
person per condition). For a face more likely assigned to the
“god guy” category (we denote these faces as g), the law of total
probability is approximately satisfied (pT (A|g) = 0.36, p(A|g) =
0.37). However, for a face more often assigned to the “bad guy”
category (we denote these faces as b), the probability of “attack”
(i.e., the optimal decision with respect to the average payoff)
is systematically greater for the D-alone condition as compared
to the C-D condition” violating the law of total probability
(p(A|b) = 0.62 > pT (A|b) = 0.56)10 . More surprising, the
probability of “attack” for the D-alone condition (which leaves
the “good” or “bad” guy category unresolved) was even greater
than the probability of “attack” given that the person previously
categorized the face as a “bad guy” (p(A|b) = 0.62 > p(A|b, B) =
0.61) on a C-D trial! For some reason, the overt categorization
response interfered with the decision by reducing the tendency to
“attack” faces that most likely belonged to the “bad guy” category.
These violations of the law of total probability contradict the
predictions of the Markov model proposed by Townsend et al.
[93] for this task.
A detailed quantum-like model for the categorizationdecision task is presented in [17], and here we only present a
brief summary following the paper [92]. The human decision
system is represented by a unit length state vector |ψi belonging
to 4-dimensional Hilbert space spanned by four basis vectors.
(Here we use Dirac’s symbolic notations, see Appendix 2 in
Supplementary Material).
Each basis vector represents one of the four combinations of
categories and actions (e.g., |GAi is a basisvector
 corresponding
to category G and action A). The state ψf = ψGA |GAi +
ψGW |GWi + ψBA |BAi + ψBW |BWi is prepared by the face
stimulus f that is presented during the trial. The question about
the category is represented by a pair of projectors for good
and bad categories CG = |GAi hGA| + |GWi hGW| , CB =
(I − CG ). The question about the action is represented by a
pair of projectors for attack and withdraw actions DA =
†
†
UDC |GAi hGA| UDC
+ UDC |BAi hBA| UDC
, DW = (I − DA ),
where UDC is a unitary operator of transformation from the
categorization basis to the decision basis.
10 This difference is statistically significant: t

(245) = 4.41, p = 0.0004. Also the same
effect was replicated in 4 independent experiment

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Quantum-like modeling of cognition

Following [92], we obtain that the probability of first
categorizing the face as a “bad guy” and then “attacking” equals

 
2
p(B, A|f ) = p (B) · p (A|B) =
CB ψf
· kDA |ψB ik2 , with

C ψ

B  f 
,

C B  ψ f

and combining the terms in the product we

 
2
obtain p(B, A|f ) =
DA · CB · ψf
; similarly, the probability
of first categorizing the face as a “good guy” and then “attacking”

 
2
equals p(G, A|f ) =
DA · CG · ψf
; and so the total probability
of attacking under the C-D condition equals pT (A|f ) =

 

 

DA · CG · ψf
2 +
DA · CB · ψf
2 .
The probability of attack in the D-alone condition equals

 
2

 
2
[92] p(A|f ) =
DA · ψf
=
DA · (CG + CB ) ψf
=

 
 

 
2

DA · CG ψf + DA · CB ψf
2
=
DA · CG ψf
+

 




DA · CB ψf
2 + Int, where Int = 2 · Re ψf |CG DA CB |ψf . If
the projectors for categorization commute with the projectors
for action (e.g., UDC = I), then the interference is zero, Int = 0,

 
2
 
2
and we obtain p(A|f ) =
DA · CG ψf
+
DA · CB ψf
=


pT A|f , and the law of total probability is satisfied. However, if
the projectors do not commute (e.g., UDC 6= I), then we obtain
an interference term. We can select the unitary operator UDC
which produces an inner product Int = −0.06, and account for
the observed violation of the law of total probability.
|ψB i =

9. Conclusion
We demonstrated that the mathematics developed to solve
QM problems is highly suitable to solving particular problems

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Conflict of Interest Statement: The author declares that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2015 Khrennikov. This is an open-access article distributed under the
terms of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) or licensor
are credited and that the original publication in this journal is cited, in accordance
with accepted academic practice. No use, distribution or reproduction is permitted
which does not comply with these terms.

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ORIGINAL RESEARCH
published: 02 August 2017
doi: 10.3389/fphy.2017.00030

The Physics of Teams:
Interdependence, Measurable
Entropy, and Computational Emotion
William F. Lawless *
Math & Psychology, Paine College, Augusta, GA, United States

Edited by:
Emmanuel E. Haven,
University of Leicester,
United Kingdom
Reviewed by:
Ignazio Licata,
ISEM- Institute for Scientific
Methodology, Italy
Nicolas Francisco Lori,
LANEN, INCYT, INECO Foundation,
Argentina
*Correspondence:
William F. Lawless
w.lawless@icloud.com
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 30 April 2016
Accepted: 05 July 2017
Published: 02 August 2017
Citation:
Lawless WF (2017) The Physics of
Teams: Interdependence, Measurable
Entropy, and Computational Emotion.
Front. Phys. 5:30.
doi: 10.3389/fphy.2017.00030

Frontiers in Physics | www.frontiersin.org

Most of the social sciences, including psychology, economics, and subjective social
network theory, are modeled on the individual, leaving the field not only a-theoretical, but
also inapplicable to a physics of hybrid teams, where hybrid refers to arbitrarily combining
humans, machines, and robots into a team to perform a dedicated mission (e.g., military,
business, entertainment) or to solve a targeted problem (e.g., with scientists, engineers,
entrepreneurs). As a common social science practice, the ingredient at the heart of the
social interaction, interdependence, is statistically removed prior to the replication of
social experiments; but, as an analogy, statistically removing social interdependence to
better study the individual is like statistically removing quantum effects as a complication
to the study of the atom. Further, in applications of Shannon’s information theory to
teams, the effects of interdependence are minimized, but even there, interdependence
is how classical information is transmitted. Consequently, numerous mistakes are made
when applying non-interdependent models to policies, the law and regulations, impeding
social welfare by failing to exploit the power of social interdependence. For example,
adding redundancy to human teams is thought by subjective social network theorists to
improve the efficiency of a network, easily contradicted by our finding that redundancy is
strongly associated with corruption in non-free markets. Thus, built atop the individual,
most of the social sciences, economics, and social network theory have little if anything
to contribute to the engineering of hybrid teams. In defense of the social sciences,
the mathematical physics of interdependence is elusive, non-intuitive and non-rational.
However, by replacing determinism with bistable states, interdependence at the social
level mirrors entanglement at the quantum level, suggesting the applicability of quantum
tools for social science. We report how our quantum-like models capture some of the
essential aspects of interdependence, a tool for the metrics of hybrid teams; as an
example, we find additional support for our model of the solution to the open problem
of team size. We also report on progress with the theory of computational emotion for
hybrid teams, linking it qualitatively to the second law of thermodynamics. We conclude
that the science of interdependence advances the science of hybrid teams.
Keywords: social reality, hybrid teams, Von Neumann entropy, interference, interdependence

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Lawless

INTRODUCTION
One of the major conclusions from modern game theorists,
based on findings in the laboratory, is that the societies that
cooperate have better social welfare [[1], p. 7–8]. The evidence
from the field, however, does not support this claim: Cooperation
between competitors is often considered by the judiciary to be
collusion [2]; consensus-seeking permits a minority faction to
control a majority (e.g., in European Union politics, see [3],
p. 29); and central decision-making promotes corruption [4].
Unexplained by traditional theory, misallocations of resources by
corrupt activities abound across the globe [5]. In our research,
we have concluded that corruption is more likely unchecked in
countries, businesses and teams that impede the interdependence
spontaneously arising among citizens in a nation with functional
checks and balances; China is an example of the corruption
that occurs from blocking interdependence (e.g., censorship),
replaced by central decision-making [6]:

Much about the Hong’ao dump was not as it appeared on paper,
a reconstruction of the disaster shows. The duplicity, involving
doctored documents and false identities, illustrates systemic
gaps in China’s efforts to prevent industrial and transportation
accidents, which claim tens of thousands of lives annually and
have galvanized public anger over official corruption . . . like the
deadly explosions last year at a toxic chemical storage site in
Tianjin . . . the disaster in Shenzhen suggests that dark pools
of mismanagement and corruption persist even in the most
developed parts of the country.

Conceptually, interdependence has been known for some time.
According to Smith’s [7] “invisible hand,” a service provided by
one worker exploiting an opportunity is interdependent with
another worker providing food, housing, transport, and on in
an endless iteration of services across a market free to respond
to market demands and signals by the movement of capital and
labor sufficient to satisfy demand. But free movement is impeded
by barriers established by centralized commands, decisions or
procedures (e.g., Dodd-Frank rules in the USA), authoritarian
governments (e.g., China), or violent gangs (e.g., Palestine’s
Hamas; Lebabon’s Hezbollah; the US’s Mara Salvatrucha).
What we know so far from this our work-in-progress is that
reducing interdependence increases errors and the misallocation
of resources [8]. We also know from the National Academy of
Sciences ([9], p. 33) that while interdependence is important
to effective teamwork, the size of a team for a given problem
remains an open question. The Academy then contradicted itself
by stating “many hands make light work,” indicating its belief
that redundancy has a positive effect on teams. Traditional
models of subjective social network theory also predict that an
increase in redundancy in social networks increases efficiency
[10]. We approach team size with our quantum-like model of
interdependence. By treating oil firms as teams [11], we theorized
that the best size for teams is the least size possible that maintains
interdependence across a team to solve a problem identified by a
society when its labor and capital are free to move. We discovered

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Physics of Teams

that by overstaffing, redundancy reduces interdependence. In this
paper, we extend our finding to the size of a nation’s military.
Even in American bureaucracies, consensus-seeking,
corruption, and mismanagement appear to go hand in hand (e.g.,
for a cover-up by the Veterans Affairs, see [12]; for unjustified
rule-making by the US Treasury, see [13]; for Department
of Energy guidance that citizen advisors should “strive for
consensus,” see [14]). As an example of the mismanagement
associated with consensus-seeking (i.e., minority control; in
Lawless [15]), DOE planned to vitrify high-level radioactive
wastes into glass for its eventual geologic disposal starting in
the 1980s at both its Hanford facility in Washington State and
at its Savannah River Site in South Carolina. However, the
consensus-seeking Hanford Citizens Advisory Board has not
formally motivated DOE to accelerate its Hanford vitrification
facility, a project plagued by gross mismanagement now delayed
until 2033 [16]. Compare that with the majority-ruled Savannah
River Site’s Citizen Advisory Board that formally motivated SRS
to start its vitrification facility in 1996 and has overseen its safe
operations for the more than 20 years hence.
In the literature, Khrennikov [17] suggests now is the time to
apply quantum-like models to address open questions in many
fields; e.g., business, psychology, and social systems. Busemeyer
and Wang ([18], p. 43) add that “Quantum cognition is an
emerging field that uses mathematical principles of quantum
theory to help formalize and understand cognitive systems
and processes.” Wang and Busemeyer [19] reintroduce the
concept of complementarity to account for order effects; we
use complementarity to account for the stable gap between
physical (objective) measures of behavior and self-reported
(subjective) observations of behavior (e.g., [20]), as well as for the
different interpretations of reality common to individuals (e.g.,
present-day supporters of Einstein’s views on quantum theory
vs. Bohr’s Copenhagen interpretation; in Lawless [2]), and the
different skills held by members of a team, where each may
have subjective interpretations and beliefs (e.g., in the search for
justice, construing the courtroom as a team, prosecutors, and
defense attorneys work together by pursuing different theories of
a crime; [11, 21]).
The phenomenon that links these examples is the
interdependence between behavior and its interpretations;
interdependence between multiple interpretations of social
reality; and the interdependence among members of a team
multitasking to solve a problem. In its review of teams, the
National Academy of Sciences repeatedly cited the presence
of interdependence but without addressing the phenomenon
theoretically [9]. In this study, we apply quantum-like models
to the study of interdependence (e.g., [22], p. 147). From
Wendt [23], “humans live in highly interdependent societies”
(p. 150); interdependence, however, creates a measurement
problem [2], which Wendt ([23], p. 67) describes as “the
apparent impossibility of an objective measurement,” and which
we have linked to the behavior-cognition gap, for example,
between the objective measures of behavior and the self-reported
subjective accounts of behavior (e.g., [20]). Wendt ([23], p.
34) adds that a quantum-like model “offers the potential for
revealing new social phenomena,” which we demonstrate by

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determining the size of teams, heretofore an open problem
([9], p. 33).
In the 1940s, Von Neumann and Morgenstern’s ([24], Section
4.8.2) theory of games introduced to generations of social
scientists a mathematical model of static interdependence in a
configuration of arbitrary rewards and punishments promoted
as tradeoffs among the choices offered to players with values
determined by scientists, not by social reality, producing decades
of biased social and political policies from these toy models.
Unlike Smith’s [7] “invisible hand” or the physical sciences where
“reality is not as it appears” to human observers [25], game
theory and wide swaths of social science are based on, at best,
simple observations of individuals and, at worst, self-reported
observations ([26]; e.g., questionnaires, surveys, interviews). The
value of actual behaviors vs. self-reports of constructs poorly
correlate, if at all, with most of the variance between actual
behavior and self-reported behavior unaccounted [20].
Bohr, the quantum physicist, criticized game theory on
foundational grounds, leading [24] to decry that if Bohr
was correct, how to proceed was “inconceivable” (p. 148).
Generalizing from quantum theory, Bohr [27] conceived of
humans as dual agents constituted of two independent but
interdependent parts in the brain (e.g., motor control and
vision; from Rees et al. [28]), viz., a human can serve to enact
(objectively) a behavior or to observe (subjectively) a behavior; or
a human can hold belief #1 (e.g., conservative) or opposing belief
#2 (e.g., liberal), the degree of complementarity between these
two parts affecting the tradeoffs common to making decisions
in social reality [11], consequently creating a measurement
problem long ignored by social scientists [23]; viz., game theory
does not recognize the existence of a measurement problem
in social reality. Specifically, when measuring a social object
interdependent with another, both are affected (e.g., [8]). When
playing games, as scientists feed choices to subjects to test
preferences and responses, they avoid seeing this problem, one
of the reasons that game theorists lament that the “evidence
of mechanisms for the evolution of cooperation in laboratory
experiments . . . [has not yet been found in] real-world field
settings” ([29], p. 422). Later, Von Neumann ([30], p. 420,
footnote 217) grappled with the social science implications of
Bohr’s ideas for the quantum interaction: “Bohr . . . was first to
point out that the dual description which is necessitated by the
formalism of the quantum mechanical description of nature is
fully justified by the physical nature of things . . . [that] may be
connected with the principle of psycho-physical parallelism.”
Kelley [31], an eminent social psychologist who spent most
of his career studying interdependence with games, finally
abandoned the study of interdependence because he could never
bridge the gap between the game matrices presented to a pair
of players versus the “invisible” matrices subjects responded to
during games; i.e., no matter how strongly held, the preferences
self-reported by subjects before they participated in games were
repeatedly contradicted by their choices made during games.
The inability of scientists to determine the value of the social
interaction at the heart of games is mirrored across the social
sciences by practitioners who base their theories on observations
of the processes of how the best teams should operate, often
with self-reported (subjective) surveys that tell us nothing new
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Physics of Teams

(e.g., the surveys and interviews of teams at Google; in Duhigg
[32]). By infusing social science with the normative values that
happen to agree with religious beliefs, presently, social science
is, unfortunately, of no value in the engineering of hybrid teams.
An exception of sorts is the report by the National Academy of
Sciences on the value of interdependence to scientific teams [9];
but, by being non-mathematical, the Academy report offers no
guidance to engineer hybrid teams.
In comparison to game theory and other traditional
approaches to the study of interdependence in teams, we define
interdependence as responsive or reactive to signals in nature
between non-independent organisms (e.g., elk grazing in a forest
with predators leads to healthier forest grasses; from Carroll
[33]). Our physics of interdependence as mutual responsiveness
is similar to that of entanglement, where the factors that
produce interdependence cannot be factored, remaining opaque
or invisible to even well-trained observers [2]. But although the
effects of interdependence are often “invisible” to rational human
observers [7], we recognize that humans manage or exploit it with
the competition between at least two teams vying for the support
of each team’s ideological beliefs or skills before an audience of
neutral individuals freely able to choose, thereby entangling them
in one belief and then countered by its contrary belief as they
process the information generated by the opposing sides of an
argument (viz., a Nash equilibrium), exactly what dictators first
seek to suppress [2].
When measuring states of interdependence, the measurement
problem’s “apparent impossibility of an objective measurement”
([23], p. 67) makes social reality non-deterministic. As an
example, Cohen [34] reported that women with HIV partners
voluntarily participated in the trial of a new drug designed
to prevent HIV infection. Ninety-five percent of the women
self-reported to medical staff that they had faithfully taken
the medication, but, if true, because the infection spread to
many of these women, the results indicated that the trial had
failed. Inadvertently, the medical research team recalled that they
had also collected blood samples from the women during the
trial. Once investigated, researchers discovered that only about
26% of the women had actually taken the drug, saving the
trial.
From the HIV example, if “quantum-like effects exist in the
social world, expressed as interdependence” ([22], p. 147), the
interdependence should produce a complementarity in socialpsychological systems that causes interference between the two
factors of a human’s physical behavior and its very different
observation of behavior, a difference ignored by traditional
social scientists’ belief in the independence of these effects. The
interpretations of observations by individuals and scientists are
impacted by their beliefs, biases, and experience, producing, for
example, interference illusions [35]. Unlike quantum systems
where angles of separation between two beams of light produce
replicable effects, and whereas we can reliably reproduce
Adelson’s interferences to create his checkerboard illusion, at
this early stage of social application, much remains unknown
and surprising as in the example above reported by Cohen [34].
It is likely the reason that wide swaths of social science have
recently come under suspicion for being unable to be replicated
[36]. The goal of our research project is thus to find a way
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to objectively study the interference between “behavior” and
“observed behavior.”
As another example of how interdependence makes social
reality non-deterministic, consider self-esteem, one of the major
foci for the clinical practice of psychology over the decades. In
the book published by the American Psychological Association
(APA), [37] began:
Although, relatively little is known about self-esteem, it is
generally considered to be a highly favorable personal attribute,
the consequences of which are assumed to be as robust as
they are desirable. Books and chapters on mental hygiene and
personality development consistently portray self-esteem as one
of the premier elements in the highest levels of human functioning
. . . Its general importance to a full spectrum of effective human
behaviors remains virtually uncontested. We are not aware of a
single article in the psychological literature that has identified or
discussed any undesirable consequences that are assumed to be a
result of realistic and healthy levels of personal self-regard.

Despite this bold claim by Bednar and Peterson under
the imprimatur of the APA, a 30-year meta-analysis of all
of the known experimental studies where self-esteem could
be measured against actual physical performance for both
academics and the workplace by Baumeister [38] found a
negligible correlation, confirming that self-reports of self-esteem
are unrelated to actual behavior.
As a result, we adopt the spirit imbued in game theory
to model interdependence, but we reject game theory as
fundamentally observational and a-theoretical. Instead, by using
Von Neumann’s model of quantum interference and Bohr, we
review herein our advances: by taking limits, we derived a
quantitative measure in the limit of what constitutes a perfect
team, another for the worst team, and another we found as a
relative metric of team performance modeled after Kullback–
Leibler divergence where redundancy in teams is characterized by
the divergence in team size from comparable free market teams
[11]. Finally, we review our past research to lay the groundwork
for a computational model of emotion in teams characterized as
a phase shift between overstaffed and rightly-sized teams.
In his theory of self-replicating automata, Von Neumann
[39] addressed energy costs and thermodynamics; Shannon
information theory; an individual, traditional, and rational
perspective of reality; stability (p. 70); errors (p. 71); parts of
self-replicating automata (p. 74); the difficulty of choosing the
parts of a self-replicating automata in the right order (p. 76); and
common sense in assembling the parts (p. 77). In contrast [11],
we use a phase shift in the production of maximum entropy to
demarcate teams with good allocations of resources from those
that misallocate; interdependence between ideologically opposed
power centers reflected as a point of social stability that drives
information processing (what we have named as Nash equilibria;
e.g., Republican and Democrat political parties; defense and
district attorneys; Einstein’s and Bohr’s view of quantum reality);
and a metric for the assembly of teams measured by a decrease
in structural entropy production. Regarding Nash equilibria, we
exemplify them as checks and balances, the source for the best
possible government. Contradicting the results of toy games by

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Physics of Teams

game theorists ([1], p. 7) and social scientists, Madison [40]
established that good governance occurs where “Ambition must
be made to counteract ambition.”
In summary, briefly, our goal is to apply our findings to
determine mathematically the performance of hybrid teams.
Traditional, but normative, models centered around cooperation,
while of value in the creation of stories or religious homilies, are
of little value for the engineering of hybrid teams. By extending
our research to team emotion, we hope to generalize our research
where our most recent goal was to use hybrid team performance
as a guide to minimize human error (e.g., [41]).

REVIEW OF PRIOR RESEARCH.
MATHEMATICAL PHYSICS
Martyushev [42] theorized that maximum entropy production
(MEP) drove the evolution of systems. Wissner-Gross and Freer
[43] added that intelligence maximizes the entropic force with
Equation (1),
F(X0 ) = T∇ x S(X)|X0

(1)

where F is the entropic force associated with macrostate X, T the
temperature and S the Shannon entropy for state X. To apply
Equation (1) to a social system, say with a team of scientists
seeking to operate at MEP, we would expect a scientific team to
use its intelligence to be able to devote its available energy to
the fullest exploration of its chosen problem space in the search
for a solution, but barriers encumber exploration, reducing
MEP, motivating the need for intelligence to overcome barriers
(e.g., bureaucracy; corruption; arbitrary rules; censorship; etc.).
We conclude that teams use their collective intelligence to
seek MEP to overcome barriers; e.g., to seek the path where
multitasking applies the maximum effort to solve a difficult
problem. Building on Wissner–Gross and Freer that barriers
impede MEP, intelligence in a team is needed to navigate around
or to overcome these barriers, helping top teams to better
compete to succeed. If, as we hypothesize, redundancy acts as a
barrier that increases destructive interference in a team, reducing
the “force” in Equation (1), then overstaffing in a team is a barrier
that frustrates the application of intelligence to decisions. As one
of our steps, we will adopt a method that helps us to look for a
sign of the collective effects of intelligence.
Our theory is that excluded spaces are governed by the politics
in play operating in a social reality, with bistable interpretations
of social reality determined by neutral supporters [44]. In
contrast to our approach with bistability, others have suggested
that stable beliefs could be implemented with epistemic logics,
comprising a Hilbert space semantics of belief states that could
lead to a formal derivation of social entanglement. Instead, we
let the beliefs held by one subgroup attempting to force its
interpretation of social reality on the whole group to be |0>, and
its complementary, orthogonal view held by a second subgroup to
be |1>, giving as the state (Equation 2) for the combined group:
√
|9 >= 1/ 2(|0 > +|1 >);

(2)

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Physics of Teams

the factors in Equation (2) of non-separable states [22] become
separable by measurement [2], but the measurement problem
([23], p. 67) means that as we determine the state of one factor,
we no longer have reliable information on the state of the other
factor. If a state’s subsystems are not separable, it is entangled;
however, if a state is, or has been made, separable, it cannot be
entangled [45, 46].
A social system that controls, stops or blocks the bistable
interdependence in Equation (2) should be modeled by Shannon
entropy. Pure states are product states, where S(ρ) = 0. Product
states are uncorrelated; e.g., , where AB is the Hilbert space of a
composite system [46]. The measurement of one subsystem in
a composite, product state system has no affect on its second
subsystem ([45], [[47], p. 61–3]). If the interdependence among
skill sets does not exist in two or more subsystems, Shannon
information governs, the data is iid1 , no correlations exist,
and joint entropy is likely greater than the contributions of
subsystems, i.e.,
H(X, Y) ≥ H(X), H(Y)

(3)

To reflect correlations caused by interference among the sources
of information, unlike Shannon entropy, interdependence can be
destructive or constructive, captured by Von Neumann’s density
matrix, ρ, with entropy depicted by Equation (4):
S = −Tr(ρ log ρ)

(4)

If a team is successful in producing a team with members who
multitask together to form what appears to be a team with “single
mind,” its degrees of freedom (dof ) go to 1 (from the equation
for cognitive interdependence by Kenny et al. [48], p. 235),
accounting for “invisibility,” giving:
S = limdof−>1 log(dof) = 0

(5)

Interestingly, Einstein was the first to discover the reduction in
dof at the quantum level, a critical insight that he shared with
Schrodinger ([49], p. 238–9). Like the “single mind” of a team,
an example of constructive interference occurs with the melding
of the brain into a single mind was given in an interview of
Donald Hoffman [50], a cognitive scientist with an evolutionary
perspective,
We have two hemispheres in our brain . . . [that form a] unified
single mind. . . . But when you do a split-brain operation, a
complete transection of the corpus callosum, you get clear
evidence of two separate consciousnesses.

Interference may be constructive, as when the members of a
team work well together. In contrast to Equation (3), to represent
Hoffman’s “unified single mind” and to further account for the
invisibility of interdependence effects, we use subaddivity to get:
S(ρAB )≤S(ρA ) + S(ρB )
1 iid: independent

and identically distributed random variables

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(6)

Working from Von Neumann’s perspective, correlations in joint
entropy can become greater or equal to their differences, reflected
by Equation (7):
S(ρAB )≥|S(ρA ) − S(ρB )|

(7)

Equation (7) implies that social groups engage in tradeoffs
to choose the more fit members of a team, where the best
fit is signified by a reduction in joint entropy. Shannon
states for subadditivity

in a composite system can also be
expressed as: H x, y
≤ H (x) + H(y)([51], p. 515–6).
From our perspective, subadditivity holds when subsystems are
correlated, indicating that the components are interdependent
with offsetting entropies, justifying our comments that teams
need coaches to compensate for a team’s invisible information
as it multitasks. At the atomic level, the trace of a density
matrix,
ρ, isTr(ρ) = 1; for a pure state, ρ 2 = ρ(idempotent).
IfTr ρ 2
= 1, ρis pure and |ψ>AB is separable; however,
ifTr ρ 2 < 1, ρis mixed and |ψ>AB is entangled ([52],
p. 207–8). The degree of mixing determines the departure
from a pure state. Based on these considerations, we theorize
that interdependence among teammates produces subadditivity,
where interdependence specifically means a lack of separability.
Returning to Equation (2), if the two factions in a group,
represented by the operators A and B, have reached a single
consensus without compromise, the eigenvalues for the operators
representing both factions in the group are the same ([[53], p.
256), giving:
[A,B] = AB - BA = 0

(8)

But interference from social interaction may be destructive; e.g.,
the rupture of a sports team; a married couple undergoing a
divorce; the splitting apart of a business striving to survive a
market turndown, like the Maersk Conglomerate [54]. When
a group with two factions holds opposed viewpoints, a gap
occurs in the group’s interpretations of (social, physical) reality.
In social-psychological systems, if a binary operator fails to
commute ([[53], p. 343], [55] and [56],), it may produce order
effects (e.g., [19], p. 2), uncertainty or incompleteness [11],
giving:
[A,B] = AB - BA = iC

(9)

where C represents a gap, a quantum gap at the atomic level
[27] and the incommensurable political gap at the social level,
the latter relabeled by us as a Nash equilibrium [4]. This gap at
the social level offers a rich, new view of social reality. When the
gap is fully driven by both factions with no neutrals on either
side, conflict becomes likely [44]. But when neutrals must be
wooed by both sides to win a debate, the solicitation of neutrals
compels a compromise for the two sides to reach a decision,
magnifying the power of neutrals freely able to influence both
sides of a debate by helping to avoid a rupture [57]. As a merger
of ideas, a compromise satisfies the decision at hand in the
heat of the moment, releasing the emotional energy pent up
by both factions (emotions are discussed later), energy that had

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been reserved to continue an intellectual battle associated with a
decision under uncertainty. The self-interests of the two sides of
a Nash equilibrium act as a quasi-team with neutrals to process
information that serves to check (control) the ambitions of both
sides [40].
As they form an audience, neutrals, we argue, are the only
social element to enter into a superposition (Equation 2), driven
into the superposition by the Nash equilibrium that acts like the
two cylinders of an engine. As they are wooed to and fro, once
neutrals are measured, the trail they leave behind forms limit
cycles [4]. Other than the trail left behind, neither side fully grasps
the social reality sufficiently well enough to control the neutrals,
why dictators, gangs and command economies suppress Nash
equilibria and free speech [4]. But in a free society, the result is
multiple tradeoffs that a free society exploits to evolve [2], such as
finding the optimum size of a team.
Wang and Busemeyer [19] use the concept of
complementarity to account qualitatively for order effects;
we use complementarity to account for the gap between behavior
and self-reported observations of behavior (e.g., [20]), as well as
the different interpretations of reality by members of opposing
teams (e.g., present-day supporters of Einstein’s views on
quantum theory vs. Bohr’s acausal Copenhagen interpretation;
in Lawless [2] and Bohr [27]).
Cohen [58] revised Equation (10) in signal detection theory
to arrive at transformations between Fourier pairs, concluding
that a “narrow waveform yields a wide spectrum, and a wide
waveform yields a narrow spectrum and that both the time
waveform and frequency spectrum cannot be made arbitrarily
small simultaneously” (p. 45), giving:

σA σB ≥1/2,

(10)

where σA is the standard deviation of variable A (often events),
σB for variable B (often the time for when events occur) both
modeled in Figure 1.
In quantum theory, the uncertainty relation Equation (10)
follows directly from the non-commutativity of Hilbert space
operators (Equation 9). Similar relations appear for Fourier
pairs in classical field theory as well. By itself, the application
of Equation (10) to what follows for the action of teams
(Equations 10, 11) can be criticized as a mere analogy and not
formally motivated. However, a new discovery of redundancy or
overstaffing among oil producers as teams coupled with another
discovery (e.g., flawed DOE nuclear waste management teams of
scientists and their managers with Equation 14 below; in Lawless
[15]) add credibility to our formulation (e.g., [59]).
Based on our prior findings, when the goal of tradeoffs is to
find the team members with the right skills for the best team
fit, we begin to extend our findings with a revision of Equation
(10) to:
σA σB − > σskills σinterpretations ≥1/2

(11)

Along with the claims of Smith [7] and Bohr [27], the
struggles of Kelley [31] and the findings of Zell and Krizan
[20], Equations (5, 11) help us to see that, based strictly on
physics, reducing the standard deviation for skills (action) to
improve teamwork increases the standard deviation for the
interpretations (observations) of the performance of a team’s
members, accounting for the “invisible” loss of awareness.
Accordingly, if the skills of a team approach perfection, the
width of different interpretations widens, making it difficult
to see what makes a team effective, motivating the need for
a coach to improve the performance of a poorly performing
team in the search for more successful outcomes for a team’s

FIGURE 1 | As a notional example, the wide Gaussian is Fourier transformed to the narrow one; the Standard deviation for the latter one is 0.33, that for the wide one
about 5.0; the two multiplied together is >1/2.

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actions (e.g., efforts have recently begun at NSF to train teams
of scientists to become better teams of scientists; e.g., [9, 60]).
To study the implications of Equation (10), we decompose a
team into a (static) structure that directs its efforts, and its efforts
at performing its mission (i.e., dynamic skill roles; actions based
on those roles). Assume that the structure of a team is functioning
perfectly, allowing the team to use an optimum amount of its
available energy to solve the problems that the team was designed
to address. Building on our prior success, but speculating, we
convert Equation (11) into two components representing the
least entropy production (LEP) for the structure of a team and
maximum entropy (MEP) to perform a team’s mission:
σLEP σMEP ≥1/2

(12)

Taking limits with the variables in Equation (12) gives us an
equation that captures a team’s excellence; i.e., as a team’s
consumption of energy by its structure goes to zero, it’s ability to
maximize its ability to problem-solve itself becomes a maximum:
S = limσLEP−>0 σMEP = ∞

(13)

With Equation (13) in hand, by inverting it, an account is
discovered for what happens when a team fails, splits apart, or
implodes [2], giving
S = limσMEP−>0 σLEP = ∞

(14)

The teams represented by Equation (14) might be a couple
undergoing divorce; a business team failing (e.g., Maersk
Shipping; in Chopping [54]); or a team of scientists forced
by managers to not follow rigid scientific practices, exactly
what [61] was concerned about. Such an example of scientific
malfeasance, driven by management, happened in 1983 with the
Department of Energy (DOE) at its Savannah River Site (SRS)
in Aiken, SC. Despite its many scientific claims to Congress that
DOE waste management practices were safe and equivalent to
commercial ones, the file photograph in Figure 2 from SRS points
out that from the 1950s until 1983, DOE’s waste management
practices permitted 90% of its military solid radioactive wastes
to be disposed of in ordinary cardboard boxes, allowing these
boxes to sit in open trenches exposed to the weather for
months at a time, becoming one of the primary sources of
radioactively contaminated groundwater across DOE’s complex.
Public awareness stopped DOE’s use of cardboard boxes in 1985.
After DOE had openly admitted its past errors and had begun to
rectify them, during the cleanup in 2000, renewed public support
for DOE accelerated the closure of the same radioactive waste
burial ground at SRS [15].

MATERIALS AND METHODS
The National Academy of Sciences [9] concluded that the
problem of team size was an open question, yet implicitly
supported redundancy with their consensus speculation that
“more hands make light work” (Ch. 1, p. 13). In contrast, to
our examples of excluded volumes we add redundancy as a

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FIGURE 2 | At DOE’s Savannah River Site from the 1950s until 1985, DOE’s
waste management permitted 90% of its military solid radioactive wastes to be
buried in ordinary cardboard boxes, allowing these boxes to sit in open
trenches exposed to the weather for months at a time.

cause of poorly performing oil firm teams [11]. We had found
that GDP/capita, our surrogate for the competitiveness of a
nation’s oil firm teams, was significantly related to its freedom
index, less teamwork redundancy, and less redundancy in the
number of employees per oil firm. Then with divergence for a
distribution of oil firms compared to another for a comparable
freedom index, we found a significant regression to indicate that
worker redundancy decreased per unit of oil produced as the oil
firms were freer to optimize their teams to deploy their capital
and labor as they saw fit when drilling for oil. For example,
Exxon’s production with 15.5 employees/M BBL of oil compares
to Sinopec’s 124.6 employees/M BBL of oil produced, illustrating
that redundancy creates inefficiency.
We first define our four factors: redundancy, economic
freedom, military power, and corruption. These factors are mixed
objective and subjective, meaning the results will include varying
levels of subjectivity.

Redundancy
Redundancy is a quantity measured for interacting human
autonomous systems and interfering with other autonomous
systems [11]. Redundant are any number of mates that exceed
the minimum number of members of a team designed to
solve the problem assigned to a team; e.g., a baseball team
with more than 9 members on a baseball field has redundant
members by that many. In quantum theory, redundant copies of
quantum states violate the no-cloning rule ([62], p. 77), and, we

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argue, interdependent states [11]; e.g., compare Sinopec’s 124.6
employees/M BBL of oil produced with Exxon’s 15.5, illustrating
that authoritarian regimes creates inefficiency with redundancy.

Military Power Ranking
We used the ranking devised by Global Firepower (http://
www.globalfirepower.com). Its ranking is based on a nation’s
weapon diversity and conventional forces without relying on
nuclear stockpiles. It includes geographic factors, logistics,
natural resources, and industry.

Economic Freedom

in this problem are footnoted below2 , as is the data for each of
them3 .
Example:
As an example of the calculations with Equation (15), for
China’s Military Power Distribution (MPD, or PMPD ), we divided
its military power rank (3) by its population in billions (1.374)
= 2.183 and the result we divided by 8.1, China’s GDP/capita in
thousands = 0.370; we summed this result for our top 22 nations
= 82.28, which we divided into 0.370 to get the fraction for
China, PChina =.0045. We repeated to calculate the Free Market
Distribution (FMD) for China (59.4) by dividing by the sum
(1296.9) for our Q1 . Next we entered the calculations stepwise
into Equation (15) to get for China the following calculation:

An index established by the Heritage foundation based on four
broad factors to measure liberty and free markets for 186 nations:
rule of law; government size; regulatory efficiency; and open
markets. Each factor has three sub-factors (http://www.heritage.
org/index/ranking).

In addition, as one of our methods, we will look for a sign of the
collective effects of intelligence.

Corruption

RESULTS

An index of nations established by Transparency International
(https://www.transparency.org). Its factors determine the abuse
of power for private gain, and whether the abuse is covert or
concealed. The assessment is first to the branches of government,
then the public sector, law enforcement, media, businesses, and
then other factors.
We measure redundancy with divergence from a Kullback–
Leibler-type equation for relative entropy, where DKL (Q||P) is
the Kullback–Leibler’s divergence of probability distribution Q
from P:
DKL (Q||P) =

X

i

Q(i)log(Q(i)/P(i))

(15)

The sum of Equation (15) reflects the divergence of distribution
Qi , from distribution Pi , with both distributions normalized.
For example, log (P(i)/P(i)) = log 1 = 0. Thus, the more
divergence, the larger the separation between two distributions.
Based on Equation (15), assuming that a relatively perfect team is
possible to solve the problem at hand, we also assume that some
structures for desired teams may be closed-ended for a solved
problem like those that exist for sports teams; e.g., for a baseball
team, designated members take the role of pitcher, catcher, first
baseman, etc. Unlike the relatively simple problem of designing a
sports team, most business and scientific teams are open-ended
whenever competition or innovation are factors. To solve this
kind of a structural problem, in business, we look to an industry
leader for the best team structure possible for the problem at
hand.
To extend these findings to militaries, we hypothesize that
redundancy is associated with less freedom in the marketplace
and with more corruption. We test this hypothesis with
correlations and Kullback–Leibler divergence. We expect that
distributions in the real world range from minimum to
maximum redundancy; from minimum to maximum freedom;
and from minimum to maximum corruption. The nations used

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FMD∗ Ln(FMD/MPD) = 0.044∗ ln(0.044/0.0045) = 0.101.

For a pilot run, we used a convenience sample of 12 nations
consisting of some of the largest militaries in the world4 . We
assumed that a country’s military could be represented as a
team. We compared military size with a country’s economic
freedom index and its corruption index. For this sample, we
first calculated correlations to obtain the following results: As
a country’s economic freedom index increased positively, its
military size per GDP decreased significantly (r = −0.78, p <
0.005) and its corruption level decreased significantly (r = −0.59,
p < 0.05). We also found that economic freedom and corruption
were inversely correlated significantly (r = −0.77, p < 0.025),
indicating that an increase in freedom was associated with a
decrease in corruption.
Heartened by these pilot results, we were ready to test our
hypothesis with Equation (15). For Q1 , we summed the result of
FMD versus MPD to get 1.78. We repeated the process for Q2
to get another distribution for corruption levels versus MPD for
a sum of 1.95. Then we regressed the FMD results individually
nation by nation versus MPD (Q1 ) with the divergence of
2 Nations:

the top 20 militaries in the world plus Cuba and North Korea were
used: China, USA, Russia, Brazil, UK, India, France, Japan, Turkey, Germany,
Italy, South Korea, Egypt, Pakistan, Indonesia, Israel, Vietnam, Poland, Taiwan,
and Iran.
3 For the actual study, we used the top 20 militaries in the world per capita (from
http://www.globalfirepower.com) and GDP per capita from the IMF (https://
en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal)_per_capita) versus
national Free Market Economy ranking (http://www.heritage.org/index/ranking)
and Transparency International’s corruption index (https://www.transparency.
org/news/feature/corruption_perceptions_index_2016).
4 For the pilot study, we used the following sample: USA, China, Cuba, North
Korea, India, Israel, Iran, Japan, Mexico, Pakistan, Russia & Turkey; economic
freedom index from 2016 http://www.heritage.org/index/ranking; b: the number
of military personnel was derived from the 2014 edition of “The Military Balance”
published annually by the International Institute for Strategic Studies, except for
Cuba, North Korea and Pakistan, with data from http://www.tradingeconomics.
com/; and the corruption index was from Transparency International at https://
en.wikipedia.org/wiki/Transparency_International.

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corruption from MPD (Q2 ) and plotted the result in Figure 3.
The result is significant (R2 , 0.926, p < 0.0001).
As a side issue, we also looked for signs of intelligence.
We found it in our calculations. Consider an abstract from
our data in column 4 of the Table 1 below where we see a
demarcation between authoritarian and democratically governed
militaries. Although fuzzy, we argue that the results in the
table’s column 4 are signs of intelligence based on information
processing; i.e., the military power rank per capita per gdp per
capita for China is 0.370 vs. that for the USA of 0.017 and
for the UK of 0.15, indicating greater protection per capita
by the USA and UK compared to China, Russia, Brazil, and
Cuba.

DISCUSSION
We had hypothesized and found in two separate distributions,
one for the divergence of GDP for a country’s Index of Economic
Freedom with its military power ranking per capita, and the
other for the divergence of a country’s corruption index versus its
military power ranking per capita, a significant regression. This
indicates that, even with real-world data containing subjective
estimations, redundancy increases the more authoritarian is a
country’s decision-making. As a corollary, the collective effects of
intelligence in a society operate best under the freedom to allocate
capital and labor for its best uses.
Our results for this study, also backed with correlations,
support theory and justify our use of quantum-like models. We
found less divergence with our hypothesis for military team size
and economic freedom, but more divergence with military team
size and corruption, indicating that National Defense improves
under the collective effects of intelligent decisions at the level of
the team in free markets. It means that a military is leaner and
more effective under democracies that under autocracies.
We suspect that redundancy in the market of teams
isolates excess teammates from interdependent effects, reducing
responsiveness, and converting co-workers into featherbedders.
Barriers, like authoritarian leadership and corruption, impede
reaching MEP by intelligent teams. And, as we have found,
redundancy increases under authoritarian governments, for
the possible but corrupt political payoffs that may become
necessary to keep civil peace. For example, corruption has
stymied the reform of scientific practices in Russia [63] and
the transformation of Russian businesses attempting to reduce
redundancy [64].
Our model is different from the traditional model, specifically,
the cognitive model. As a representative example of the influence
of the cognitive model transported from social science to history
in the hands of a popular historian5 , Harari [65] concludes
that human groups of no more than 150 can be held together,
primarily with gossip, but that larger groups, like Peugeot SA, are
“a figment of our collective imagination” (p. 29) based on shared
stories, a social construct that forms the “imagined realities” of
the cognitive model. But if Harari’s account is true, the differences
in team distributions between those domiciled in authoritarian
5 His book is a New York Times Bestseller

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FIGURE 3 | In this figure, we regressed the divergence of freedom from a
military distribution with the divergence of corruption from a military distribution.
The result indicates a significant regression (R2 = 0.926, p < 0.0001). The
nations used in this regression are listed in section Materials and Methods.

TABLE 1 | Data rounded off to three significant decimals.
Mil power rank,
http://www.globalfirepower.com

Mil rank/
billion capita

Mil pwr rank/
gdp/capita

China

3

2.183

0.370

USA

1

3.086

0.017
0.225

Russia

2

14.084

Brazil

17

82.927

1.954

UK

6

93.75

0.150

Cuba

78

7090.909

10.196

(Military power ranking and population from Global Firepower; Freedom Index from
Heritage Foundation; GDP per capita from the International Monetary Fund; and
Corruption Index from Transparency International, footnoted and defined above.)

regimes versus democracies should be random. Nor would there
be any path forward to build teams of machines or robots that
could reasonably be expected to advance social welfare in any
meaningful way. If Harari’s perspective is true, the success of any
one’s story may be no more than a matter of taste, preference or
culture, not a matter for physics or engineering.
That is not what we have found. Our results establish the
meaningful differences that interdependent information plays
in the interactions and affairs of humans under any and every
form of government. Information constraints (barriers) under
authoritarian regimes are less able to direct the movement of
labor and capital to best solve targeted problems, an added
constraint for innovation, one reason the Chinese rely on the
theft of intellectual property (see the interview of General M.
Hayden, the former CIA and NSA chief, by the editor-in-chief
of the Wall Street Journal, [66]). Certainly, obstacles exist in
democracies, especially when they become less free to allocate
resources to solve the problems targeted (e.g., the Department of
Energy’s practices included cardboard boxes, seepage basins and
other shortcuts to dispose of its radioactive wastes to save money
that may eventually cost DOE well over two hundred billions of
dollars to remediate its Hanford Site and its Savannah River Site;
in Lawless [15]).

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Unlike Google’s survey of teams [32], guided as it was
by traditional social science, we have conjectured and found
evidence that an improved theory of human behavior includes
both cognitive (subjective surveys like Google’s) and behavioral
(physical) data which our quantum-like model handles well. By
reporting that interdependence is a factor in the best scientific
teams, the National Academy has made a nice corrective ([9];
also see [67]). While we agree with the Academy about the
value of interdependence, it would have been better for it to
have addressed the theoretical value of interdependence as we
did with our quantum-like models to shift the focus from
individuals to teams, to how teams disarm “imagined realities”
to improve their, and their society’s, situational awareness of
reality, and to better justify the social tools that humans use
to produce superior decisions (e.g., political debate). From our
perspective, independent individuals or neutrals are valued as
critical to the determination of the winners and losers in a
contest where the uncertainty associated with an outcome is
high and depends on the persuasion of an audience of neutrals
(e.g., in the competition between two equally competent teams
competing against each other in politics, in courtrooms, or for the
philosophical meaning of quantum reality). The added benefit is
that we can generalize these results to mathematical metrics for
hybrid teams.

NEW WORK: EMOTION
In the HRI community, a lot of research with reinforcement
learning (RL) is designed to assist in social interaction where
“emotions obviously are important for social interaction” ([68],
p. 29). For their research, RL agents require few assumptions,
are easy to apply in all kinds of domains, and allow for learning.
In contrast, our theory is designed to determine when teams are
working well and when not.
In his magisterial review of the literature on emotion,
supported by our theory, Zajonc [69] saw that emotion may
be interpersonal rather than individual (p. 593), especially
during communication (p. 604); emotion exists independently
of cognition or is even disconnected from awareness (p. 607)
and correlates poorly with self-reports (p. 612), supporting the
concept of a mind-body duality (p. 596); habituation indicates
a low level of emotion (p. 614); positive emotions lower
temperature, T, negative ones increase T (p. 616); and deaf
subjects respond more emotionally to spoken texts than normals
(p. 619), an effect that, ceteris paribus, suggests expressing a skill
is less evident to observers than its absence (p. 619).
In addition, a rise in T occurs with cognitive or social
dissonance [2]; energy doubles when expressing a statement
in a normal versus an angry voice [70]. Emotions reflect an
individual’s self-interest ([71], p. 439; i.e., less dissonance) and
serve to guide social behavior (p. 442) by minimizing marginal
expenditures of energy (see also [69], p. 592 and 622).
What if judgments about reality are not rational but guided
primarily by experience (where a culture has been ushered into
being and molded by experiential learning; [35])? Letting LEP
represent the ground state for the structure of a team, a team

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Physics of Teams

with its structure at its ground state can devote its availability
energy to solving problems, giving experience time to develop
into a successful culture. For example, a perfect business team
is able to devote its available energy to addressing the problems
life offers to it. By way of contrast, when a team’s structure exists
at an excited state, a business team splitting apart is expending
most of its free energy on ripping apart the culture and structure
of its team (e.g., Maersk; in Chopping [54]), leaving little available
energy to solving the problems it encounters.
Applied to teams by integrating Zajonc and others, we can see
that the structure of a team is in a relatively stable state (dof −>
1), and that independent, asocial individuals are in a freer state
(maximum dof ) than team members. Based on the second law of
thermodynamics, comparing a solid substance (ice) and its liquid
form (water), energy must be emitted by a group of individuals
as a team is formed (e.g., those mergers that reduce redundant
employees; in Bunge [72]) and absorbed by a team if it breaks
apart.
Initially, we use a sigmoid function to model the effort
required to hold a team together (see Figure 4). In Equation (16),
the effort, f(effort), applied to a team’s structure to channel its
interactions into a single whole becomes
f(effort) = 1/(1 + exp(-effort))

(16)

Results from a Monte Carlo simulation of Equation (16) shown
in Figure 4 below indicates that as effort to hold together a team’s
structure increases beyond a critical point, the team’s structure
begins to fail. In this simple model, we consider the effort as

FIGURE 4 | A Monte Carlo simulation of Equation (16) with the y-axis intercept
at (0, ½) in the center, with y ranging from 0 to +1 (listed vertically on the far
left side). From the y intercept to the right along the x-axis (with x = 0, +1, +2
units) represents increasing effort and emotion; from the left of the intercept
along the x-axis represents stability and a team’s ground state (where x = -2,
-1, 0 units). As the effort to maintain a team’s structure approaches zero in the
middle of the graph, a critical point is reached. As more effort is required to
hold the team’s structure together (i.e., moving to the right), it begins to break
apart as team members begin to act more and more like redundant,
independent individuals.

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the potential energy of the team; a well-fitted team then has
negative potential energy. Conversely, the less successful the team
at solving its designated problems, the more its teammates begin
to act as individuals, the more the strength of the team’s structure
becomes random, causing the cohesion of the team to dissipate.
Once the team reaches a critical point (near the y-intercept) in
its dissipation, a phase shift has occurred, requiring more and
more energy to maintain the team, offsetting a team’s successes,
destroying the team’s structure. Once that happens, joint entropy
begins to resume a Shannon-like nature (i.e., Equation 3; e.g., the
coming collapse of Sears; in Halzack [73]).
We have also found evidence that a well-fitted team
having success at solving the problems it was designed to
solve exhibits more intelligence than an under-fitted or overfitted team with redundant members. The well-fitted team
generates less entropy than its individual contributors, an
indication that a state of maximum interdependence exists
inside of the team, where each member is responsive to
every other team member and to the team’s mission as well.
The state of maximum interdependence, however, can be
reversed or blocked. Like quantum computations, the state
of interdependence is a resource for a team but also for
the society within which the well-fitted team is embedded
and to which the well-fitted team contributes. Once a wellfitted team establishes a point of stability, an emotionless
baseline, it is operating in a ground state (Figure 5, bottom
left). If the joint entropy generated by the team begins
to exceed the entropy of any single contributor, the team’s
interdependence and structure have begun to dissipate (Figure 5,
upper right).
We have not addressed the characteristics of the problem
targeted, but we suspect that a team must be designed to match
its designated problem (e.g., a well-fitted 5-member baseball team
is of value in playing against an equally competent 5-member
baseball team, but of little value when playing against an equally
competent but nine-person baseball team).

FIGURE 5 | A notional aspect of the transactions modeled by the sigmoidal
function in Equation (16).

Frontiers in Physics | www.frontiersin.org

Physics of Teams

CONCLUSIONS
Significant impediments exist in the formulation of a science
of teams using traditional theories. Specifically, Shannon’s
information theory and the social sciences, including economics,
assume that the human observation of human behavior
records the actual behavior that has occurred, even for selfreports of self-observed behavior. In computational social
science, this phenomenon has been labeled informally as
the “god’s eye view,” indicating that the “computer” within
which computational action occurs knows immediately whatever
action a computational agent takes. In the social sciences,
this phenomenon manifests as an observational bias; it allows
social scientists to assume that self-reported behavior is
actual behavior (e.g., if this assumption was true, deception
or denial, such as alcoholic denial or spying, would not
exist). We claim that this assumption is unsupported by the
evidence, as is the “knowledge” gathered in support, such
as the conclusion consonant with widespread religious beliefs
that cooperation provides for the best social good. At the
heart of these rational, but false models, interdependence
is seen as a constraint (information or communication
theory) or experimental confound (cognitive science) that
must be overcome by traditional social scientists to confirm
a theoretical models based on methodological individualism
(MI; [26]).
By replacing MI with quantum-like models, we have
found computational metrics for good and poor teamwork
performance, and a third finding that redundancy is associated
with corruption by using relative entropy to model divergence
from an oil market leading team, now supported in this study
by the size of a nation’s military. We have also proposed a new
model for a team’s emotion as it shifts from a ground state to an
excited state. We conclude that, like entanglement at the atomic
level, interdependence at the social level is the primary social
resource that ordinary humans exploit to innovate and promote
social welfare.
Wendt [23] said that quantum-like models should be given a
chance to make new discoveries. Who would have even thought
that redundancy is a problem, or that it could give insight into
the structure of what constitutes the best team. The National
Academy of Sciences report on teams points out that team size
is an open problem, but while it did not mention redundancy [9],
it did speculate that “many hands make light work,” a speculation
faulted by our results.
We reject the traditional model of redundancy (e.g., [10]).
Cummings [67] found that the more interdisciplinary a science
team, the least productive it was as a science team; however,
he also found that the best science teams were highly
interdependent; i.e., highly responsive to each other. We agree
with Cummings, and our results support him.
Excessive team emotion is observable to external observers;
e.g., a divorce; a business breakup; a team’s collapse. More
difficult to observe is the critical point, the transition from a team
arguing appropriately [74] over an “invisible” structural issue
that, if not resolved, may represent a transition from being a wellfitted team past the critical point until “visible” to those observing

August 2017 | Volume 5 | Article 30 | 85

Lawless

a team’s transition along the path to becoming ill-fitted as a team’s
structure breaks down.
For a mathematical physics of teams, a significant impediment
has too long existed from accepting the traditional belief that
social truth can be established by observing individuals. As
exemplars, both built around the statistics of independent,
identically distributed data (i.i.d.), information theory and
social science, including economics, assume that self-reports
record actual behavior, especially self-observed behavior. But the
traditional social science model simply does not generalize to
hybrid teams; to evolve, to design hybrid teams, this idea that
“self-observations record actual behavior” must be rejected.
In contrast, based on our model where interdependence
reduces a team’s degrees of freedom (dof ), thereby obscuring
this effect by making it “invisible” to viewers, we propose that
ordinary teamwork is characterized by the search for an optimum
in the tradeoff between maximum entropy production (MEP)
and least entropy production (LEP), where MEP reflects team
performance (dynamics; e.g., productivity), LEP determines team
structure (statics), and, unexpectedly, the tradeoff generalizes to
represent a new and computational model of team emotion. With
our theory, we are able to draw several conclusions. First, as
a resource, social humans exploit interdependence to innovate
and promote social welfare, suggesting that, by increasing and
aligning the MEP density across teams, a culture of competition
among teams predictably improves social intelligence, innovation
and social welfare. Second, however, interdependence precludes
replication, causality and truth, exactly what is commonly found
in social reality, including social science. And, finally, our

Physics of Teams

local theory of teams appears to scale without limit, limiting
the value of independent individuals; but, we theorize, value
returns when independent individuals enter into states of
superposition driven by the opposed worldviews of competing
teams, interdependently entangled until these now individuals
superposed to both views are measured to determine the winner
of the competition that they are most responsive to.
The best teams have the least redundancy so that they are
maximally interdependent among teammates to be responsive
to each other as they multitask to solve the problems that they
face intelligently. In conclusion, we have found support for our
quantum-like model with the solution to the open problem of
team size.

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
approved it for publication.

FUNDING
Some of the work was performed while the corresponding author
was a senior faculty researcher at the Naval Research Laboratory
over the past 2 years, including the Summer 2016 and 2017.

ACKNOWLEDGMENTS
The author thanks the reviewers of his manuscript for their very
helpful comments, suggestions and corrections.

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Conflict of Interest Statement: The author declares that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Lawless. This is an open-access article distributed under the terms
of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) or licensor
are credited and that the original publication in this journal is cited, in accordance
with accepted academic practice. No use, distribution or reproduction is permitted
which does not comply with these terms.

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ORIGINAL RESEARCH
published: 18 July 2017
doi: 10.3389/fphy.2017.00028

Nilpotent Quantum Mechanics:
Analogs and Applications
Peter Marcer 1 and Peter Rowlands 2*
1
Independent Researcher, St. Raphael, France, 2 Oliver Lodge Laboratory, Department of Physics, University of Liverpool,
Liverpool, United Kingdom

Edited by:
Emmanuel E. Haven,
University of Leicester,
United Kingdom
Reviewed by:
Raimundo Nogueira Costa Filho,
Federal University of Ceará, Brazil
Diego Lucio Rapoport,
Universidad Nacional de Quilmes
(UNQ), Argentina
*Correspondence:
Peter Rowlands
p.rowlands@liverpool.ac.uk
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 12 April 2016
Accepted: 28 June 2017
Published: 18 July 2017
Citation:
Marcer P and Rowlands P (2017)
Nilpotent Quantum Mechanics:
Analogs and Applications.
Front. Phys. 5:28.
doi: 10.3389/fphy.2017.00028

Frontiers in Physics | www.frontiersin.org

The most significant characteristic of nilpotent quantum mechanics is that the quantum
system (fermion state) and its environment (vacuum) are, in mathematical terms, mirror
images of each other. So a change in one automatically leads to corresponding
changes in the other. We have used this characteristic as a model for self-organization,
which has applications well beyond quantum physics. The nilpotent structure has also
been identified as being constructed from two commutative vector spaces. This zero
square-root construction has a number of identifiable characteristics which we can
expect to find in systems where self-organization is dominant, and a case presented
after the publication of a paper by us on “The ‘Logic’ of Self-Organizing Systems” [1],
in the organization of the neurons in the visual cortex. We expect to find many more
complex systems where our general principles, based, by analogy, on nilpotent quantum
mechanics, will apply.
Keywords: universal rewrite system, self-organization, nilpotent quantum mechanics, renormalization group

INTRODUCTION
Three main developments form the background to this work. The first is a universal rewrite
system, which is a scale-independent and fractal computational process of generating zerototality alphabets, with seemingly very general application [1], The most immediate applications
of the rewrite structure have been found in physics and biology, which brings us to the second
development. Nilpotent quantum mechanics is a form of relativistic quantum mechanics/quantum
field theory which can be derived from the rewrite system and which minimalizes the whole
quantum apparatus to a single operator acting on a universal environment, which is its mirror
image. The third development is that both the rewrite structure and nilpotent quantum mechanics
require a combination of two vector spaces, each dual to the other, which provides a powerful model
for self-organization [1].
Nilpotent quantum mechanics is the most immediately successful application of the universal
rewrite system and serves as an almost perfect model for other applications. It is not so much
that these applications derive directly from nilpotent quantum mechanics, rather that they derive
from the structure which makes this form of quantum mechanics possible. Many characteristics
can be described as identifiers of both the rewrite and the nilpotent structures, whether at the
quantum mechanical level, or applicable in mathematics, chemistry, biology, or other areas of
physics. We have proposed a number of such features as being detectable in systems of very different
kinds and as thus being signatures of quantum-like organization or behavior, especially where selforganization is dominant [1], and we have identified a new one in the organization of the neurons
in the visual cortex.

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Marcer and Rowlands

Nilpotent Quantum Mechanics: Analogs and Applications

THE UNIVERSAL REWRITE SYSTEM

TABLE 1 | The universal rewrite process.

The universal rewrite system provides a computational approach
to both mathematics and physics based on the idea of a
zero totality alphabet [1]. We successively create alphabets in
which “conjugation” represented by ∗ ensures totality zero, and
in which new creation is ensured by the regular appearance
of anticommutative pairs A, B; C, D; etc., each of which is
commutative to all the others:
∗

∗

∗

∗

(R, R )(R, R ) ⇒ (R, R , A, A )

(1)

(R, R∗ , A, A∗ )(R, R∗ , A, A∗ ) ⇒ (R, R∗ , A, A∗ , B, B∗ , AB, AB∗ )

(2)

Successive alphabets absorb the previous ones in the sequence, so
creating a new cardinality. We may start at any arbitrary zerototality alphabet but there is no natural beginning or end to the
process, which can be summarized in Table 1.

An Algebra for the Rewrite Process
The rewrite process is more general than any particular
mathematical interpretation, but such interpretations include
both binary integers and digital logic, in addition to the algebraic
series:
(1, −1)
(1, −1) × (1, i1 )
(1, −1) × (1, i1 ) × (1, j1 )
(1, −1) × (1, i1 ) × (1, j1 ) × (1, i2 )
(1, −1) × (1, i1 ) × (1, j1 ) × (1, i2 ) × (1, j2 )
(1, −1) × (1, i1 ) × (1, j1 ) × (1, i2 ) × (1, j2 ) × (1, i3 ) ...

In this algebraic structure, the unit vectors i, j, k have the
multiplication rules

Frontiers in Physics | www.frontiersin.org

1b

1c

...

1n

0

00

01a

01b

01c

01n

1a

1a 0

1a 1a

1a 1b

1a 1c

1a 1n

1b

1b 0

1b 1a

1b 1b

1b 1c

1b 1n

1c

1c 0

1c 1a

1c 1b

1c 1c

1c 1n

1n 0

1n 1a

1n 1b

1n 1c

1n 1n

:
1n

The ∆ symbols, here, represent the alphabets:
∆a (R)
∆b (R, R*)
∆c (R, R*, A, A*)
∆d (R, R*, A, A*, B, B*, AB, AB*)
∆e (R, R*, A, A*, B, B*, AB, AB*, C, C*, AC, AC*, BC, BC*, ABC, ABC*)
...

TABLE 2 | The units of Clifford algebra.
Vector

i, j, k

Bivector

ii, ij, ik

Pseudovector

Quaternion (i, j, k)

Trivector

i

Pseudoscalar

Complex algebra

Scalar

1

(ii)2 = (ij)2 = (ik)2 = 1

(1, −1)
(1, −1) × (1, i)
(1, −1) × (1, i) × (1, j)
(1, −1) × (1, i) × (1, j) × (1, i)
(1, −1) × (1, i) × (1, j) × (1, i) × (1, j)
(1, −1) × (1, i) × (1, j) × (1, i) × (1, j) × (1, i) ...

ij = −ji = ik; jk = −kj = ii; ki = −ik = ij

1a

which are essentially those of complexified quaternions, with
multiplication rules

In this interpretation, the anticommutative pairs A, B; C, D;
E ... are expressed as quaternion units, i1 , j1 ; i2 , j2 ; i3 ...,
each of which is commutative to all the others. By the fourth
stage, we have repetition, which then continues indefinitely. An
incomplete set of quaternion units (for example, i3 in the sixth
alphabet) becomes equivalent to the algebra of complex numbers.
Mathematically, we can see the process of the creation of the
zero totality alphabets as one of conjugation, followed by repeated
cycles of complexification and dimensionalization (where each i
is paired with a j).
At the point where the cycle repeats, we have what can be
recognized as a Clifford algebra—the algebra of 3-D space, where
the vectors i, j, k are constructed from i1 i2 , j1 i2 , i1 j1 i2 , and i1 ,
j1 , i1 j1 = k1 and i2 , j2 , i2 j2 = k2 are (mutually commutative)
quaternion algebras of the form i, j, k.

i2 = j2 = k2 = 1

0

(3)
(4)

(5)

compared to those for pure quaternions
i2 = j2 = k2 = ijk = −1.

(6)

In the Clifford vector algebra, there is a full product between
vectors a and b which combines vector and scalar products
ab = a.b + ia × b

(7)

It has been shown by Hestenes [2] and others, that using a
Clifford vector algebra is a natural way of incorporating spin into
quantum mechanics as an automatic consequence of the vector
structure of space and momentum. The units are, significantly,
isomorphic to those of Pauli matrices.
Clifford vector algebra produces three subalgabras from
the products of its basic units (see Table 2). Bivectors (for
example, area and angular momentum, in physics) are products
of two orthogonal vector units (such as i and j); they are
also called pseudovectors and are isomorphic to quaternion
units. Trivectors (for example, volume) are products of three
orthogonal vector units (i, j, k), and are also called pseudoscalars;
their full algebra is that of complex numbers.
Standard Clifford vector algebra notably produces these
subalgebras in the reverse order to the universal rewrite system,
which generates, in its first four alphabets, scalars, pseudoscalars,
quaternions and vectors, along with the scalar subalgebras of
pseudoscalars and quaternions.

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Significantly, if we take all these algebras as independently
true, and hence commutative, as the rewrite structure seems
to suggest we should, since each is a complete description of
zero totality, then we require an algebra that is a commutative
combination of vectors, bivectors, trivectors and scalars, or
vectors, quaternions, pseudoscalars and scalars. This turns out
to be equivalent to the algebra of the sixth alphabet, a group
structure of order 64 with elements, as in Table 3.
The algebra represented by these group elements is
isomorphic to the gamma matrix algebra of the Dirac equation,
which defines conventional relativistic quantum mechanics.

TABLE 3 | The units of the Dirac algebra.
i

j

k

ii

ij

ik

i

j

k

ii

ii

ik

ii

ij

ik

iii

iii

iik

ji

jj

jk

iji

iji

ijk

ki

kj

kk

iki

iki

ikk

–i

–j

–k

–ii

–ij

–ik

–i

–j

–k

–ii

–ii

–ik

–ii

–ij

–ik

–iii

–iii

–iik

–ji

–jj

–jk

–iji

–iji

–ijk

–ki

–kj

–kk

–iki

–iki

–ikk

i

1

–i

–1

Characteristics of the Rewrite Process
The universal rewrite process is characterized by duality,
self-similarity, scale-independence and holism; it proceeds by
bifurcation at every stage. It can be distinguished from nonuniversal rewrite processes by the fact that it has no fixed
starting or ending point, and by the endless reconstruction
of both alphabet and production rules at every bifurcation.
The self-similarity follows immediately from the absence of
a fixed starting point. It implies that physical applications at
some particular level will be matched by applications at other
levels, and that the scaling up from small to larger and more
complex systems is governed by a principle analogous to the
renormalization group in physics, where the structures which
the rewrite process generates are maintained by new emergent
physical principles.
The observer, who is always placed within the zero
totality system (variously described as “the universe,” “nature,”
or “reality”), must necessarily start from the minimum
representation of a zero totality alphabet, say (R, R∗ ). The stages
in the process are all zeros. During the process we go from
one zero cardinality or totality to the next. The cardinalities are
like Cantor’s cardinalities of infinity, but are cardinalities of zero
instead. We ensure that they are cardinalities by always including
the previous cardinality or alphabet. So (R, R∗ , A, A∗ ) includes
(R, R∗ ). Because they are cardinalities or zero totality alphabets
(descriptions of the universe in physical terms), the process is
always holistic. We have to include everything.
Now, we have to assume that (R, R∗ ) is not necessarily the
beginning, though it is the point where we as observers start
from. So, this already “bifurcated” state will have started from
a previous alphabet, which we assume we can’t access directly.
If we describe this as R, then the ∗ or R∗ character creates the
doubling process. Before we create (R, R∗ ), we have to assume
that (R) is a zero totality alphabet, but it is a zero we can’t access
because we have no structure for it. In effect, we are trying to posit
an ontology that exists before the epistemology or observation,
begins with (R, R∗ ). So, we assume that it must happen without
being able to observe it.
Duality is intrinsic to the process. The operation () () ⇒ (,)
describes how we proceed from one description of the entire
or universe zero totality alphabet to the next alphabet in the
hierarchy. The (,) becomes a “bifurcation” or “doubling.” So
(R, R∗ )(R, R∗ ) ⇒ (R, R∗ , A, A∗ )
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(8)

can be expressed as (R, R∗ , A, A∗ ), where (R, R∗ ) (R, R∗ ) is
the bifurcation that creates the new zero cardinality (R, R∗ , A,
A∗ ), in effect transforming the second (R, R∗ ) into the new (A,
A∗ ). All the “doublings” or “bifurcations” in the process are, in
this sense, similar to the initial creation of (R, R∗ ), even when
they involve complexification or dimensionalization rather than
conjugation. So
(R, R∗ , A, A∗ )(R, R∗ , A, A∗ ) ⇒ (R, R∗ , A, A∗ , B, B∗ , AB, AB∗ ) (9)
can also be written as
(R, R∗ , A, A∗ )(R, R∗ , A, A∗ ) ⇒ (R, R∗ , A, A∗ ,

B, B∗ , AB, AB∗ )
(10)
where the original alphabet (R, R∗ , A, A∗ ) and its dual (B, B∗ , AB,
AB∗ ) are created by a process similar to the one which created R
and R∗ . For practical purposes, a new character (B) is introduced.

Application to Physics
In applying to physics, we note that the universal rewrite process
creates successive models for a zero-totality universe. This is what
we mean by physical parameters. We can recognize the algebras
of the fundamental parameters mass, time, charge and space as
being, respectively, scalar, pseudoscalar, quaternion, and vector,
exactly as are generated by the first four alphabets of the universal
rewrite system. Here, mass is the source of gravity and includes
energy, while charge is a term used to represent the sources of the
three non-gravitational interactions [3]. The four alphabets seem
to be independent descriptions of the universe which must be
simultaneously true, as should all subsequent alphabets following
these.
i
i,j,k
1
i,j,k
time
space
mass
charge
pseudoscalar vector scalar quaternion
Now, if we combine the algebras of these quantities, we obtain
the 64-part algebra isomorphic to the Dirac gamma algebra that
we tabulated in Section An Algebra for the Rewrite Process. But
the 8 units of time, space, mass, and charge are not the minimum
number of starting units to generate the 64-part algebra. This is,
in fact, 5 and its construction always involves the breaking of the
symmetry of one of the two 3-D components, space or charge.

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Typically, we “combine” one of the units of charge with each of
those of time, space and mass, to obtain:
ik
energy
ikE
pseudoscalar

ii, ij,ik
momentum
iipx , ijpy , ikpz
vector

j
rest mass
jm
scalar

(operator acting on unique phase term)2 = amplitude2 = 0

(12)

The units ik, ii, ij, ik, j correspond to the five base units of
the gamma algebra, and the new terms, energy, momentum and
rest mass, can be seen to take on aspects of the original charge
units, together with the pseudoscalar, vector and scalar properties
of their other parent quantities. Now, when we combine the
momentum terms into a single vector p and take the complete
package (ikE + ip + jm) to represent the properties of a
fundamental physical unit (particle or fermion), we find that we
can, in this case, immediately solve the problem of indefinite
extension of the alphabets since (ikE + ip + jm) is a nilpotent
or square root of zero. The equation
(ikE + ip + jm)(ikE + ip + jm) = E2 − p2 − m2 = 0

(11)

is then simply the relativistic and quantum mechanical
conservation of energy and momentum. So, if we take (ikE +
ip + jm) as incorporating all the alphabets needed to create a
repetitive sequence, when we seek to generate the next alphabet
by squaring, we will find that it is zeroed automatically, so zeroing
all higher alphabets which incorporate it, and we can describe the
world through an indefinite succession of such units.
Simultaneously when we create energy, momentum and rest
mass as concepts, the combination of time, space, mass, and
charge breaks the symmetry between the weak, strong and
electric charges, which then take up the algebraic characteristics
of their associated parameters:
ik
weak charge
‘ pseudoscalar

ii, ij, ik
strong charge
vector

j
electric charge
scalar

The reduction of the original 8 units to the composite set of 5 is,
in fact, a characteristic symmetry-breaking operation in nature,
which is found also in mathematics, chemistry and biology as well
as several other aspects of physics. In all these areas, 5 seems to
be the number at which symmetry is necessarily broken.

NILPOTENT QUANTUM MECHANICS
Nilpotent quantum mechanics is founded on a nilpotent operator
which can be expressed in the form (± ikE ± ip + jm), which
is an abbreviated expression for a row or column vector, whose
4 terms encompass the four sign variations in E and p. We can
use a canonical quantization procedure to replace E and p as
operators by E → i∂ /∂t, p → – i▽, for a free fermion, or by
covariant derivatives such as E → i∂ /∂t + e8 ..., p → – i▽ +
eA +..., for a fermion constrained by any number of potentials
of any type or even by curvature terms. The structure of the
operator then determines both the complete quantum behavior
of the fermion and also that of its environment or “vacuum” by

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defining a unique phase term which, when operated on, produces
an amplitude which squares to zero:

The process incorporates both Dirac and Klein-Gordon
equations in the form
(± ikE ± ip + jm) (± ikE ± ip + jm) → 0

(13)

where (± ikE ± ip + jm) can stand for either operator or
amplitude. This would make the Dirac equation for a free
fermion




∓ k∂/∂t ∓ ii∇ + jm ± ikE ± ip + jm e−i(Et−p.r) = 0 (14)
For a fermion under the constraint of a potential or any number
of potentials, the phase factor would take a different form but the
result would still be a term with the same structure as (± ikE ±
ip + jm) being squared to zero.

Characteristics of Nilpotent Quantum
Mechanics
Nilpotent quantum mechanics is relativistic and is concerned
with fermions. It shares all the standard characteristics of
relativistic quantum mechanics using the more conventional
formalisms of the Dirac equation, and can be easily transformed
into these formalisms using the one-to-one correspondence
between the algebraic operators and gamma matrices. However, it
also has some characteristics which only become apparent in this
mathematical form, but which are necessary for understanding
how the process can be scaled up in higher order systems.
Spin ½ and zitterbewegung are among the shared
characteristics and can be easily derived using variants of the
standard formalisms. Chirality (or the intrinsic left-handedness
of fermions and right-handedness of antifermions) emerges in
the same way. Fermion uniqueness or Pauli exclusion is obvious
in the nilpotent formalism, as any combination state of identical
fermions will automatically vanish. However, the nilpotent also
creates a completely new meaning for the concept. Because the
totality of experience is defined always to be zero, if we take a
fermion in any state, say (ikE + ip + jm), subject to any number
of constraints that can be built into its operator, and imagine
that we can create it from absolutely nothing, then the “vacuum”
which defines the rest of the universe for that fermion, must
be a kind of mirror image, −(ikE + ip + jm), so that both the
superposition and the combination of vacuum and fermion
remain at zero:
−(ikE + ip + jm) + (ikE + ip + jm) = 0
−(ikE + ip + jm) (ikE + ip + jm) = 0

(15)
(16)

To maintain this zero totality in all circumstances, any change
in either the fermion or its environment must be reflected in
a corresponding change in the other. In effect, this creates a
principle of self-organization which can be imagined in systems

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Marcer and Rowlands

on a much larger scale, and which will be identifiable by strongly
characteristic features which originate in the nilpotent structure
and the universal rewrite process.
Another significant aspect of quantum mechanics is that it
involves both locality and non-locality. The distinction between
the two processes is clear in the nilpotent form. Everything inside
the bracket is local; everything outside the bracket is non-local.
So the conservation laws of energy and angular momentum are
local; superpositions and combination states and interactions
with vacuum are non-local. Both processes, however, are holistic
in requiring the cooperation of the entire universe, and each
produces consequences which affect the other. In nilpotent
quantum mechanics, the individual fermion conserves its energy
only with respect to the rest of the universe. The fermion is
an open system and intrinsically dissipative. The first law of
thermodynamics must be accompanied by the second.
Fermions are, in a very fundamental way, incomplete. They
have half-integral spin, are only observable when interacting in
a pairing with other fermions, and are square roots of algebraic
operators which only have meaning when multiplied with other
objects of the same kind. In the nilpotent formalism, bosons
of spin 1 and spin 0 are formed from fermion-antifermion
combinations of the form (± ikE ± ip + jm) (∓ ikE ± ip
+ jm) and (± ikE ± ip + jm) (∓ ikE ∓ ip + jm), while a
fermion-fermion combination can exist in the form (± ikE ±
ip + jm) (± ikE ∓ ip + jm) in Cooper pairs, Bose-Einstein
condensates and other applications of Berry phase. All these
expressions become scalars when multiplied out. All the tendency
for aggregation in nature can be seen as stemming from the need
for fermions to acquire partners to remove this incompleteness,
and it can be linked to the action of a harmonic oscillator, of
which the zitterbewegung is a special instance. The same pattern
emerges at higher levels, suggesting that the nilpotent model
applies well beyond the direct application of quantum principles.
It is very likely that a major role in providing the “staircase”
that we hope to show leading from the smallest systems to
the largest will be provided by the renormalization group
procedure.

Dual Space
The most significant aspect of the structure is that it incorporates
two full vector spaces with the full Clifford algebra of each.
The 64-part algebra requires a Clifford vector algebra for space
commutatively combined with its three subalgebras, representing
time, mass, and charge. If we take these three subalgebras
together, we find that they have the mathematical characteristics
of another vector space, entirely commutative to the first. This
“space,” however, as a composite of three other parameters, is
not an observable quantity. So, the nilpotent structure emerges
from combination of two vector spaces, only one of which is
observable.
We can call the unobserved space “vacuum space,” and its
effects are immediately apparent in spin ½ and the 4-component
structure of the fermion wavefunction. Here, the fermion also
includes two terms associated with antifermion states. These are
a manifestation of the fermion’s vacuum, and are responsible for
the fermion spending half its time as a real particle and half as a

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Nilpotent Quantum Mechanics: Analogs and Applications

vacuum particle (zitterbewegung), which is also one of many ways
of accounting for the fermion’s ½ spin.
Another way of looking at this is to relate it to Berry phase, and
to attribute this to the fact that the fermion is a singularity with
respect to ordinary space. As is well-known, Berry or geometric
phase can be described in purely topological terms. If we parallel
transport a vector around any complete circuital path in ordinary
or simply-connected space, we can expect it to leave the vector
pointing in the same direction at both beginning and end of the
circuit. However, if the space of the circuit contains a singularity
or is multiply-connected, then the vector will gain a phase change
of π and end up pointing in the opposite direction from its
starting position.
Spin ½ could be seen as indicating that the fermion singularity
rotates in its own multiply-connected space. So, we can attribute
the same effect to the fact that the fermion is defined as a
singularity and that it is defined by a nilpotent connection
between two spaces, leading to the conclusion that the dual
space structure is actually responsible for the existence of discrete
matter in the form of physical singularities. In our understanding,
the Berry phase/spin ½/zitterbewegung results from defining a
localized point particle simultaneously with defining the nonlocalized vacuum that determines its relation to the rest of
the universe, and that carries the information about its future
evolution. We can consider Berry phase to be a particularly
significant indicator of the presence of some kind of dual space,
nilpotent-related behavior, especially in systems subject to selforganization.

The Holographic Principle
The nilpotent dual spaces are genuinely dual, in that they contain
precisely the same information, though in different forms. This
duality has many manifestations. For example, the uniqueness
of the nilpotent (ikE + ip + jm) and Pauli exclusion could be
determined by the “direction” of a line drawn from the origin if
iE, p, and m re represented as coordinates on the quaternion axes
k, i, and j. Alternatively, we could express Pauli exclusion by the
more conventional method of defining fermion wavefunctions as
antisymmetric. This leads to a truly remarkable result if we take
(ψ 1 ψ 2 – ψ 2 ψ 1 ) for two fermions in the nilpotent formalism:
(± ikE1 ± ip1 + jm1 ) (± ikE2 ± ip2 + jm2 )

(17)

−(± ikE2 ± ip2 + jm2 ) (± ikE1 ± ip1 + jm1 )
= 4p1 p2 − 4p2 p1 = 8ip1 × p2 ,

for this only has a non-zero value if the fermion spins are oriented
in different directions. In effect, the complete information about
a fermion state is contained in its instantaneous spin direction,
or in the plane to which this is perpendicular. In principle, the
orientation of the fermion in real space and in the “vacuum
space” created by the quaternion axes k, i, and j carries the same
information. Exactly the same duality occurs in the derivation
of spin ½ either from the anticommutativity of the momentum
operator, which uses real space, or from the Thomas precession,
which uses vacuum space, and the duality again informs the
holographic principle.

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Marcer and Rowlands

The holographic principle, in which the entire information
about a system is found on the bounding area, is thought to
be a significant organizing principle for many systems. We have
already considered it as “a characteristic signature of a nilpotent,
self-organizing system with its planar fractality” [1]. Essentially,
it uses the information coded in the E and p terms of the operator
(ikE + ip + jm), that is, in two components of the vacuum space,
as nilpotency makes the third term redundant. This then becomes
equivalent to using the information coded in two components of
the dual real space. Significantly, this can also be coded in one
dimension of space and one of time, which would be equivalent to
using the vacuum space. As space and momentum are conjugate
variables, area is also a conjugate of angular momentum, and
(ikE + ip + jm) is recognizably an angular momentum operator,
with the E term determining the handedness, p the direction
and m the magnitude. Since any system which conserves angular
momentum or which operates according to the holographic
principle (for example, galaxies acting collectively under gravity)
can be expressed in this form, then any such system can be seen
as a direct analog of the nilpotent fermion, even though it is not
intrinsically quantum or relativistic.
The application of the nilpotent operator to the holographic
principle also suggests that it can itself be regarded as a quantum
hologram, with phase ikE, amplitude ip and reference phase jm.
Again, we can recover the entire structure from just two terms,
for example, phase and reference phase. Quantum holography
has now been officially recognized as occurring in the case of
“quantum holographic encoding in a two-dimensional electron
gas” [4], but the work of Walter Schempp has already shown
that it has extensive practical application in Magnetic Resonance
Imaging based on harmonic analysis on the 3D Heisenberg Lie
group [5]. The universal rewrite system shows that the repeating
unit that we need for the description of a quantum or quantumlike system is a double vector space. The two three dimensional
spaces make quantum holography possible via Fourier transform
action, and relate to the 3D Heisenberg Lie Group and its
nilpotent Lie algebra and their dual/inverses [1].
The holographic paradigm is particularly significant in that
the wavefunction is defined only up to an arbitrary fixed phase,
which provides a direct meaning for the quantum vacuum in
quantum field theory, as ensuring that only relative phases, which
encode the 3 + 1 space-time geometries, can be measured.
This phase becomes the fixed, though arbitrary, measurement
standard for all subsequent measurements, and acts as the
holographic basis for a universal and self-organized quantum
process in which new fermionic states of matter are produced.
After each new emergence, a new arbitrary standard is created,
providing a complete history, i.e., hologram or holographic
record of past events, which our senses perceive as an unending
irreversible evolution. Nature allows us to use our arbitrary
standard as the new beginning of a rewrite process (as in
Section Characteristics of the Rewrite Process) and to conceive
of how self-organization can take place at each new level of
complexity. The universal rewrite system implies that the only
valid mathematical representations of nilpotent quantum-like
systems are all automorphisms of the universe itself, and that this
is the mathematical meaning of quantum entanglement [6].

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Nilpotent Quantum Mechanics: Analogs and Applications

SELF-ORGANIZATION: A NEW
APPLICATION
The universal rewrite system provides a blueprint for selforganization mediated through the nilpotent relation between
the defined system and the rest of the universe, which
emerges in this form of quantum mechanics. The particular
characteristics of nilpotent quantum mechanics provide us with
a number of identifiers which we have already linked to selforganization, citing specific examples, and which include double
3-dimensionality; a five-fold broken symmetry; geometric phase;
spin ½ or equivalent double helical structure; uniqueness of the
objects and unique birthordering; irreversibility; dissipation;
chirality, harmonic oscillator mechanism, zitterbewegung;
fractality of dimension 2; the holographic principle and quantum
holography [1]. Self-organization appears to be very general in
nature and an almost obvious consequence of a universal rewrite
system which reappears at each new level of complexity, so we
can expect to find increasing evidence of such identifiers in
systems that have been shown to be self-organizing.
A completely new application emerged immediately after our
last QI presentation [1]. Our previous work has indicated that
geometrical or Berry phase is a particularly significant identifier
of nilpotent-like behavior in a system that need not necessarily
be quantum. In an earlier paper, we wrote, concerning nilpotent
structure (where X 2 = 0): “each X 2 signifies a return (in terms
of a corresponding unique dual Dirac annihilation operator) to
the quantum mechanical vacuum state which takes the form of a
universal attractor of fractal dimension 2 ..., where the uniqueness
of each of the nilpotent quantum mechanical Dirac operators is
carried by means of quantum phase, in the form a unique gauge
invariant Berry/geometric phase able to encode the requisite
relativistic 3 + 1 space time geometric information about the
unique fermion state vector, and is ‘scale free”’ [7].
Now, new research, by Kaschube et al. published just after
“The Logic of Self-Organizing Systems,” shows that the neurons
in the visual cortex in the brain of three distantly-related
mammals have a quasiperiodic structure. Orientations of the
neurons in the flat sheets of the cortex change continuously,
repeating over a length known as the “map period” (λ), while
appearing to converge on centers known as “pinwheels,” while
the pinwheel density per λ2 appears to equal π to within a
few percent [8]. According to Miller, writing in the same issue:
“The result offers insight into the development and evolution
of the visual cortex, and strongly suggests that key architectural
features are self-organized rather than genetically hard-wired”
(our emphasis). Miller also says that “The universality of
self-organizing behavior provides a simple and compelling
explanation for the arrival of widely divergent evolutionary lines
at this common design [9].”
In our interpretation, π might well appear in the density of the
squared “map period” of the neurons because the spatial structure
of the system requires a geometrical phase. If the “pinwheel” is
taken as a “singularity” in the physical space, then we need a
double circuit through the “map period” or cycle of orientations
to re-establish the original phase state. The singularity would
then generate a double map period (2λ) in any direction of the

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Marcer and Rowlands

two-dimensional cortical sheet, and each pinwheel singularity
would be situated in a circle with radius length λ in this twodimensional space, creating a pinwheel density/λ2 of π. This
would coincide directly with our proposal that a characteristic
structure for the space of self-organizing systems at all levels
of complexity results from a dual vector system, or equivalent,
for which a geometrical phase of π becomes an identifying
feature.
This is referred to in many publications and very explicitly in
the biological context, with the relevant identifying structures, in
“A Computational Unification of Scientific Law” and references
therein [10]. The underlying Clifford algebra suggests that
analogous mathematical models are also possible, one of which
is the Klein bottle structure proposed by one of us in earlier
work [3]. This has been developed further by Rapoport, along
with the appearance of pinwheel structures and the appearance
of the identifying π which we associate with the Berry phase,
and Rapoport has proposed extensions of the analogy to many
seemingly unrelated areas [11, 12]. The fundamental dualities
involved can be expressed in many different ways and have
been discussed by the authors as the universal basis for physical,
chemical, biological, and other systems and their organization
in many previous publications. Essentially, where energy is
a conserved or near-conserved quantity in any structure, or
where there is a definable energy flow, a nilpotent relation
can be found between that structure and the rest of the
universe, and the analogies presented in Zero to Infinity and
earlier works will automatically apply, together with certain
identifying characteristics [3]. The structure of the visual cortex
is just one example where our prior predictions appear to
have been vindicated in a seemingly unexpected and visually
striking way. It is our belief that examination of other selforganizing systems will reveal the presence of other characteristic
identifiers of a structure analogous to nilpotent quantum
mechanics.

CONCLUSION
Self-organization in Nature has been posited as resulting from
a universal rewrite system, which manifests itself at each new
structural level. In this system, the totality of the entire universe
or anything that can be applied universally is taken to be
zero. New zero totality structures or alphabets emerging from a
previous one always include it, leading to what has been described
here as a succession of alphabets, with zero cardinality by
analogy with the well-known succession of infinite cardinalities
in mathematics. The process is universal, so is not confined
to specific interpretations, but one such interpretation is an
algebraic series which becomes a form of Clifford algebra, or
an infinite series of sets of quaternion units, with the full set
of terms produced by squaring out or multiplying to a higher
order. The series of zero totalities has a fractal quality in that
combinations of all the alphabets in the series up to any order,
as independent units, leads to an alphabet higher up in the series.
This appears to be applicable to physics where the first few terms

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Nilpotent Quantum Mechanics: Analogs and Applications

in the series correspond to the successive algebraic properties
of the fundamental physical parameters mass, time, charge and
space.
A combination of these leads to a higher algebra which
appears to correspond to that used in the Dirac equation of
relativistic quantum mechanics, which describes the fermionic
state, the only known fundamental entity in physics. Interpreting
this algebra as a group of order 64, allows us to select sets of
5 generators for the entire combination, which we can show
correspond precisely to the algebraic terms that define the Dirac
state and that we conventionally identify as the gamma matrices.
In addition, the combination of the terms as used in physics
has only nilpotent solutions, squaring to zero, suggesting that all
higher alphabets incorporating these will automatically produce
zero squared and higher order products as well.
The higher order alphabet which incorporates all the alphabets
corresponding to the parameters appears to be equivalent to
that which would be produced by a double vector or dual
space. The nilpotent structure of the fermion (which can also
be derived from the conventional form of the Dirac equation
using gamma matrices) immediately explains Pauli exclusion
and interprets vacuum as corresponding to the “rest of the
universe” (zero totality—fermion) which allows a fermion to
exist in any particular state, with the fermion and vacuum
occupying the two “spaces” required by our algebra. Quantum
mechanics can then be structured as the interaction between
a nilpotent fermion and the rest of the universe acting like a
mirror image creating a totality of zero. The many powerful
applications of this kind of quantum mechanics have already
been extensively described [3]. If we now interpret the nilpotent
fermion plus vacuum combination as an example of a more
universal condition, produced by the universal rewrite system, we
can extend the application to self-organizing systems in general,
and we suggest, among other things, that it is the explanation of
the holographic principle being applicable to such systems. We
also suggest how it should apply to biological systems, giving an
example from the structure of the visual cortex, which we propose
is an example of the Berry phase which results from the system
and its entire environment occupying two different mathematical
“spaces.”

AUTHOR NOTE
This paper is a revised and expanded version of an AAAI
technical report on “The ‘Logic’ of Self-Organizing Systems”
(2010-08-020). The authors hold the copyright and no
permission is required from AAAI for the use and reproduction
of material from this report.

AUTHOR CONTRIBUTIONS
PR: The original ideas of universal rewrite system, the associated
algebra and nilpotent quantum mechanics. Joint contribution
with PM on recognizing the wide application of these ideas in
particular to many areas outside of physics.

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Marcer and Rowlands

REFERENCES
1. Marcer P, Rowlands P. The ‘Logic’ of Self-organizing Systems. AAAI Technical
Reports 2010-08-020 (2010).
2. Hestenes D. Space-Time Algebras. New York, NY: Gordon and Breach (1966).
3. Rowlands P. Zero to Infinity: The Foundations of Physics. Singapore;
Hackensack, NJ: World Scientific (2007).
4. Moon CR, Mattos LS, Foster BK, Zeltzer G, Manoharan HC.
Quantum holographic encoding in a two-dimensional electron
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08.415
5. Schempp W. Quantum holography and neurocomputer architectures. J Math
Imaging Vis. (1992) 2:279–326.
6. Marcer P, Rowlands P. The Grammatical universe and the laws of
thermodynamics and quantum entanglement. AIP Conf Proc. (2010)
1303:161–7. doi: 10.1063/1.3527151
7. Marcer P, Rowlands P. Further evidence in support of the universal nilpotent
grammatical computational paradigm of quantum physics. AIP Conf Proc.
(2010) 1316:90–101. doi: 10.1063/1.3536456
8. Kaschube M, Schnabel M, Löwel S, Coppola DM, White LE, Wolf
F. Universality in the Evolution of orientation columns in the
visual cortex. Science (2010) 330:1113–6. doi: 10.1126/science.11
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9. Miller KD. π = visual cortex. Science (2010) 330:1059–60.
doi: 10.1126/science.1198857
10. Marcer P, Rowlands P. A Computational Unification of Scientific Law: spelling
out a universal semantics for physical reality. In: Amoroso RL, Kauffman
LH, Rowlands P, editors. The Physics of Reality Space, Time, Matter, Cosmos.
Singapore; Hackensack, NJ: World Scientific (2013), p. 50–9.
11. Rapoport DL. Klein bottle logophysics, self-reference, heterarchies, genomic
topologies, harmonics and evolution. Quantum Biosyst. (2016) 7:1–73,
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J Phys. (2013) 437. doi: 10.1088/1742-6596/437/1/012024
Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Marcer and Rowlands. This is an open-access article distributed
under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original
author(s) or licensor are credited and that the original publication in this journal
is cited, in accordance with accepted academic practice. No use, distribution or
reproduction is permitted which does not comply with these terms.

July 2017 | Volume 5 | Article 28 | 96

REVIEW
published: 28 June 2016
doi: 10.3389/fphy.2016.00026

Quantum Probabilistic Models
Revisited: The Case of Disjunction
Effects in Cognition
Catarina Moreira * and Andreas Wichert
Instituto Superior Técnico/INESC-ID, Oeiras, Portugal

Edited by:
Emmanuel E. Haven,
University of Leicester, UK
Reviewed by:
Jan Broekaert,
Vrije Universiteit Brussel, Belgium
Irina Basieva,
General Physics Institute, Russia
*Correspondence:
Catarina Moreira
catarina.p.moreira@ist.utl.pt
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 22 March 2016
Accepted: 10 June 2016
Published: 28 June 2016
Citation:
Moreira C and Wichert A (2016)
Quantum Probabilistic Models
Revisited: The Case of Disjunction
Effects in Cognition. Front. Phys. 4:26.
doi: 10.3389/fphy.2016.00026

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Recent work in cognitive psychology has revealed that quantum probability
theory provides another method of computing probabilities without falling into the
restrictions that classical probability has in regard to modeling cognitive systems and
decision-making. This enables the explanation of paradoxical scenarios that are difficult,
or even impossible, to explain through classical probability theory. In this work, we
perform an overview of the most important quantum models in the literature that are used
to make predictions under scenarios where the Sure Thing Principle is being violated
(the Quantum-Like Approach, the Quantum Dynamical Model, the Quantum Prospect
Theory and Quantum-Like Bayesian Networks). We evaluated these models in terms of
three metrics: interference effects, parameter tuning and scalability. The first examines if
the analyzed model makes use of any type of quantum interferences to explain human
decision-making. The second is concerned with the assignment of values to a large
number of quantum parameters. The last one consists of analyzing the ability of the
models to be extended and generalized to more complex scenarios. We also studied
the growth of the quantum parameters when the complexity and the levels of uncertainty
of the decision scenario increase. Finally, we compared these quantum models with
traditional classical models from the literature. We conclude with a discussion of the
manner in which the models addressed in this paper can only deal with very small
decision problems and why they do not scale well to larger, more complex decision
scenarios.
Keywords: quantum cognition, quantum-like approach, quantum dynamical model, quantum prospect theory,
quantum-like Bayesian networks

1. INTRODUCTION
The process of decision-making is a research field that has always triggered a vast amount of
interest among several fields of the scientific community. Throughout time, many frameworks
for decision-making have been developed, namely the Expected Utility hypothesis, which is
characterized by a specific set of axioms that enable the computation of the person’s preferences
with regard to choices under uncertainty [1]. Later, Savage [2] proposed an extension to this theory:
the Subjective Expected Utility theory. In this extension, uncertainty is described by subjective
probabilities, since not all uncertainty can be described using an objective probability distribution.
However, human behavior tends to violate the axioms of Expected Utility, leading to the well known
Allais paradox [3]. Human behavior also tends to violate the axioms of the Subjective Expected
Utility framework, leading to the Ellsberg paradox [4].

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Moreira and Wichert

1.1. Background
In the 70s, the cognitive psychologists Amos Tversky and Daniel
Kahneman decided to put to the test the axioms of the Expected
Utility hypothesis. They performed a set of experiments in
which they demonstrated that people usually violate the Expected
Utility hypothesis and the laws of logic and probability in
decision scenarios under uncertainty [5–9]. This means that,
when people need to make a decision under scenarios with high
levels of uncertainty, ambiguity and risk, they tend to violate the
laws of probability theory, leading to decision paradoxes [3, 4].
One of these paradoxes was demonstrated in the article
of Tversky and Shafir [10] and corresponds to the violation
of Savage’s Sure Thing Principle, also known as disjunction
effects, under the Prisoner’s Dilemma Game. This principle is
fundamental in classical probability theory and states that, if one
prefers action A over B under the state of the world X, and if
one also prefers A over B under the complementary state of the
world X, then one should always prefer action A over B even
when the state of the world is unspecified [2]. Violations of the
Sure Thing Principle imply violations of the classical law of total
probability [11].
Quantum cognition has emerged as a research field that aims
to build cognitive models using the mathematical principles
of quantum mechanics. Given that classical probability theory
is very rigid in the sense that it poses many constraints and
assumptions (single trajectory principle, obeys set theory, etc.),
it becomes too limited (or even impossible) to provide simple
models that can capture human judgments and decisions since
people are constantly violating the laws of logic and probability
theory [12–14].
In this sense, psychological (and cognitive) models benefit
from the usage of quantum probability principles because they
have many advantages over classical counterparts [15]. They
can represent events in vector spaces through a superposition
state, which comprises the occurrence of all events at the same
time. In quantum mechanics, the superposition principle refers
to the property that particles must be in an indefinite state.
That is, a particle can be in different states at the same time.
Under a psychological point of view, a quantum superposition
can be related to the feeling of confusion, uncertainty or
ambiguity [16]. This vector space representation does not obey
the distributive axiom of Boolean logic and to the law of total
probability. This enables the construction of more general models
that can mathematically explain cognitive phenomena such as
violations of the Sure Thing Principle [17, 18], which is the
focus of this study. Quantum probability principles have also
been successfully applied in many different fields of the literature,
namely in biology [19, 20], economics [21, 22], perception [23,
24], jury duty [25], etc.
One of the pioneering contributions to the Quantum
Cognition field comes from the works of Aerts and Aerts
[26]. The authors designed a quantum machine, which consists
in a particle that can move across the surface of a sphere.
An elastic, representing some experiment is introduced in
this sphere. The particle then moves orthogonally to the
elastic and the elastic breaks uniformly into two parts. With

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Quantum Probabilistic Models Revisited

this geometric representation, one can easily compute the
probabilities of the particle falling into each side of the elastic.
The model was extended with an ǫ parameter that represents
the evolution from a quantum structure to a classical one.
This parameter varies between [0, 1], where 0 corresponds
to maximum lack of knowledge (quantum structure) and
1 to zero lack of knowledge (classical knowledge). Between
this interval, there is the possibility of exploring other types
of structures that are neither classical nor quantum. The
authors also made several experiments to test the variation
of probabilities when posing yes/no questions. According to
their experiment, most participants formed their answer at the
moment the question was posed. This behavior goes against
classical theories, because in classical probability, it would be
expected that the participants have a predefined answer to the
question and not form it at the moment of the question. A
further discussion about this study can be found in the works
of Aerts [27–29], Gabora and Aerts [30], and Aerts et al.
[31].
In other subsequent works, namely in Aerts [32], the
author uses the formalisms of quantum mechanics in order
to accommodate disjunction effects. The author, represents
concepts as vectors and membership weights as quantum
weights, in a complex Hilbert Space. By using quantum
interference effects and quantum superpositions, the author was
able to model accurately the disjunction of concepts present in
experimental data.

1.2. The Article’s Main Statement
In this article, we provide an overview and discussion of the
most important state-of-the-art quantum cognitive models that
are able to explain the paradoxical findings of experiments
that violate the Sure Thing Principle (ex: the Prisoner’s
Dilemma game [33]). We conduct a deep comparison of
and discussion on several quantum models: the QuantumLike Approach [34], the Quantum Dynamical Model [35],
the Quantum Prospect Decision Theory [36] and Quantum
Bayesian Networks [37–40]. We discuss these models in terms
of three metrics: (1) incorporation of quantum interference
effects, (2) how to find values for quantum parameters,
and (3) scalability of the model for more complex decision
problems.
The first metric checks if the model uses quantum interference
effects to predict actions chosen under uncertainty. Following the
work of Yukalov and Sornette [36], toward uncertainty, human
beings tend to have aversion preferences. They prefer to choose
an action that brings them a certain but lower propensity/utility
instead of an action that is uncertain but can yield a higher
propensity/utility [41]. This can be simulated through quantum
interference effects, in which one outcome is enhanced (or
diminished) toward the opposite outcome.
The second metric takes into account the problem of finding
values for quantum parameters. In quantum mechanics, a
quantum state is modeled by probability amplitudes [42]. These
amplitudes are a component of the wave function and this
wave function represents a quantum state. Associated with each

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Moreira and Wichert

probability amplitude is a quantum parameter representing the
phase of the wave. The interpretation of this parameter under
the psychology literature is still not clear, although various works
have presented interpretations [17]. Moreover, when applying
quantum principles to cognition (or to any other subject), one
will need to set these quantum parameters in such a manner
that they will lead to accurate predictions. In this metric, we
will check how easy it is for the analyzed models to set these
parameters.
The third and last metric consists of determining if the model
can be extended to more complex scenarios. Although there
are many experiments that report violations of the Sure Thing
Principle [17, 35, 43, 44], these experiments consist of very small
scenarios that are modeled by, at most, two random variables.
Therefore, many of the proposed models in the literature are only
effective under such small scenarios and become intractable (or
even cannot be applied) under more complex situations. These
metrics will be analyzed with more detail in Section 8 of the
present work.
It is important to note that the goal of this work is
the following: we have collected a set of models from
the literature that attempt to tackle violations of the
Sure Thing Principle in a quantum fashion, and then we
compare the collected models. For this comparison, we just
show, through a mathematical description of each model,
their advantages and disadvantages. That is, we compare
these models with the three metrics proposed: number
of parameters involved in the model, the scalability of
the quantum interference effects and their usage. We will
also show that classical models also suffer from the same
parameter growth problem as quantum approaches. However,
because these models must obey set theory and the laws of
classical probability, it is not possible to use them to make
predictions in situations where the Sure Thing Principle is being
violated.

1.3. Outline
We will start this article with a motivational problem, in which
the Sure Thing Principle is found to be violated under the
Prisoner’s Dilemma Game (Section 2). In Section 3, we will show
that a classical approach cannot accommodate violations of the
Sure Thing Principle because these approaches obey set theory
and consequently the laws of probability theory. We will make
a full step-by-step description of the most influential models of
the literature. We will show how one could apply them to predict
the results concerned with violations of the Sure Thing Principle
in the Prisoner’s Dilemma Game. In Section 4, we will cover the
Quantum-Like Approach [34]. In Section 5, we will analyze the
Quantum Dynamical Model [17]. In Section 6, we will describe
the Quantum Prospect Decision Theory [36]. In Section 7, we will
provide an overview of Quantum-Like Bayesian Networks [37–
40]. We then engage in a deeper discussion of these approaches
and give thought to the advantages/disadvantages of each
model in Section 8. We finish this article by presenting the
main conclusions of this work by providing some insights
regarding various trends in quantum probabilistic models
(Section 9).

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Quantum Probabilistic Models Revisited

2. VIOLATION OF THE SURE THING
PRINCIPLE: THE PRISONER’S DILEMMA
GAME
The Prisoner’s Dilemma game corresponds to an example of
the violation of the Sure Thing Principle. In this game, there
are two prisoners who are in separate solitary confinements
with no means of speaking to or exchanging messages with
each other. The police offer each prisoner a deal: they can
either betray each other (defect) or remain silent (cooperate).
For understanding purposes, we provide an example of a payoff
matrix for the Prisoner’s Dilemma Game (Figure 1). The payoff
matrix represents the rewards that each player receives for a given
action.
The dilemma of this game is the following. Taking into
account the payoff matrix, the best choice for both players
would be to cooperate. However, the action that yields a bigger
individual reward is to defect. If player A has to make a choice,
he has two options: if B has chosen to cooperate, the best
option for A is to defect because he will be set free; if B has
chosen to defect, then the best action for A is also to choose
to defect because he will spend less time in jail than if he
cooperates.
To test the veracity of the Sure Thing Principle under
the Prisoner’s Dilemma game, several experiments were
performed in the literature in which three conditions were
tested:
• Participants were informed that the other participant chose to
defect.
• Participants were informed that the other participant chose to
cooperate.
• Participants had no information about the other participant’s
decision.
Table 1 summarizes the results of several works in the literature
that have performed this experiment using different payoffs. Note
that all entries of Table 1 show a violation of the Sure Thing
Principle and, consequently, the law of total probability. In a

FIGURE 1 | Example of a payoff matrix for the Prisoner’s Dilemma
Game.

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Quantum Probabilistic Models Revisited

classical setting, assuming neutral priors, it is expected that:




Pr P2 = Defect | P1 = Defect ≥ Pr P2 = Defect


≥ Pr P2 = Defect | P1 = Cooperate
However, this is not consistent with the experimental results
reported in Table 1. Note that Pr(P2 = Defect | P1 = Defect)
corresponds to the probability of the second player choosing
the Defect action given that he knows that the first player chose
to Defect. In Table 1, this corresponds to the entry Known to
Defect. In the same manner, Pr(P2 = Defect | P1 = Cooperate)
corresponds to the entry Known to Cooperate. The observed
probability during the experiments concerned with player 2
choosing to defect, Pr(P2 = Defect), corresponds to the
unknown entry of Table 1 because there is no evidence regarding
the first player’s actions. Finally, the entry Classical Probability
corresponds to the classical probability Pr(P2 = Defect), which
is computed through the law of total probability assuming neutral
priors (a 50% chance of a player choosing either to cooperate or
to defect):




Pr P2 = Defect = Pr P1 = Defect


·Pr P2 = Defect|P1 = Defect




+ Pr P1 = Cooperate · Pr P2 = Defect|P1 = Cooperate
For simplicity, we will use the following notation. The probability
of Player 2 choosing to defect will be Pr ( P2 = D ). In the same
way, the probability of Player 2 choosing to cooperate will be
Pr (P2 = C).
In the next sections, we will introduce the most representative
models in the quantum cognition literature that are able to solve
problems concerning violations of the Sure Thing Principle and
also show that a classical model cannot accommodate violations
of the Sure Thing Principle. We will also demonstrate how
quantum models work when trying to predict the probabilities
of the average results of the Prisoner’s Dilemma Game, reported
in Table 1.

3. A CLASSICAL MARKOV MODEL OF THE
PRISONER’S DILEMMA GAME
A Markov Model can be generally defined as a stochastic
probabilistic undirected graphical model that satisfies the Markov

property. This means that the probability distribution of the
next state depends on the current state and not on previous
states. These probabilistic models are very useful for modeling
systems that change states according to a transition matrix that
specifies some probability distribution or some transition rules
that depend solely on the current state.
One can apply a dynamical Markov process to model the
Prisoner’s Dilemma Game in the following manner. Having as
reference the work of Pothos and Busemeyer [17], the Prisoner’s
Dilemma is a 2-person game and can be modeled in a fourdimensional classical Markov model. Initially, the states can be
represented by all possible actions of the players: Cooperate (C)
and Defect (D). These are represented in a state vector in which
all possible actions are equally likely to be chosen:


  
DD
1
 D C   1 1
  
PI = 
 C D  =  1 · 4
CC
1
The probability of the second player choosing to Defect given
that the action of the other player is unknown is given by
Equation (1) and consists of the multiplication of this initial
probability state PI by a transition function T(t):
PF = T(t) · PI

(1)

The transition function T(t) is represented by a matrix
containing positive real numbers and with the constraint that
each row must sum to one (normalization axiom). In other
words, this matrix represents the new probability distribution
across the player’s possible actions over some time period t [17].
d
T(t) = K · T(t) ⇒ T(t) = eK.t
dt

(2)

In Equation (2), the matrix K corresponds to an intensity matrix.
It is a matrix representation of all payoffs of the players. A
solution to the above equation is given by T(t) = eK.t , which
allows one to construct a transition matrix for any time point
from the fixed intensity matrix. These intensities can be defined
in terms of the evidence and payoffs for actions in the task.
In other words, the intensity matrix performs a transformation

TABLE 1 | Works of the literature reporting the probability of a player choosing to defect under several conditions.
Literature

Known to defect

Known to cooperate

Unknown

Shafir and Tversky [33]

0.9700

0.8400

0.6300

0.9050

Crosson [45]a

0.6700

0.3200

0.3000

0.4950

Li and Taplin [46]b

0.8200

0.7700

0.7200

0.7950

Busemeyer et al. [47]

0.9100

0.8400

0.6600

0.8750

Hristova and Grinberg [48]

0.9700

0.9300

0.8800

0.9500

Average

0.8700

0.7400

0.6400

0.8050

a
b

Classical probability

corresponds to the average of the results reported in the first two payoff matrices of the work of Crosson [45].
corresponds to the average of all seven experiments reported in the work of Li and Taplin [46].

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Quantum Probabilistic Models Revisited

on the probabilities of the current state to favor defection or
cooperation, which are represented by the parameters µd and µc ,
respectively [17].
  

  

1 0
µ
1
0 0
µ 1
KAd =
⊗ d
KAc =
⊗ c
(3)
0 0
1 −µd
0 1
1 −µc

−1 µD
 1 −µD
KA = KAd + KAc = 
0
0
0
0


0
0
0
0 

−1 µC 
1 −µC

(4)

In the work of Pothos and Busemeyer [17], the authors proposed
the incorporation of dissonance effects to simulate the change
of mind to dissolve contradictory beliefs that a player can
experience. This is given by the parameter γ and corresponds to
the payoffs of the players (Equation 5).


−1 0 γ 0
 0 −γ 0 1 

KB = 
(5)
 1 0 −γ 0 
0 γ 0 −1
Thus, the final intensity matrix K is given by:


−2
µD
γ
0

 1 −γ − µD
0
1

K = KA + KB = 
1
0
−1 − γ
µC 
0
γ
1
−µC − 1

(6)

(7)

In Equation (7), we do not need to perform any normalization in
the end because the operation in Equation (1) together with the
intensity matrix K ensures that the values computed are already
probability values. Moreover, there is no possible combination
of parameters resulting from Equation (7) that will satisfy the
results observed in Table 1. This occurs because, although we
have parameterized the Markov Model, the model will always
satisfy the laws of classical probability theory. Thus, there is no
possible optimization that can predict the violation of the Sure
Thing Principle in such situations. This was already noticed in
the previous works of Pothos and Busemeyer [17] and Busemeyer
et al. [35].
In the next sections, we explain several quantum approaches
proposed in the literature that can accommodate violations ofthe
Sure Thing Principle.

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The Quantum-Like Approach has its roots in contextual
probabilities. This model was proposed by A. Khrennikov and
corresponds to a general contextual probability space from which
the classical and quantum probability models can be derived [34,
49].

4.1. Contextual Probabilities: The Växjö
Model
In the Växjö Model, the context relates to the circumstances that
form the setting for an event in terms of which it can be fully
understood, clarifying the meaning of the event. For instance, in
domains outside of physics, such as cognitive science, one can
have mental contexts. In social sciences, we can have a social
context. The same idea is applied to many other domains, such
as economics, politics, game theory, and biology.
Associated with a context, there is a set of observables. In
quantum mechanics, an observable corresponds to a self-adjoint
operator on a complex Hilbert Space. Under the Växjö Model,
these observables correspond to the set of possible events with
their respective values.
Prcontext = (C , O, π )

To compute the final probability of a player defecting, we
need to sum the components of the column vector PF that
correspond to the second player choosing the action Defect. Note
that the four components of the column vector PF correspond
to [ DD DC CD CC ], where C corresponds to Cooperate and D
to Defect. The first letter represents the action chosen by the
first player, and the second letter corresponds to the action of
the second player. Thus, the probability of player 2 choosing the
action Defect corresponds to the summation of the first and the
third components of the column vector PF :
Pr( P2 = Defect ) = PF[1st_dim] + PF[3rd_dim]

4. THE QUANTUM-LIKE APPROACH

(8)

For instance, for a context C ∈ C and for an observable a ∈ O
having values α, the probability of the value of one observable
is expressed in terms of the conditional (contextual) probability
involving the values of an observable. That is, the probability
distribution π is given by:
π(O, C ) = Pr( a = α | C )

(9)

If we move into the quantum mechanics realm, Equation (9) can
be interpreted as the selection with respect to the result a = α of
a measurement performed in a.
For the contextual probability model, the Växjö framework
corresponds to a model M described by M = (C , O, π(O, C )).
Again, C is a set of contexts, O is the set of observables,
and π(O, C ) corresponds to a probability distribution of some
observables belonging to a specific context.
In addition, assume for a context C ∈ C that there are two
dichotomous observables a, b ∈ O and that each of these
observables can take some values α ∈ a and β ∈ b, respectively.
The Växjö Model can be built from the general structure
of the quantum law of total probability. That is, the formula
is a combination of the classical probability theory with a
supplementary term called the interference term (Equation 10).
This term does not exist in classical probability and enables the
representation of interferences between quantum states.
Pr(b = β) = Classical_Probability(b = β) + Interference_Term
(10)
Under this representation, we can replace Classical_Probability
by the classical total probability and also replace the quantum
Interference_Term by a supplementary measure, represented by

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δ(β | a, C). Under the Växjö Model, the term δ(β | a, C)
corresponds to:

δ(β|a, C) = Pr(b = β) −

X
α∈a

Pr(a = α|C)Pr(b = β|a = α, C)

(11)
Equation (11) can be written in a similar way to the classical
probability in the following manner:

Pr(b = β|C) =

X
α∈a

Pr(a = α|C)Pr(b = β|a = α, C) + δ(β|a, C)

(12)
If we perform the normalization of the probability measure of
supplementary δ(β | a, C) by the square root of the product of all
probabilities, we obtain:

δ(β|a, C)
λθ = pQ
2
α∈a Pr(a = α|C)Pr(b = β|a = α, C)

(13)

From Equation (13), the general probability formula of the Växjö
Model can be derived. For two variables, it is given by:

Pr(b = β|C) =

X
α∈a

Pr(a = α|C)Pr(b = β|a = α, C)

+2λθ

sY

α∈a

Pr(a = α|C)Pr(b = β|a = α, C)
(14)

If we look closely at Equation (14), we can see that the first
summation of the formula corresponds to the classical law of
total probability. The second term of the formula (the one that
contains the λθ parameter) does not exist in the classical model
and is called the interference term.

4.2. The Hyperbolic Interference
Although the Quantum-Like Approach provides great
possibilities compared with the classical one, it appears that
it cannot completely cover data from psychology and that a
quantum formalism was not enough to explain some paradoxical
findings (see [50]), so hyperbolic spacesPwere proposed [51–53].
From Equation (14), if Pr(b = β) − α∈a Pr(a = α|C)Pr(b =
β|a = α, C) is different from zero, then various interference
effects occur. To determine which type of interference occurred,
one tests the Växjö Model for quantum probabilities. This can
be determined by normalizing the supplementary measure in a
quantum fashion, just as presented in Equation (13).
If we are under a quantum context, then the quantum
interference term will be:

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δ(β|a, C) = 2

sY

α∈a

Pr(a = α|C)Pr(b = β|a = α, C) cos(θ)

(15)
In a quantum context because the supplementary term δ(β | a, C)
is being normalized in a quantum fashion, then we automatically
know that the indicator term λθ will always have to be smaller
than 1 to obtain quantum probabilities, λθ
≤ 1. Thus,
under trigonometric contexts, the Växjö Model for quantum
probabilities becomes:
λθ = cos(θ ) → Pr(β|C) =
+2

sY

X

Pr(α|C)Pr(β|α, C)

α∈a

Pr(α|C)Pr(β|α, C) cos(θ)

(16)

α∈a

If, however, the probability Pr(b = β) was not computed
in a trigonometric space (that is, it is not quantum), then,
it is straightforward that the quantum normalization applied
in Equation (13) will yield a value larger than 1. Because we
are not in the context of quantum probabilities, the quantum
normalization factor will fail to normalize the interference
term and will produce a number larger than the normalization
factor. Under these circumstances, the Växjö Model incorporates
the generalization of hyperbolic probabilities, arguing that
the context in which these probabilities were computed was
Hyperbolic [49, 53, 54].
Under Hyperbolic contexts, the Växjö Model contextual
probability formula becomes:
λθ = cosh(θ ) → Pr(β|C) =
±2

sY

X

Pr(α|C)Pr(β|α, C)

α∈a

Pr(α|C)Pr(β|α, C) cosh(θ )

(17)

α∈a

In summary, according to the values computed by the indicator
function λθ , the Växjö Model enables the computation of
probabilities in the following contexts:
• If | λθ | = 0, then there is no interference, and the Växjö
Model collapses to classical probability theory.
• If | λθ | ≤ 1, then we fall into the realm of quantum
mechanics, and the context becomes a Hilbert space. The
indicator function is then replaced by the trigonometric
function cos( θ ).
• If | λθ | > 1, then we fall into the realm of hyperbolic
numbers, and the context becomes a hyperbolic space. The
indicator function is then replaced by the hyperbolic function
cosh(θ ).

4.3. Quantum-Like Probabilities as an
Extension of the Växjö Model
The probabilities that emerge from the Växjö model for
trigonometric spaces (i.e., quantum probabilities), do not provide
a complete description of a quantum system because it can violate
the positivity axiom of probability theory [49].

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In this sense, an algorithm was proposed in the literature
that extends the Växjö model and is able to accommodate the
positivity axiom. The algorithm proposed is the Quantum-Like
Representation Algorithm (QLRA), and it was proposed by
Khrennikov [55–59].
As already mentioned, quantum complex amplitudes can be
obtained from classical probability by using Born’s rule [60,
61]. In the QLRA, for any trigonometric context C, one
can simplify Born’s rule for two dichotomous variables using
(Equation 19) [49].
Pr(β|C) = Pr(α1 |C)Pr(β|α1 , C) + Pr(α2 |C)Pr(β|α2 , C) +
p
+2 Pr(α1 |C)Pr(β|α1 , C)
p
Pr(α2 |C)Pr(β|α2 , C) cos θ
(18)
Equation (18) can be simplified in the following manner:
p

Pr(β|C) =  Pr(α1 |C)Pr(β|α1 , C)
2
p

+ eiθβ|α,C Pr(α2 |C)Pr(β|α2 , C)

(19)

Equation (19) corresponds to the representation of the quantum
law of total probability through the Växjö model. In this equation,
the angle θβ|α,C corresponds to the phase of a random variable
and incorporates the phase of both A = α1 and A = α2 in the
following manner: θβ|α, C = θβ|α1 − θβ|α2 .
One should note that the Quantum-Like Approach can be
extended to more complex decision scenarios, that is, with
more than two random variables. However, this will lead to
the very difficult task of tuning an exponential number of
quantum θ parameters. Peter Nyman noticed this problem when
he generalized the Quantum-Like Approach for 3 dichotomous
variables [52, 62–64].

4.4. Modeling the Prisoner’s Dilemma using
the Quantum-Like Approach
If we want to compute the average probabilities reported in
Table 1 for the Prisoner’s Dilemma game, then we would need
to make the following substitutions to Equation (18):


Pr (α1 |C) · Pr (β|α1 , C) = Pr P1 = Defect|C


· Pr P2 = Defect|P1 = Defect = 0.5 × 0.87 = 0.435


Pr (α2 |C) · Pr (β|α2 , C) = Pr P1 = Cooperate|C


· Pr P2 = Defect|P1 = Cooperate
= 0.5 × 0.74 = 0.37

The main problem of the Växjö model and the Quantum-Like
Approach is that it can only address very small decision scenarios
and the fitting of the θ parameter has to be done fitted to
data. To compute

the probability of a player choosing to defect,
Pr P2 = Defect , one would proceed as follows:
Pr(P2 = Defect) = 0.435 + 0.37 + 2 ·

√
√
0.435 · 0.37 · cos(θ )

To achieve the observed result, θ must be equal to 1.7779 to
achieve the final probability Pr(P2 = Defect) = 0.64. However,
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this method does not provide any other means to find this θ
parameter except by extrapolating the observed data.

5. THE QUANTUM DYNAMICAL MODEL
In the works of Busemeyer et al. [11], Pothos and Busemeyer [17],
and Busemeyer et al. [35], the authors present a model to perform
quantum time evolution. This model requires the creation of a
doubly stochastic matrix, which represents the rotation of the
participants’ beliefs. The double stochasticity is a requirement to
preserve unit length operations and to obtain a probability value
that does not require normalization. The participants’ actions are
represented by a superposition vector with all possible actions:
[ψDD ψDC ψCD ψCC ], where C corresponds to Cooperate and D
to Defect.
The doubly stochastic matrix that the model requires can only
be computed by the use of an auxiliary Hamiltonian matrix,
which needs to be self-adjoint. For instance, to explain the
average results of the Prisoner’s Dilemma game, the Hamiltonian
matrix is given by Equation (20), where µD and µC correspond
to parameters representing the payoffs of the defect and cooperate
actions, respectively.
  

1
1 0
µ
1
q
⊗ D
0 0
1 −µD
1 + µ2D
  

1
0 0
µ
1
q
HAc =
⊗ C
0 1
1 −µC
1 + µ2C

 µD
√ 2 √1 2
0
0
1+µD
1+µD

 1

 √ 2 − √ µD 2
0
0

 1+µD
1+µD
HA = HAd + HAc = 
µ
1
C
√ 2 √ 2 
0

 0
1+µC
1+µC 

µ
1
√ 2 −√ C 2
0
0

HAd =

1+µC

1+µC

(20)
The dynamical model also takes dissonance effects into account.
That is, the participants might have been confronted by some
information that conflicted with his/her existing beliefs to
simulate the dissonance effect when the participants had to
decide on an action. Thus, the Quantum Dynamical Model makes
use of a second Hamiltonian matrix, HB.




+1 0 +1 0
0 0 0 0
 0 0 0 0 −γ
0 −1 0 +1 −γ



HBd = 
+1 0 −1 0 · √2 HBc = 0 0 0 0  · √2
0 0 0 0
0 +1 0 +1
 −γ

√
√
0 −γ
0
2
2


√ 
 0 √γ 0 −γ
2
2
HB = HBd + HBc = 
(21)
γ
 −γ

 √2 0 √2 0 
√
√
0 −γ
0 −γ
2

2

The general Hamiltonian matrix combines the matrices from
Equations (20) and (21). In the end, the final matrix needs to be
self-adjoint and, consequently, symmetric. To explain the average

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results of the Prisoner’s Dilemma game, the final Hamiltonian
matrix is given by:
H = HA + HB =
 −γ
√ + √ µD
√1 2
1+µ2D
1+µD
 2
γ

√
√1 2
− √ µD 2

2
1+µD
1+µD

−γ

√
0

2

−γ
√
0
2

γ
√

2

−γ
√
2

0

0

−γ
√
2
√1 2
1+µC
−γ
√ − √ µC
2
1+µ2C

+ √ µC
√

1+µ2C
1
1+µ2C










(22)
Next, we need to create a unitary matrix. In quantum mechanics,
a unitary matrix restricts the allowed evolution of quantum
systems, ensuring that the sum of probabilities of all possible
outcomes of any event is always 1. This means that the matrix
must be doubly stochastic (all rows and columns sum to 1).
In the Quantum Dynamical Model, this matrix encodes all
state transitions that a person can experience while choosing a
decision. A unitary matrix is computed by a differential equation
called Schrödinger’s equation:
δ
U(t) = −i · H · U(t) ⇒ U(t) = e−i·H·t
δt

(23)

The parameter t corresponds to the time evolution. Under the
Dynamical Quantum Model, this parameter was set to π/2,
corresponding to the average time that a participant takes to
make a decision (approximately 2 seconds) [17, 35]. Also, in the
book of Busemeyer and Bruza [16], the authors state that the time
parameter was set to π/2, because it produces a probability that
reaches its maximum.
The initial belief state corresponds to a quantum state
representing a superposition of the participant’s beliefs.
 
1
1
1

Qi = 

2 1
1

(24)

By multiplying the unitary matrix with the initial superposition
belief state, one can compute the transition of the participants’
beliefs at each time. The final vector Qf represents the amplitude
distribution across states after deliberation.
 
1
1 1

(25)
QF = U · Qi = U · 
1· 2
1
Having the final state QF , one can compute probabilistic
inferences by computing the sum squared magnitude of the
rows of interest in the final belief state. Note that the four
components of the column vector QF respectively correspond
to [ DD DC CD CC ], where C corresponds to Cooperate and D
to Defect. The first letter represents the action chosen by the
first player, and the second letter corresponds to the action of
the second player. Thus, the probability of player 2 choosing

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FIGURE 2 | Illustration all possible probabilities, Pr(P2 = Defect), that
can be obtained by varying the parameters γ and µC .

the action Defect corresponds to the summation of the squared
magnitude of the first and the third components of the column
vector QF :

2 
2




Pr(P2 = Defect) = QF[1st_dim]  + QF[3rd_dim] 

2 
2




Pr(P2 = Cooperate) = QF[2nd_dim]  + QF[4th_dim]  (26)
To explain the average results observed in the Prisoner’s Dilemma
Game, in the work of Pothos and Busemeyer [17], the authors
chose the following parameters:
• µD = 0.51. This parameter corresponds to a participant
choosing the defect action.
• µC = 0.51. This parameter corresponds to a participant
choosing the cooperate action.
• γ = 0.6865. This parameter corresponds to the simulation of
the dissonance effect.
Using the above parameters, one can estimate the average results
of Table 1 to be Pr(P2 = Defect) = 0.64. The Quantum
Dynamical model shows that quantum probability is a very
general framework and can lead to many different probabilities.
These probabilities just depend on the way one chooses to fit
these free parameters. This has also been shown in the previous
study of Moreira and Wichert [65]. To illustrate this concept, we
decided to fix one of the parameters µD , µC or γ and vary the
others between the interval [−1, 1]. Figures 2–4 show all possible
probabilities that can be obtained with the presented Dynamical
Quantum Model for the Prisoner’s Dilemma game1 The value of
these figures is to show how sensitive quantum parameters are
and how challenging it is to find values for these parameters.
In the Quantum Dynamical Model, the parameters used
are based on a psychological setting. The incorporation of
parameters to model dissonance effects and the payoffs of
the players provide an approximation for the psychology of
the problem that is not observed in other quantum cognitive
1 These graphs were plotted using the

Wolfram Mathematica 10.4.1 software.

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Moreira and Wichert

FIGURE 3 | Illustration of all possible probabilities, Pr(P2 = Defect), that
can be obtained by varying the parameters γ and µD .

models of the literature. However, one great disadvantage of the
Quantum Dynamical Model is related to Hamiltonian matrices.
Creating a manual Hamiltonian is a very hard problem because it
is required that all possible interactions of the decision problem
are known, and this specification must be made in such a way
that the matrix is doubly stochastic. A recent work from Yearsley
and Busemeyer [66] describes how to construct Hamiltonians for
quantum models of cognition. The Hamiltonian matrix grows
exponentially with the complexity of the decision problem, and
the computation of a unitary operator from such matrices is a
very complex process. Most of the time, approximations are used
because of the complexity of the calculations involved in the
matrix exponentiation operation.

6. THE QUANTUM PROSPECT DECISION
THEORY
The Quantum Prospect Decision Theory was developed
by Yukalov and Sornette [36, 67] and developed throughout
many other works [68–71]. The foundations of this theory
are very similar to the previously presented Quantum-Like
Approach.
In the Quantum-Like Approach, we start with two
dichotomous observables. In the Quantum Prospect Decision
Theory, these observables are referred to as intensions. An
intension can be defined by an intended action, and a set of
intended actions is defined as a prospect.
Each prospect can contain a set of action modes, which are
concrete representations of an intension. Making a comparison
with the Quantum-Like Approach, a prospect can be seen as a
random variable, and the set of action modes are the assignments
that each random variable can have. For instance, the intension to
play can have two representations: play action A or play action B.
Following the work of Yukalov and Sornette [36], two
intensions A and B have the respective representations: A = x
where x ∈ a1 , a2 and B = y, where y ∈ b1 , b2 . The corresponding
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Quantum Probabilistic Models Revisited

FIGURE 4 | Illustration of all possible probabilities, Pr(P2 = Defect), that
can be obtained by varying the parameters µD and µC .

state of mind is given by:
| ψs (t)i =

X
i,j

ci,j (t) | Ai Bj i

(27)

Equation (27) represents a linear combination of the prospect
basis states. From a psychological perspective, the state of mind
is a fixed vector characterizing a particular decision-maker with
his/her beliefs, habits, principles, etc. That is, it describes each
decision-maker as a unique subject.
The prospect states corresponding to the intensions A and B
are given by Equation (28). The ψ symbol corresponds to
quantum amplitudes associated with the prospect state. Under
the Quantum Prospect Decision Theory, these amplitudes
represent the weights of the intended actions while a person is
still deliberating about them.
|πA=a1 i = c11 |A = a1 B = b1 i + c12 |A = a1 B = b2 i

|πA=a2 i = c21 |A = a2 B = b1 i + c22 |A = a2 B = b2 i (28)
The probabilities of the prospects can be obtained by computing
the squared magnitude of the prospect states (just as in the
Quantum-Like Approach and the Quantum Dynamical Model).
Consequently, the final probabilities are given by:
Pr(πA=a1 ) = Pr(A = a1 , B = b1 ) + Pr(A = a1 , B = b2 )

+ q(πA=a1 ) = |ψ11 |2 + |ψ12 |2 + q(πA=a1 )

Pr(πA=a2 ) = Pr(A = a2 , B = b1 ) + Pr(A = a2 , B = b2 ) (29)
+ q(πA=a2 ) = |ψ21 |2 + |ψ22 |2 + q(πA=a2 )

where the interference term q is defined by:
p
q(πA=a1 ) = 2 · ϕ(πA=a1 ) Pr(A = a1 , B = b1 )
p
· Pr(A = a1 , B = b2 )
p
q(πA=a2 ) = 2 · ϕ(πA=a2 ) Pr(A = a2 , B = b1 )
p
· Pr(A = a2 , B = b2 )

(30)

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In Equation (30), the symbol ϕ corresponds to the uncertainty
factor and is given by Equation (31).


ϕ(πA=a1 ) = cos arg (ψ11 · ψ12 )


ϕ(πA=a2 ) = cos arg (ψ21 · ψ22 )

(31)

The interference term corresponds to the effects that emerge
during the process of deliberation, that is, while a person
is making a decision. These interference effects result from
conflicting interests, ambiguity, emotions, etc. [36].
One can notice that the Quantum Prospect Decision Theory
is very similar to the Quantum-Like Approach proposed
by Khrennikov [72]. Both theories end up with the same
quantum probability formula. However, the Quantum Prospect
Decision Theory provides some heuristics for how to choose the
uncertainty factors. This information will be addressed in the
next section.

6.1. Choosing the Uncertainty Factor
To accommodate the violations of the Sure Thing Principle,
the uncertainty factor must be set in such a way that it
will enable accurate predictions. Two methods were proposed
by Yukalov and Sornette [36] to estimate the uncertainty
factor: the Interference Alternation method and the Interference
Quarter Law.
• Interference Alternation - Under normalized
conditions, the


probabilities of the prospects p πj must sum to 1. This
normalization only occurs if one characterizes the interference
term as an alternation such that the interference effects
disappear while summing the probability of the prospects.
This results in the property of the interference alternation,
given by:
X


q πj = 0
(32)
j

The interference alternation property is in accordance with
the findings of Epstein [41]: the destructive interference effects
can be associated with uncertainty aversion. This leads to a
less probable action under uncertainty conditions. In contrast,
the probabilities of other actions that contain less uncertainty
are enhanced through constructive quantum interference
effects. This uncertainty aversion happens quite frequently in
situations where the Sure Thing Principle is violated. This
implies that one of the probabilities of the prospects must be
enhanced, whereas the other must be decreased.




sign ϕ(πA=a1 ) = −sign ϕ(πA=a2 )


where ϕ(πA=a ) ∈ [0, 1]
i

The probability distribution p (ξ ) is given by Equation (35)
and can be computed by taking the average of two probability
distributions.
pr (ξ ) =


1
1
pr1 (ξ ) + pr2 (ξ ) = δ (ξ ) + 2 (1 − ξ )
2
2

(35)

One of the probability distributions, (p1 (ξ )), is concentrated
in the center and is described by a Dirac function δ (ξ ).
pr1 (ξ ) = 2 · δ (ξ )

(36)

The other probability distribution,(p2 (ξ )), is a uniform
distribution in the interval [0, 1].

0, if ξ < 0
pr2 (ξ ) = 2 (1 − ξ ) where 2 (ξ ) =
(37)
1, if ξ ≥ 0
For a more detailed proof of the Interference Quarter Law, the
reader should refer to Yukalov and Sornette [36].

6.2. The Quantum Prospect Decision
Theory Applied to the Prisoner’s Dilemma
Game
In this section, we apply the Quantum Prospect Decision Theory
to try to predict the average results for the Prisoner’s Dilemma
Game reported in Table 1.
The probability of a player defecting (and cooperating), given
that one does not know what the action of the other player was,
is given by Equation (38). For simplicity, we will assume the
following notation: Defect (D) and Cooperate (C).
Pr(P2 = D) = Pr(P1 = D, P2 = D)

+ Pr(P1 = C, P2 = D) + Interferenced

Pr(P2 = C) = Pr(P1 = D, P2 = C)

+ Pr(P1 = C, P2 = C) + Interferencec (38)

The interference terms are given by:
Interferenced = 2 · ϕ (P2 = D)
p
· Pr(P1 = D, P2 = D) · Pr(P1 = C, P2 = D)
Interferencec = 2 · ϕ (P2 = C)
p
· Pr(P1 = D, P2 = C) · Pr(P1 = C, P2 = C)

(39)

(33)

• Interference Quarter Law - The interference terms generated
by quantum probabilistic inferences have a free quantum
parameter, which is the uncertainty factor (Equation 31).
The Interference Quarter Law corresponds to a quantitative
estimation of this parameter. The modulus of the interference
term q can be quantitatively estimated by computing the

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expectation value of the probability distribution of a random
variable ξ in the interval [0, 1]:
Z 1
1
q≡
ξ · pr (ξ ) dξ =
(34)
4
0

The uncertainty factors are given by:
ϕ (P2 = D) =
ϕ (P2 = D) =

interferenced
2·

p

2·

p

pr(P1 = D, P2 = D) · Pr(P1 = C, P2 = D)
interferencec

pr(P1 = D, P2 = C) · Pr(P1 = C, P2 = C)
(40)

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According to the Interference Quarter Law and to the Alternation
Law, the probabilities for acting under uncertainty are given by:
Pr(P2 = D) = Pr(P1 = D, P2 = D)

Pr(X1 , . . . , Xn ) =

+ Pr(P1 = C, P2 = D) − 0.25

Pr(P2 = C) = Pr(P1 = D, P2 = C)

+ Pr(P1 = C, P2 = C) + 0.25

(41)

For the Prisoner’s Dilemma Game,
Pr(P1 = D, P2 = D) = Pr(P1 = D) · Pr(P2 = D|P1 = D)
= 0.5 × 0.87 = 0.435

Pr(P1 = C, P2 = D) = Pr(P1 = C) · Pr(P2 = D|P1 = C)
= 0.5 × 0.74 = 0.37

Then, the final predicted probabilities are given by:
Pr(P2 = D) = 0.435 + 0.37 − 0.25 = 0.555
Pr(P2 = C) = 0.065 + 0.13 + 0.25 = 0.445

The full joint distribution [74] of a Bayesian Network, where X
is the list of variables, is given by:
n
Y
i=1

The average probability to defect for the Prisoner’s Dilemma
Game in Table 1 when the first player’s action is unknown is 0.64.
That means that, with the Quarter Interference Law together with
the Interference Alternation property, the Prospect Quantum
Decision Theory obtained an error of 13%.

7. PROBABILISTIC GRAPHICAL MODELS
In this section, we introduce the concepts of classical and
Quantum-Like Bayesian Networks, as well as some approaches in
the literature that formalized traditional Bayesian Networks into
a Quantum-Like Approach.

7.1. Classical Bayesian Networks
A classical Bayesian Network can be defined by a directed acyclic
graph structure in which each node represents a different random
variable from a specific domain and each edge represents a
direct influence from the source node to the target node. The
graph represents independence relationships between variables,
and each node is associated with a conditional probability table
that specifies a distribution over the values of a node given each
possible joint assignment of values of its parents [73].

(43)

The formula for computing classical exact inferences on Bayesian
Networks is based on the full joint distribution (Equation 43).
Let e be the list of observed variables and let Y be the remaining
unobserved variables in the network. For some query X, the
inference is given by:


X
Pr(X|e) = αPr(X, e) = α 
Pr(X, e, y)
(44)
y∈Y

Where α = P
(42)

Pr(Xi |Parents(Xi ))

1
x∈X Pr(X = x, e)

The summation is over all possible y, i.e., all possible
combinations of values of the unobserved variables y. The α
parameter corresponds to the normalization factor for the
distribution Pr(X|e) [74]. This normalization factor comes from
some assumptions that are made in Bayes rule.

7.2. Classical Bayesian Networks for the
Prisoner’s Dilemma Game
We represent the Prisoner’s Dilemma Game under a Bayesian
Network structure in which we assume neutral priors: there
is a 50% of a player choosing the actions Defect or Cooperate
(Figure 5). The decision of the first participant is then followed
by the decision of the second participant. The probability
distribution of the second player is obtained (or learned) from
the experimental data for the averaged results in Table 1 when
the actions of the first player are observed. Using this data, the
goal is to try to determine the probability of the second player
choosing to defect given that it is not known what action the first
player chose.
To compute the probability Pr(P2 = Defect), two operations
are required: the computation of the full joint probability
distribution (Equation 43) and the computation of the marginal
probability.
The full joint probability distribution can be easily computed
by multiplying all possible assignments of the network

FIGURE 5 | Bayesian Network representation of the Average of the results reported in the literature (last row of Table 1). The random variables that were
considered are P1 and P2 , corresponding to the actions chosen by the first participant and second participant, respectively.

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TABLE 2 | Classical full joint probability distribution representation of the
Bayesian Network in Figure 5.
P1

Pr( P1 , P2 )

P2

Defect

Defect

Defect

Cooperate

Cooperate

Defect

Cooperate

Cooperate

0.5 × 0.87 = 0.4350

0.5 × 0.13 = 0.0650

0.5 × 0.74 = 0.3700

0.5 × 0.26 = 0.1300

with each other. Table 2 shows the computation of these
probabilities.
The marginalization formula is used when we want to perform
queries to the network. For instance, in the Prisoner’s Dilemma
Game, we want to know what the probability is of the second
player choosing to defect given that we do not know what the
other player has chosen, Pr(P2 = Defect). This is obtained by
summing the entries of the full joint probability (Table 2) that
have P2 = Defect. That is, we sum up the first and third rows of
this table. Equation (45) shows this operation. For simplicity, we
have used the following notation: D = Defect and C = Cooperate.
Pr(P2 = D) = Pr(P1 = D) · Pr(P2 = D|P1 = D) + Pr(P1 = C)
·Pr(P2 = D|P1 = C) = 0.8050

(45)

In Equation (45), one can see that the classical Bayesian Network
was not able to predict the observed results in Table 1 using
classical inference. One might think that, if we parameterize the
Bayesian Network to take into account the player’s actions and
dissonance effects, there could be a possibility of obtaining the
required results. This line of thought is legitimate, but one must
take into account that, in the end, the probabilistic inferences
computed through the Bayesian Network must obey set theory
and the law of total probability. This means that, even if we
parameterize the network, we cannot find any closed form
optimization that could lead to the desired results. This happened
with the previous example of the Markov Model in Section 3.
Although we parameterized the player’s actions and dissonance
effects, we could not arrive at the desired results because they go
against the laws of probability theory, and Markov Models (as
well as Bayesian Networks) must obey these laws.

7.3. Quantum-Like Bayesian Networks in
the Literature
There are two main works in the literature that have contributed
to the development and understanding of Quantum Bayesian
Networks. One belongs to Tucci [37] and the other to Leifer and
Poulin [38].
In the work of Tucci [37], it is argued that any classical
Bayesian Network can be extended to a quantum one by replacing
real probabilities with quantum complex amplitudes. This means
that the factorization should be performed in the same manner
as in a classical Bayesian Network. Thus, the Bayesian Network
of Figure 5 could be represented by a Quantum Bayesian
Network with the following matrices tables (the ordering of the

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probability amplitudes in the matrices are the same as the ones in
Figure 5):
i
q

2
a · eiθ1 1 − a · eiθ1  · eiθ2


q

2
b · eiθ3 1 − b · eiθ3  · eiθ4

q
P2 = 

2
c · eiθ5
1 − c · eiθ5  · eiθ6
P1 =

h

One significant problem with Tucci’s work is related to the nonexistence of any methods to set the phase parameters eiθ . The
author states that one could have infinite Quantum Bayesian
Networks representing the same classical Bayesian Network
depending on the values that one chooses to set the parameter.
This requires that one knows a priori which parameters would
lead to the desired solution for each node queried in the network
(which we never know). Thus, for these experiments, Tucci’s
model cannot predict the results observed because one does not
have any information about the quantum parameters.
In the work of Leifer and Poulin [38], the authors argue that,
to develop a Quantum Bayesian Network, quantum versions
of probability distributions, quantum marginal probabilities
and quantum conditional probabilities are required (Table 3).
The authors performed a preliminary study of these concepts.
Generally speaking, a quantum probability distribution
corresponds to a density matrix contained in a Hilbert space,
with the constraint that the trace of this matrix must sum to 1.
In quantum probability theory, a full joint distribution is given
by a density matrix, ρ. This matrix provides the probability
distribution of all states that a Bayesian Network can have. The
marginalization operation corresponds to a quantum partial
trace [75, 76].
In the end, the models of Tucci [37] and Leifer and Poulin
[38] fail to provide any advantage relative to the classical
models because they cannot take into account interference effects
between random variables. Thus, they provide no advantages in
modeling decision-making problems that try to predict decisions
that violate the laws of total probability.
A more recent work from Moreira and Wichert [65] suggested
defining the Quantum-Like Bayesian Network in the same
manner as in the work of Tucci [37], replacing real probability
numbers by quantum probability amplitudes.
In this sense, the quantum counterpart of the full joint
probability distribution corresponds to the application of Born’s
rule to Equation (43):
N
2
Y



Pr(X1 , . . . , Xn ) =  ψ(Xi |Parents(Xi )) 



(46)

i=1

The general idea of a Quantum-Like Bayesian network is
that, when performing probabilistic inference, the probability
amplitude of each assignment of the network is propagated
and influences the probabilities of the remaining nodes. In
other words, every assignment of every node of the network
is propagated until the node representing the query variable is
reached. Note that, by taking multiple assignments and paths

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TABLE 3 | Relation between classical and quantum probabilities used in
the work of Leifer and Poulin [38].

State

Classical probability

Quantum probability

Pr(A)



 iθ 2
e ψA 

Joint probability distribution

Pr(A, B)

Marginal probability distribution

Pr(B) =

Conditional state

ρAB
P

A Pr(A, B)

Pr (B|A)
P
b∈B Pr(b|A) = 1

ρB = TrA (ρAB )
ρB|A
Tr(ρB|A ) = IA

at the same time, these trails influence each other in producing
interference effects.
The quantum counterpart of the Bayesian exact inference
formula corresponds to the application of Born’s rule to
Equation (44), leading to:

2


N
 XY

(47)
Pr(X|e) = α 
ψ(Xx |Parents(Xx ),e,y) 
 y x=1


Expanding Equation (47), it will lead to the quantum interference
formula:


N
2
|Y| Y

X


Pr(X|e) = α 
 ψ(Xx |Parents(Xx ),e,y=i)  + 2 · Interference
 x

i=1

N

|Y| Y

X


Interference =
 ψ(Xx |Parents(Xx ),e,y = i) 


i = 1 j=i + 1 x
N

Y



·  ψ(Xx |Parents(Xx ),e,y = j)  · cos(θi − θj )(48)


|Y|−1
X

x

In the Quantum Dynamical Model, because it uses unitary
operators, the double symmetric property of these operators does
not require the normalization of the computed values. However,
in this approach, because we do not have the constraints of
double stochasticity operators, we need to normalize the final
scores that are computed to achieve a probability value. In
classical Bayesian inference, normalization of the inference scores
is also necessary due to assumptions made in Bayes rule. The
normalization factor corresponds to α in Equation (48).
Note that, in Equation (48), if one sets (θi − θj ) to π/2,
then cos(θi − θj ) = 0, which means that the quantum Bayesian
Network collapses to its classical counterpart. That is, they can
behave in a classical way if one sets the interference term to
zero. Moreover, in Equation (48), if the Bayesian Network has N
binary random variables, we will end up with 2N free quantum θ
parameters. We represent each set of quantum parameters as a
single parameter of the full joint probability distribution just like
it is presented in Table 4. Approaches to tune those parameters
under a Quantum-Like Bayesian Network approach are still an
open research question.

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In the model of Moreira and Wichert [65], if there are
many unobserved nodes in the network, then the levels of
uncertainty are very high and the interference effects produce
changes in the final likelihoods of the outcomes. However, in
the opposite scenario, when there are very few unobserved
nodes, then the proposed quantum model tends to collapse into
its classical counterpart because the uncertainty levels are very
low. This work only provides a study on the impact of the
quantum parameters in complex decision scenarios. On later
works, the same authors have proposed the usage of heuristics
to automatically assign values to quantum parameters [39, 77].

7.4. Application of the Quantum-Like
Formalism to the Prisoner’s Dilemma
Game
In this section, we will demonstrate how the proposed Bayesian
Network can be applied to the average results presented in
Table 1 for the Prisoner’s Dilemma game, just as was proposed
in the work of Moreira and Wichert [65].
We begin applying the Quantum-Like formalism by creating
a Bayesian Network out of the decision problem, in which
real classical probabilities are replaced by quantum amplitudes
(Figure 6). In the Prisoner’s Dilemma Game, if nothing is told
to the participants, then there is a 50% chance of the first
participant choosing to defect or cooperate. The decision of the
first participant is then followed by the decision of the second
participant.
To compute the probability Pr(P2 = Defect), two operations
are required: the computation of the quantum version of
the full joint probability distribution (Equation 46) and the
computation of the quantum version of the marginal probability
(Equation 48).
The full joint probability distribution can be easily computed
by multiplying all possible assignments of the network with
each other. For instance, the quantum full joint probability
amplitude ψ(P1 =Defect,P2 =Defect) is given by multiplying the
prior probability amplitude ψ(P1 =Defect) with the conditional
probability amplitude ψ(P2 =Defect|P1 =Defect) . Table 4 shows the
computation of these quantum probability amplitudes.
From the quantum version of the full joint probability
distribution, one can compute the quantum version of the
marginal probability distribution by summing all the entries of
Table 4 that contain the assignment P2 = Defect (Equation 49).
For simplification purposes, we will consider the following
abbreviations: Defect = D and Cooperate = C.
2

2 
Pr (P2 = D) = α[ψ(P1 =D,P2 =D)  + ψ(P1 =C,P2 =D) 

+ 2 · ψ(P1 =D,P2 =D) · ψ(P1 =C,P2 =D) cos (θA − θB )]

(49)


2
Pr (P2 = D) = α[ψ(P1 =D) · ψ(P2 =D|P1 =D)  +


ψ(P =C) · ψ(P =D|P =C) 2 + 2 · ψ(P =D)
1
2
1
1
·ψ(P2 =D|P1 =D) · ψ(P1 =C) · ψ(P2 =D|P1 =C)

· cos (θA − θB )]

(50)

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TABLE 4 | Quantum full joint probability amplitude distribution representation of the Bayesian Network in Figure 5.
P1

P2

Defect

Defect

Defect

Cooperate

Cooperate

Defect

Cooperate

Cooperate

ψ(P1,P2)
√

0.5 · ei·θ1 ×

√

√
√
0.5 · ei·θ1 × 0.13 · ei·θ4 = 0.2550 · ei·
√
i·θ
0.5 · e 2 × 0.74 · ei·θ5 = 0.6083 · ei·
√
√
0.5 · ei·θ2 × 0.26 · ei·θ6 = 0.3606 · ei·

√




0.87 · ei·θ3 = 0.6595 · ei· θ1 +θ3 = 0.6595 · ei·θA



θ1 +θ4 = 0.2550 · ei·θB


θ2 +θ5 = 0.6083 · ei·θC


θ2 +θ6 = 0.3606 · ei·θD

FIGURE 6 | Bayesian Network representation of the Average of the results reported in the literature (last row of Table 1). The random variables that were
considered are P1 and P2, corresponding to the actions chosen by the first participant and second participant, respectively.


Pr (P2 = D) = α |0.6595|2 + |0.6083|2 + 2 × 0.6595
×0.6083 · cos (θA − θB ) ]

= α [0.8050 + 0.8023 · cos (θA − θB )]

(51)

To compute the normalization factor α, we also need to compute
Pr(P2 = C):

2 
Pr (P2 = C) = α[ψ(P1 =D) · ψ(P2 =C|P1 =D)  + ψ(P1 =C)
2
·ψ(P2 =C|P1 =C)  + 2 · ψ(P1 =D) · ψ(P2 =C|P1 =D)
·ψ(P1 =C) · ψ(P2 =C|P1 =C) · cos (θA − θB )]

(52)


Pr (P2 = C) = α |0.255|2 + |0.3606|2 + 2 × 0.255 × 0.3606
· cos (θA − θB )] = α [0.195 + 0.1839
· cos (θA − θB )]

(53)

The normalization factor α is given by Equation (54).
1
1
=
Pr (P2 = D) + Pr (P2 = C)
1 + 0.9862 · cos (θA − θB )
(54)
Equation (54) contains two quantum parameters θ . Setting these
parameters is still an open research question in the literature,
although in some works, various heuristics have been proposed
to address this problem [39, 40, 77].
α=

8. DISCUSSION OF THE PRESENTED
MODELS
The purpose of this section is to present discussion of and a
comparison between the existing quantum models in terms of
the proposed evaluation metrics: terms of interference, parameter
tuning and scalability. The discussion will be mainly focused on
the set of parameters that the current quantum cognitive models

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have that need to be fitted to match the desired predictions.
For instance, the Quantum Dynamical Model requires three
parameters for such small decision scenarios, whereas the
Quantum-Like Approach only needs one, and the Quantum
Prospect Decision Theory does not need any parameters because
it has a static heuristic to replace the interference term. Note
that the Quantum Dynamical Model uses three parameters µc ,
µd , γ to predict three probabilities to defect when { known to
defect, known to cooperate, unknown }. While the Quantum-Like
Approach uses one chosen parameter and two probabilities to
defect { known to defect, known to cooperate } to predict one
probability to defect when { unknown }. In the end, we will
see that the problems that we note for the quantum models are
similar to many other classical cognitive models.

8.1. Discussion in Terms of Interference,
Parameter Tuning and Scalability
In this section, we analyze the presented works in the literature
regarding three different metrics: interference effects, parameter
tuning, and scalability.
• Interference Effects. Many works from the literature state
that, through quantum interference effects, one could simulate
the paradoxical decisions found across many experiments
in the literature. Without interference effects, quantum
probability converges to its classical counterpart. This metric
examines if the analyzed model makes use of any type of
quantum interference to explain human decision-making.
• Parameter Tuning. The problem of applying quantum
formalisms to cognition is concerned with the number of
quantum parameters that one needs to find. These parameters
grow exponentially with the complexity of the decision
problem, and thus far, very few works in the literature have
suggested ways to automatically find these parameters to make
accurate predictions.

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Moreira and Wichert

• Scalability. Most problems of the current models of the
literature are concerned with their inability to scale to more
complex decision scenarios. Most of these models are built
to explain very small paradoxical findings (for example,
the Prisoner’s Dilemma Game and the Two-Stage Gambling
Game). Therefore, this metric consists of analyzing the
presented models with respect to their ability to extend and
generalize to more complex scenarios.
Table 5 presents a summary of the evaluation of the models
presented in this work with respect to the three metrics described
above. The parameter growth column is based on the number
of parameters that each model generates when we increase the
number of unknown random variables in the decision model
Starting the discussion with the classical models presented in
Sections 3 and 7.1, the probabilistic inferences computed through
Bayesian/Markov Networks must obey set theory and the law
of total probability. This means that, even if we parameterize
the networks, we cannot find any closed form optimization
that could lead to the desired results. These networks can be
modeled with no parameters (just as was presented in Sections 3
and 7.1), or they can be parameterized. This parameterization
can end up with the same size as the full joint probability
distribution of the networks. Although these models do not
make use of any quantum interference effects and consequently
cannot accommodate violations of the Sure Thing Principle, it
is worth noting that one can always classically explain behavioral
results through appropriate conditionalizations and extensions of
classical probabilistic models [16].
The Quantum-Like Approach [72] is based on the direct
mapping of classical probabilities to quantum probability
amplitudes through Born’s rule. This means that one can perform
inferences for more complex decision-making scenarios by using
the quantum counterpart of the classical marginal probability
formula. Thus, the model generates quantum interference effects.
The main problem of the Quantum-Like approach concerns the
quantum parameters. The current works of the literature do not
provide any means to assign values to these quantum parameters.
They have to be fitted to explain the observed outcome. Thus,
the Quantum-Like approach, although it can be (mathematically)
extended to more complex decision scenarios, does not provide
any means to assign quantum parameters. Note that, in the
Quantum-Like approach, just like in many other models, it is
required a mathematical fitting of a set of parameters to make
an optimal prediction of the probabilities. So, the Quantum-Like
Approach is considered to be a predictive model.
The Quantum Dynamical Model proposed by Pothos and
Busemeyer [17] and Busemeyer et al. [35] incorporates quantum
interference effects not from the quantum law of probability
but by the usage of unitary operators and Hamiltonians. One
of the main disadvantages of this model concerns the definition
of the Hamiltonian matrices. Creating a Hamiltonian is a very
hard problem. It is required that all possible interactions of
the decision problem are known, and this specification must be
made in such a way that the matrix is doubly stochastic. The
unitary matrix also grows exponentially with the complexity of
the decision problem, and the computation of a unitary operator

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Quantum Probabilistic Models Revisited

from such matrices is a very complex process. Most of the
time, approximations are used because of the complexity of the
calculations involved in the matrix exponentiation operation.
Just as in the Quantum-Like Approach, one needs to fit the
quantum parameters so that the final model can give the observed
outcome. It is important to note that, in the Quantum Dynamical
Model, the parameters used are based on a psychological setting.
The incorporation of parameters to model dissonance effects
and the payoffs of the players provide an approximation to the
psychology of the problem that is not observed in other quantum
cognitive models in the literature.
Finally, the Quantum Prospect Theory proposed by Yukalov
and Sornette [36] also incorporates quantum interference effects
from the quantum law of total probability. This model is very
similar, from a mathematical point of view, to the QuantumLike Approach, with the difference that it proposes laws to
compute the quantum interference parameters: the alternation
and the quantum quarter laws. Although the model is very
precise for very small decision problems (such as the Prisoner’s
Dilemma), it is not clear how the quantum quarter law and the
alternation law would work for more complex problems. For this
reason, the Quantum Prospect Theory is a model that enables the
usage of quantum interference terms to make predictions under
paradoxical scenarios and also provides an automatic mechanism
to set the quantum parameters under very small scenarios with a
static interference term (q = ±0.25). That is, the interference
term is always the same, even for different contextual problems.
For this reason, the model is not able to generalize well for more
complex decision scenarios.
Regarding Bayesian Networks, it is hard to apply the model
proposed in the work of Tucci [37] in paradoxical findings
that violate the Sure Thing Principle because the author makes
no mention of how to set these parameters. He even argues
that a classical Bayesian Network can be represented by an
infinite number of quantum Bayesian Networks depending on
how one tunes the quantum parameters. Because the model
is a Bayesian Network, one is able to perform inferences for
any scenario by using the quantum counterpart of the classical
marginal probability formula. Thus, in the end, the quantum
Bayesian Network proposed by Tucci [37] is scalable and takes
into account quantum interference effects; however, it does not
give any insights into how to set the quantum parameters that
result from the interference.
In the work of Leifer and Poulin [38], the authors create a
direct mapping from classical probability to quantum theory.
Because they made a quantum Bayesian Network, this model
enabled probabilistic inference, and consequently, it can be
generalized for any number of random variables through the
use of the quantum part of the marginal probability formula.
By making the direct mapping from classical to quantum
probabilities, the full joint probability distribution is mapped
into a density matrix. This means that the interference terms are
canceled. The authors also take into account the order in which
the operations are performed. Because, the commutativity axiom
is not valid in quantum mechanics, we obtain different outcomes
if the calculations are performed in a different order. Thus, the
quantum Bayesian Network proposed by Leifer and Poulin [38]

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TABLE 5 | Comparison of the different models proposed in the literature.
Approach
Bayesian/classical theory

Bayesian/Markov networks

Interference effects Parameter tuning Parameter growth
No

Nperson

Manual

Nactions

Comments
Number of parameters varies
for different models

Khrennikov [72]
Pothos and Busemeyer [17]

Quantum-like approach

Yes

Manual

Quantum dynamical model

Yes

Manual

Nperson

Nactions

Nactions

Hamiltonian Size exponential:
Nperson

Busemeyer et al. [35]
Yukalov and Sornette [36]

Grows exponentially large

Nactions

Quantum prospect decision theory

Yes

Automatic

Nperson

Nactions

Static heuristic
Quarter Law of Interference

Moreira and Wichert [65]

Quantum-like Bayesian networks

Yes

Automatic

Nperson

Nactions

Dynamic heuristic

Moreira and Wichert [39, 40, 77]

is scalable and takes into account quantum interference effects;
however, by making a direct mapping from classical to quantum,
these interference effects will cancel because the network will
collapse into its classical counterpart. Thus, in the end, this model
does not take advantage of quantum interferences to explain
paradoxical decision scenarios.
In the work of Moreira and Wichert [65], the authors
also make a direct mapping from classical theory to quantum
probability by replacing classical real probability values by
complex quantum probability amplitudes using Born’s rule. They
also applied the same mechanism to derive a quantum-like
full joint probability distribution formula and a quantum-like
marginal probability distribution for exact inference. In the end,
the model is very similar to the Quantum-Like Approach, and
it can be modeled for more complex decision-making scenarios
very easily due to its graphical structure. Because this model uses
quantum probability amplitudes, quantum interference effects
arise from the quantum-like exact inference formula. However,
the number of parameters grows exponentially large when the
levels of uncertainty are high, that is, when there are many
unobserved nodes in the network. Although the authors have
proposed some dynamic heuristics to address this problem in
recent works [39, 40, 77], one needs to take into account that
they are heuristics, which means that it can lead to the expected
outcome, but it can also lead to completely inaccurate results.
Note that we are aware that the problems that we note
in this discussion section about the quantum models are the
same in many cognitive science models. However, we are not
claiming that it is difficult to find the parameters for a game
such as the Prisoner’s Dilemma. What we are claiming is that the
several models analyzed in this work (Quantum-Like Approach,
Quantum Dynamical Model, Quantum-Like Bayesian Networks)
contain a set of parameters that need to be fitted to match the
desired predictions.
For instance, the Quantum Dynamical Model requires three
parameters for such a small decision scenario, whereas the
Quantum-Like Approach only needs one, and the Quantum
Prospect Decision Theory does not need any parameters, because
it has a static heuristic to replace the interference term. The

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purpose of this discussion section is simply to compare the
existing quantum models in terms of the evaluation metrics
specified in Table 5.

8.2. Discussion in Terms of Parameter
Growth
All models analyzed in this work present different growth rates
in what concerns parameters. For instance, the Dynamical Model
parameterizes the player’s actions plus an additional parameter
to model cognitive dissonance effects. Thus, the number of
parameters would be static if we consider the N-Person Prisoner’s
Dilemma Game. That is, instead of having only 2 players, it
is extended to N players. In the case of the Quantum-Like
Approach, we would have 2N parameters for the N-Person
Prisoner’s Dilemma Game. The number 2 comes from the fact
that each player has two actions (either defect or cooperate).
The same applies to the Classical Networks, the Quantum-Like
Bayesian Networks and the Quantum Prospect Theory Model.
However, because the authors of this last model presented the
Quantum Quarter Law of Interference as a static heuristic, this
model does not require any parameters.
At this point, the reader might be thinking that the Quantum
Dynamical Model provides great advantages vs. the existing
models because the number of parameters required corresponds
to the player’s actions with an additional cognitive dissonance
parameter. Although this line of thought is correct, one should
also take into account how the model unfolds. Although the
numbers of parameters do not grow exponentially large as
in the Quantum-Like Approach, the size of the Hamiltonian
does. In fact, it grows exponentially large with the following
Nplayers

Nplayers

size: Nactions × Nactions , where Nactions represents the number of
actions of the players and Nplayers corresponds to the number of
players.
We conclude this section by clarifying that most of the
quantum cognitive models proposed in the literature have
been directed toward small decision scenarios because of the
scarcity of datasets representing complex decision scenarios
and violations of the Sure Thing Principle. Consequently,

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the models proposed are simply overfitting simple decision
scenarios. Moreover, we believe that the violations of the Sure
Thing Principle tend to diminish with the complexity of the
decision scenario. Imagine, for instance, a Three-Stage Gambling
game. It will be very hard to find significant data that shows
a player wishing to play the last gamble given that he has lost
the two previous gambles. More experimental data and more
studies are needed for more complex decision scenarios to test
the viability of quantum models vs. their classical counterparts.

9. CONCLUSION
Recent work in cognitive psychology has revealed that quantum
probability theory provides another method of computing
probabilities without falling into the restrictions that classical
probability has in modeling cognitive systems of decisionmaking. Quantum probability theory can also be seen as a
generalization of classical probability theory, because it also
includes the classical probabilities as a special case (when the
interference term is zero).
Quantum probability has the particularity of enabling the
representation of events in a geometric structure. The main
advantage of this geometrical representation is the ability to
rotate from one basis to another to contextualize and interpret
events. This ability does not exist in the classical probability
theory and provides great flexibility for decision-making systems.
Consequently, quantum probability can be more expressive than
its traditional classical counterpart. Under quantum theory, these
paradoxical findings can simply be seen as consequences of the
geometric flexibility that quantum probability theory offers.
We have collected a set of models from the literature that
attempt to tackle violations of the Sure Thing Principle in a
Quantum fashion, and then we compared the collected models.
To illustrate this comparison, we provided a mathematical
description of each model and how they could be applied in a
decision scenario. We compared the models in terms of three
proposed metrics: the number of parameters involved in the
model, the scalability and the usage of the quantum interference
effects. We have also performed a more detailed study concerning
the growth of the number of quantum parameters when the
complexity and the levels of uncertainty of the decision scenario
increase. We have also performed this comparison with classical
models, namely a Markov Model and a Bayesian Network. The
main statement of this work is not to express that quantum
models are preferred with respect to the classical models. With
this work, we have concluded that purely classical models suffer
from the same exponential parameterization growth as quantum
models, with the added difficulty that they are not capable of
simulating results that violate the Sure Thing Principle. It is worth

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Quantum Probabilistic Models Revisited

noting that one can always classically explain behavioral results
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AUTHOR CONTRIBUTIONS
All authors listed, have made substantial, direct and intellectual
contribution to the work, and approved it for publication.

ACKNOWLEDGMENTS
This work was supported by national funds through
Fundação para a Ciência e a Tecnologia (FCT) with
reference UID/CEC/50021/2013 and through the PhD grant
SFRH/BD/92391/2013. The funders had no role in study design,
data collection and analysis, decision to publish, or preparation
of the manuscript.

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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2016 Moreira and Wichert. This is an open-access article distributed
under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original
author(s) or licensor are credited and that the original publication in this journal
is cited, in accordance with accepted academic practice. No use, distribution or
reproduction is permitted which does not comply with these terms.

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ORIGINAL RESEARCH
published: 14 February 2017
doi: 10.3389/fphy.2017.00004

Topological and Orthomodular
Modeling of Context in Behavioral
Science
Louis Narens *
Department of Cognitive Sciences, University of California, Irvine, Irvine, CA, USA

Edited by:
Emmanuel E. Haven,
University of Leicester, UK
Reviewed by:
Irina Basieva,
Graduate School for the Creation of
New Photonics Industries, Russia
Tomas Veloz,
University of British Columbia, Canada
*Correspondence:
Louis Narens
lnarens@uci.edu
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 26 May 2016
Accepted: 18 January 2016
Published: 14 February 2017
Citation:
Narens L (2017) Topological and
Orthomodular Modeling of Context in
Behavioral Science. Front. Phys. 5:4.
doi: 10.3389/fphy.2017.00004

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Two non-boolean methods are discussed for modeling context in behavioral data and
theory. The first is based on intuitionistic logic, which is similar to classical logic except
that not every event has a complement. Its probability theory is also similar to classical
probability theory except that the definition of probability function needs to be generalized
to unions of events instead of applying only to unions of disjoint events. The generalization
is needed, because intuitionistic event spaces may not contain enough disjoint events
for the classical definition to be effective. The second method develops a version of
quantum logic for its underlying probability theory. It differs from Hilbert space logic used
in quantum mechanics as a foundation for quantum probability theory in variety of ways.
John von Neumann and others have commented about the lack of a relative frequency
approach and a rational foundation for this probability theory. This article argues that its
version of quantum probability theory does not have such issues. The method based on
intuitionistic logic is useful for modeling cognitive interpretations that vary with context,
for example, the mood of the decision maker, the context produced by the influence of
other items in a choice experiment, etc. The method based on this article’s quantum
logic is useful for modeling probabilities across contexts, for example, how probabilities
of events from different experiments are related.
Keywords: non-boolean methods, Hilbert space, intuitionistic logic, quantum logic, event lattices

1. INTRODUCTION
Probability functions are special kind of functions on event algebras. Following Birkhoff and von
Neumann [1], a lattice event algebra is a structure of the form,
X = hX , ⊆, ⋒, ⋓, X, ∅i ,
where X is a nonempty set, X is a set of subsets of X, ⊆ is the set-theoretic subset relation, X and
the empty set ∅ are in X , and for all A and B in X , A ⋒ B is the ⊆-least upper bound in X of A
and B, and A ⋓ B is the ⊆-greatest lower bound in X of A and B. X is said to be complemented if
and only if for all A in X there exists a B in X , called the complement of A, such that A ⋒ B = X
and A ⋓ B = ∅. (Throughout this article, ⋒ and ⋓ will always denote, respectively, the ⊆-least upper
bound and ⊆-greatest lower bound operators on some collection of sets. The complement of A will
often be denoted by A⊥ .) A special kind of lattice event algebra has been used throughout science
and mathematics to describe the domain of finitely additive probability functions. It is where

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Modeling of Context in Behavioral Science

X = hX , ∪, ∩, −, X, ∅i
i.e., where ⋒ = set-theoretic union, ∪, ⋓ = set-theoretic
intersection, ∩, and set-theoretic complementation, −, is a
complementation operation for X. This special event algebra is
called a set-theoretic boolean algebra.
Probability theory began in the seventeenth century with the
study of gambling games. Part of the assumptions underlying
such games was that the occurrence of each event that was the
basis of a wager could be determined to have happened or could
be determined not to have happened. The non-happening of
an event A was viewed as the occurrence of another event, the
complement of A, −A. Ambiguous or indefinite outcomes were
not allowed. In the nineteenth century Boole formulated the
logical structure underlying such gambling situations as a settheoretic boolean algebra. One principle of this algebra is the Law
of the Excluded Middle: For each event A, either A happens or
−A happens, or in algebraic notation, A ∪ −A = X, where X is
the sure event. Another is the Distributive Law, for all A, B, C,
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
During the late 1920s to early 1930s, the validity of the Law
of the Excluded Middle and the Distributive Law were called
into question as general logical principles: The mathematician
Brouwer concluded that the Law of the Excluded Middle was
improper for some kinds of mathematical inference, and the
mathematician von Neumann found the Distributive Law to be
too restrictive for the structure of events in quantum physics.
Both Brouwer and von Neumann constructed new logics that
generalized boolean algebras.
Brouwer’s logic became known as intuitionistic logic. This
article uses the special form of it that is a topology open sets.
Brouwer developed his intuitionistic logic for philosophical
considerations in the foundations of mathematics. Here
intuitionistic logic is used for entirely different purposes: It has
a more flexible algebraic structure than boolean algebras, and
this flexibility is exploited to described how context can affect
probability in organized manners.

2. TOPOLOGICAL ALGEBRA OF EVENTS
T = hT , ∪, ∩, –˙ , X, ∅i is said to be a topological algebra if and
only if T is a topology of open subsets with universal set X, and
for each A in T ,
–˙ A is the ⊆-largest element of T that is in X − A,
that is,
–˙ A =

[

B.

B∩A = ∅

–˙ A is called the pseudo complement of A. For the special case
where T is a boolean algebra (and thus each element of T is both
an open and closed set), –˙ is set-theoretic complementation, −.
A “topological probability function” is defined on T as follows:
Definition 1. A topological probability function P is a
function from T into the closed interval [0,1] of the reals
such that for all A and B in T ,

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• P(X) = 1, P(∅) = 0,
• if A ⊆ B then P(A) ≤ P(B), and
• topological finite additivity: P(A ∪ B) = P(A) + P(B) −
P(A ∩ B) .
If T is a boolean algebra, then topological finite additivity is
logically equivalent to the usual concept of finite additivity for
probability functions. In this article, a finitely additive probability
function on a set-theoretic boolean algebra is called a boolean
probability function.
A topology with a topological probability function is a
generalization of a set-theoretic boolean algebra with a finitely
additive probability function. Topologies are much richer
algebraically than boolean algebras, and this richness is useful for
describing probabilistic concepts that are difficult or impossible
to formulate in a boolean algebra, for example, various concepts
of ambiguity, vagueness, and incompleteness. This article uses
topologies to formulate a specific concept of “context” that
applies to some decision situations. This is done through the use
of properties of the pseudo complementation operation –˙ .
Definition 2. Let T = hT , ∪, ∩, –˙ , X, ∅i be a topological
algebra. Then A in T is said to be a refutation if and only
if there exists a B in T such that A = –˙ B.
One interpretation of –˙ is based on the operations of
“verification” and “refutation” used in the philosophy of science.
For this interpretation, an underlying empirical domain is
assumed along with a scientific theory about its events. An event
is said to be “verified” if its occurrence is empirically verified or
it is a direct consequence of the underlying theory. An event
A is said to be “refuted” if and only if the assumption of its
occurrence is inconsistent with known facts and theory about its
occurrence. Event A can be refuted by verifying an event B such
that A ∩ B = ∅. A can also be refuted by showing its occurrence
is inconsistent with known verifiable events and fundamental
tenets of the theory underlying the empirical domain. Under this
interpretation, the refutation of A is the largest open set S in the
topology that refutes A. It follows that A ∩ S = ∅ and thus
S = –˙ A. The refutation of –˙ A, –˙ –˙ A, is the largest open set T
that refutes –˙ A. Because A ∩ –˙ A = ∅, A refutes –˙ A. However,
is often the case that –˙ –˙ A is not verifiable—i.e., it is only the case
that –˙ A is refutable. In such a situation A ⊂ –˙ –˙ A. Because of
this, it is often the case that for verifiable A, A ∪ –˙ A is not the sure
event. This reflects that in most cases that verifiability should not
be identified with truth and refutation with falsehood.
Refutations play a different role in defining context for
topological algebras. Their key properties for this are given in the
following theorem.
Theorem 1. Let T = hT , ∪, ∩, –˙ , X, ∅i be a topological algebra.
Then the following six statements hold for all A and B in X .
1.
2.
3.
4.
5.
6.

if A ⊆ B then –˙ B ⊆ –˙ A .
A ⊆ –˙ –˙ A .
–˙ A = –˙ –˙ –˙ A .
–˙ (A ∪ B) = –˙ A ∩ –˙ B and –˙ A ∪ –˙ B ⊆ –˙ (A ∩ B) .
A ∩ B = ∅ iff (–˙ –˙ A ∩ –˙ –˙ B = ∅) .
There does not exists C in X such that A ⊂ –˙ C ⊂ –˙ –˙ A .
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Proof. Statements 1–4 follow from Theorem 3.13 of Narens
[2]. Statements 5 and 6 follow from Theorem 8 of Narens [3].
The key difference between set-theoretic boolean algebras and
topological algebras is that in set-theoretic boolean algebras
B = −−B
for all B in the algebra, whereas for topological algebras that are
not boolean it can be shown that there exist events A and D such
that
A ⊂ D ⊂ –˙ –˙ A .

(1)

By Statement 5 of Theorem 1, such a D in Equation 1 cannot be a
refutation.
Let T = hT , ∪, ∩, –˙ , X, ∅i be a topological algebra. Define
≡ on T as follows: For all A and B in T , A ≡ B if and only if
–˙ –˙ A = –˙ –˙ B. Then ≡ is an equivalence relation on T , and
each ≡-class is called a contextual class. The ≡-class to which A
belongs, A≡ , is called the contextual class for A. Note if A∩B = ∅,
then A and B belong to different contextual classes by Statement 5
of Theorem 1.
In psychology, context viewed as an operation that changes
an event’s interpretation. This is often done in formalizations by
making a distinction between a description of an event E (gamble,
etc.) given in an instruction and the interpretation of that
description in a context C , EC , that can vary with instructions,
emotional states, or other forms of context.
The contextual classes of a topological algebra are highly
structured. In particular, each contextual class A≡ has a ⊆maximal element, the refutation –˙ –˙ A, and that these maximal
elements form the following boolean algebra.
Theorem 2. Define ⊎ on the set of refutations R as follows: For all
A and B in R, A ⊎ B = –˙ –˙ (A ∪ B). Then
R = hR, ⊎, ∩, –˙ , X, ∅i
is a boolean lattice, that is, for all A, B, C in R, A ∩ (B ⊎ C) =
(A ∩ B) ⊎ (A ∩ C).
Proof. Theorem 3.16 of Narens [2].
In general R is not a set-theoretic boolean algebra, because there
may exists an element A in R such that A ∪ –˙ A is a proper subset
of X. When –˙ A ∪ A = X for all A in R, R is called a stone algebra,
and it can be shown that –˙ = − on R, that is, R is a set-theoretic
boolean algebra. Stone algebras are useful in applications, because
a topological probability function P on a stone algebra T is also
a finitely additive probability function on R, and for each A
in T , P(–˙ –˙ A) can be viewed as the upper boolean probability
of the topological probabilities of the events in the contextual
class A≡ .
There are many ways contextual classes can be used in
psychology. One way is to provide generalizations of the standard
theory for rational decision making, SEU (Subjective Expected
Utility.) For gambling situations, SEU assumes a gamble g =
(a1 , A1 · · · an , An ) is composed of a series of terms of the form
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ai , Ai , where ai , Ai stands
S for receiving outcome ai if the event
Ai occurs, and where ni= 1 Ai is a partition of the sure event
X. In determining the utility of gambles in SEU, the subjective
probability P(Ai ) of Ai is independent of the outcome ai across
gambles. That is, SEU requires that if bi , Ai is a term in another
gamble h that partitions X, then P(Ai ) is also the probability
assigned to Ai in the computation of h. Some in the literature
have question whether this is a valid rationality principle. In any
case, one might want to investigate psychological models where
such independence is violated. This is done in a model of Narens
[3] called “DSEU” (“Descriptive Subjective Expected Utility”). In
DSEU, the nature of the outcome a in a term a, A can influence
the implied subjective judgment of the probability of the event
A, e.g., where a is a catastrophe such as losing one’s life vs. a
is winning $5. Narens models the various interpretations of an
event occurring in different gambles as events in a contextual
class of a topological algebra. Strong disjointness (i.e., –˙ –˙ C ∩
–˙ –˙ D) guarantees that contextual interpretations of gambles
remain gambles. Narens [3] shows that subjective judgments
of the utilities of the contextual interpretations of gambles and
their associated subjective probability of events are rational
in the sense that there is a SEU model that has a submodel
that is isomorphic to the judgments made on the subjective
interpretations of gambles. The existence of such a submodel
shows that any irrationality observed in the DSEU model by
standard tests (e.g., making a Dutch Book) will transfer to SEU,
making SEU irrational by such tests, which is impossible by
known results.

3. A BEHAVIORAL QUANTUM
PROBABILITY THEORY
3.1. Orthomodular Event Lattices
In making decisions involving probabilistic phenomena, people’s
behavior often violate economic and philosophic principles
of rationality. Various theories in economics and psychology
have been developed to account for these violations, Prospect
Theory of Kahneman and Tversky [4] being currently the most
influential. Almost always the accounts assumed an underlying
boolean algebra of events. The deviations from SEU are modeled
by changing or generalizing characteristics of a finitely additive
probability function. Relatively recently, a different approach
has been taken: Change the event space to accommodate the
violations of economic and philosophic rationality. Topological
event spaces of the previous section are one example of such
an approach. More commonly in the literature are modeling
techniques inspired by von Neumann’s approach to quantum
mechanics, for example, Busemeyer and Bruza [5].
In
his
classic
Mathematische
Grundlagen
der
Quantenmechanik, von Neumann [6] modeled probabilistic
quantum phenomena using closed subspaces of a Hilbert space
as events. The seminal article by Birkhoff and von Neumann
[1], “The Logic of Quantum Mechanics,” isolated the algebraic
properties of the event spaces that von Neumann thought
underly the probability theory inherent in quantum phenomena.
The logic consisted of the following:

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• A lattice event algebra X = hX , ⊆, ⋒, ⋓,⊥ , X, ∅i with a
complementation operation ⊥ .
⊥
• satisfies the properties of DeMorgan’s Laws, that is, for all
A and B in X ,
(A ⋒ B)⊥ = A⊥ ⋓ B⊥ and (A ⋓ B)⊥ = A⊥ ⋒ B⊥ .

(2)

A complementation operation that satisfies DeMorgan’s
Laws is called an othrocomplementation operation.
• X satisfies the modular law, that is for all A, B, and C in X ,
if B ⊆ A then B ⋒ (A ⋓ C) = (B ⋒ A) ⋓ (B ⋒ C) .
For lattice algebras, the modular law is a generalization of the
distributive law, B ⋒ (A ⋓ C) = (B ⋒ A) ⋓ (B ⋒ C). Thus,
the von Birkhoff-von Neumann logic is a generalization of a
boolean lattice algebra, that is, of an orthocomplemented lattice
algebra satisfying the above distributive law. It applies to the
lattice algebra of all subspaces of a finite dimensional Hilbert
space. However, as Husimi [7] pointed out, the lattice algebra
of closed subspaces of an infinite dimensional Hilbert space
does not satisfy the modular law. He suggested replacing the
modular law with the following consequence of it that he called
the orthomodular law: For all A, B, C,
if B ⊆ A then A = B ⋒ (B⊥ ⋓ A) .
Today, Husimi’s suggestion won out and the term quantum logic
applies to lattice event algebras with orthocomplementation
satisfying the orthomodular law. In this article, lattice
terminology is used instead, and such lattices are called
orthomodular lattices.
In psychology, ideas derived from quantum mechanics have
been implemented in various ways, from borrowing methods
that assume some physics, to using only Hilbert space probability
theory, to using only orthomodular lattices. All of these have
foundational issues: Why should methods based on physical laws,
e.g., methods based on the conservation of energy, apply to
psychology? How does one derive the geometrical properties of
Hilbert space used in quantum probability from psychological
considerations? What does orthomodularity have to do with how
experiments are designed and conducted? To my knowledge,
the first two questions has not been adequately addressed in the
literature. This article makes some progress on the third.

3.2. Counterfactuals in Behavioral
Experiments
The behavioral modeling described in this section concerns a
simplified experimental situation. It differs from a similar model
presented in Narens [8] in that minor errors and ambiguities in
the construction of that model are eliminated and the material is
presented in a more clear manner. The version presented here is
also more general.
The assumed simplified situation makes for easier
mathematical modeling and philosophical analysis, which
are the principal goals of this article. The cost for this is a
loss of realism and a design that may require much larger

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numbers of subjects than is practical for usual psychological
experimentation.
The experimental situation under consideration has a large
population of subjects, where each is put into exactly one of
a finite number of experiments. In psychology, this is called a
between-subject paradigm.
Each experiment has a finite, nonempty set of choices—called
outcomes—and each of an experiment’s subjects must choose
exactly one of the experiment’s outcomes. Different experiments
are assumed to have different outcomes. Thus, each outcome
occurs only in one experiment.
To simplify the presentation, only a specific case involving two
experiments is considered throughout most of this article. The
definitions, concepts, and methods of proof developed for this
specific case are formulated in manners so that they generalize
to the case of finitely many experiments. Such a generalization is
briefly discussed in Section 3.5.
The (experimental) paradigm (P) has two experiments, (A)
and (B). Experiment (A) has a set of 3 outcomes, OA = {a, b, c},
and experiment (B) has a set of 3 outcomes, OB = {d, e, f }.
(P), which spans (A) and (B), has the set six outcomes, O =
{a, b, c, d, e, f }. The set of (P)’s subjects, S , is randomly divided in
half, with one of the halves participating in (A) and the other in
(B). In each experiment, the identity of each subject is recorded
along with the outcome she chose. Thus, the number of subjects,
N, and the number Nx of subjects who chose outcome x, x =
a, b, c, d, e, f , are known. This is the collected data of (P).
Paradigm (P) also has a theory that connects its experiments
(A) and (B). This connection is described counterfactually, for
example,
If subject s in experiment (A) chose an outcome in event E in the
power-set ℘(OA ) of OA were instead originally put in experiment
(B), then she would have chosen an outcome in event F in the
power-set ℘(OB ) of OB .

Such counterfactuals exist only in theory, not in data: For a
subject s who chose some outcome of E in ℘(OA ) and E 6= OA
in experiment (A), it is not possible to determine from (P)’s
data alone whether or not s’s choice would have been in F ∈
℘(OB ), where in experiment (B), F 6= OB and F 6= ∅. Such a
determination must be a consequence of the theory posited by
paradigm (P).
Definition 3. Let s be a subject in paradigm (P) and o be
an outcome in O. Then s is said to have actually chosen o
if and only if o is an outcome in an experiment of (P), s is
a subject in that experiment, and s chose o. s is said to have
counterfactually chosen o if and only if
(i) s is a subject in (A), o is an outcome in (B) and s would have
chosen o if she were placed in (B) instead of (A), or
(ii) s is a subject in (B), o is an outcome in (A) and s would
have chosen o if she were placed in (A) instead of (B).
Let E be an event in ℘(O). Then s is said to have
paradigmatically chosen E if and only if s actually chose
some element of E or s counterfactually chose some element
of E.

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Theoretical assumptions of (P). The following three theoretical
assumptions are made about (P):

− < H >, i.e., < −H > ⊆ − < H >. If p is in − < H >, then
her actual choice is in −H, i.e., − < H > ⊆ < −H >.

(T1) Each subject in S paradigmatically choses exactly one
outcome from each of (P)’s experiments; and each of (P)’s
outcomes is paradigmatically chosen by some subject in S .
(T2) Each subject who actually chose the outcome c in OA =
{a, b, c} would have counterfactually chosen the outcome d
in OB = {d, e, f }, and each subject who actually chose the
outcome d in OB would have counterfactually chosen the
outcome c in OA .
(T3) For x = a, b, c, d, e, f , let ≪ x ≫ be the set of (P)’s subjects
that paradigmatically chose x. Then for y = a, b and z =
e, f , ≪ y ≫ 6⊆ ≪ z ≫ and ≪ z ≫ 6⊆ ≪ y ≫. (Note by
(T2) that ≪ c ≫ = ≪ d ≫.)

Definition 6. The following notation is used throughout this
article:

(T1) is a general theoretical assumption that extends to
paradigms having finitely many experiments. (T2) and (T3) are
theoretical assumptions that are specific to properties of (P).
Section 3.5 describes modified versions of them that apply more
widely to paradigms having finitely many experiments.
By assumption, OA ∩ OB = ∅. However, there are situations
where outcomes in OA and OB are needed to be identified.
This accomplished through the use of counterfactual statements.
Assumption (T2) above is an example of this: For the purposes of
analysis and drawing conclusions about (P), it counterfactually
identifies c and d as being the same outcome.
The following notation and concepts are useful.
Definition 4. The following notation is used throughout this
article.
• < E > is the set of all subjects s of (P) who actually chose
some element e in E.
• ≪ E ≫ is the set of all subjects s of (P) who paradigmatically
chose some element e in E.
• | < E > | is the number of subjects in < E >.
• | ≪ E ≫ | is the number of subjects in ≪ E ≫.
The following definition provides a method for identifying events
across experiments.
Definition 5. Throughout this article for each G ⊆ O, σ (G)
denotes the event in ℘(O) such that G ⊆ σ (G) and for
each of (P)’s subjects p, if p has a paradigmatic choice in
G then all of her paradigmatic choices are in σ (G). (From
the latter, it follows that she has no paradigmatic choices
in O − σ (G).) H is said to be a proposition if and only if
for some K and H = σ (K). Such a H is also called the
proposition associated with K. Note that for each K in ℘(O),
the proposition associated with K, σ (K), exists.
Notation For each H ∈ ℘(O), let < −H > = < O − H > and
− < H > = < O > − < H >.
The following lemma is a simple consequence of Definition 5.
Lemma 1. Let H be the proposition. Then < −H > = − < H >.
Proof. Each subject p makes one unique actual choice. If this
choice is in −H then p is in < −H > and therefore p must be in
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• For each event F in ℘(O), F is the proposition associated with
F.
• o for an outcome in O is the proposition associated with {o}.
• It follows from (P)’s assumptions that c = d = the proposition
associated with {c, d}. Throughout this article, let k stand for
the proposition associated with {c, d}. Thus, c = d = k.
• P stands for the set of propositions in ℘(O).
Elements of P are described later in Figure 1.
It follows from P ’s theory and data that ∅ is a proposition and
O is a proposition.
It will be shown that the proposition a is {a, e, f }.
{a} ⊆ {a, e, f }. Because each subject paradigmatically selects
exactly one outcome in OA , it follows that b ∈
/ a and c ∈
/ a,
and thus by assumption (T2), d ∈
/ a. By assumption (T1) each
subject who paradigmatically chooses some element of a must
also paradigmatically choose some element in OB . This element
cannot be d. Therefore, it is either e or f . If it is e, then another
subject who paradigmatically chose a must have chosen f , for
otherwise a ⊆ e, contradicting assumption (T3). Similarly if a
subject who chose an element of a paradigmatically chose f , then
another subject who paradigmatically chose and element of a
must have paradigmatically chosen e. Thus, it has been shown
that a = {a, e, f }. Similarly b = {b, e, f }. Note that
a ∩ b = {e, f } .
However {e, f } is not a proposition. Note that the proposition that
is the ⊆-greatest-lower bound of a and b is the empty set, ∅ .
Definition 7. Let E and F be, respectively, the propositions
associated with E and F. Then the following definitions
hold:
• E⊥ = σ (< −E >) .
• E ⋒ F = σ (< E > ∪ < F >) .
• E ⋓ F = σ (< E > ∩ < F >).
Note by Lemma 1 and the meanings of “< >” and “∪” that for all
E and F in ℘(O),
< −E > = − < E > and < E > ∪ < F > = < E ∪ F > .
(3)
Lemma 2. Let C, D, and E, respectively be, respectively,
propositions associated with C, D, and E. Then C⊥ is a proposition,
C = C⊥ ⊥ , and D ⊆ E iff E⊥ ⊆ D⊥ .
Proof. C⊥ = σ (C) is a proposition by Definition 7.
By Equation (3),
C⊥ ⊥ = σ [< −(C⊥ ) >] = σ [− < C⊥ >] = σ [− < −C >]
= σ [−− < C >] = σ [< C >] = C .

Because

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D ⊆ E iff < D > ⊆ < E >

iff σ (< D >) ⊆ σ (E)

making F ⋓ G a lower bound of F and G. Suppose the proposition
H associated with H is such that

iff D ⊆ E
and

D ⊆ E iff − < E > ⊆ − < D >

< F > ⊇ < H > and < G > ⊇ < H > .
Then < F > ⊇ < H > and < G > ⊇ < H >, and thus
< F > ∩ < G > ⊇ < H >,

iff σ (< −E >) ⊆ σ (−D)

iff E⊥ ⊆ D⊥ ,

and therefore,

it follows that D ⊆ E iff E⊥ ⊆ D⊥ .
Lemma 3. ⋒ is the ⊆-least upper bound operation on P .
Proof. Let F and G, respectively, be propositions associated
with F and G. Then, because
< F > ∪ < G > = < F ∪ G >,
it follows that
F ⋒ G = σ (< F ∪ G >)
and therefore F ⋒ G is a proposition. Then
F = σ (< F >) ⊆ σ (< F > ∪ < G >)

F ⋓ G = σ (< F > ∩ < G >) = σ (H) = H ,
showing that F ⋓ G is the greatest lower bound of F and G.
Lemma 5. P = hP , ⋒, ⋓, ⊥ , O , ∅ i is a complemented lattice
event algebra.
Proof. O is clearly the ⊆-largest element of P and ∅ is clearly
the ⊆-smallest element of P .
Because, by Lemmas 3 and Lemma 4, ⋒ and ⋓ are, respectively,
the ⊆-least upper bound and ⊆-greatest lower bound operators
on P , P is a lattice event algebra. The following shows that ⊥ is a
complementation operation on P.
Let E and F, respectively, be the propositions associated with
E and F. Then, by E⊥ = σ (−E) and Equation (3),

and

E ⋒ E⊥ = σ (< E) > ∪ < −E) >)

G = σ (< G >) ⊆ σ (< F > ∪ < G >) ,
making F ⋒ G an upper bound of F and G. Suppose H is the
proposition associated with H and is such that F ⊆ H and
G ⊆ H. Then
< F > ⊆ < H > and < G > ⊆ < H > .
Thus
< F > ∪ < G > ⊆ < H >,
and therefore,
F ⋒ G = σ (< F > ∪ < G >) ⊆ σ (< H >) = H ,
showing that F ⋒ G is the least upper bound of F and G.

= σ (< E > ∪ − < E >) = σ (< O >) = O ,

and
E ⋓ E⊥ = σ (< E) > ∩ < −E) >)

= σ (< E > ∩ − < E >) = σ (< ∅ >) = ∅ ,

Lemma 6. The complemented lattice event algebra P =
hP , ⋒, ⋓, ⊥ , O , ∅ i satisfies DeMorgan’s Laws.
Proof. It is a well-known result of lattice theory (e.g.,
Theorem 2.14 of [2]) that DeMorgan’s Laws for P are equivalent
to the following: For all E and F in P ,
E⊥ ⊥ = E and (E ⊆ F iff F⊥ ⊆ E⊥ ) .

(4)

Equation (4) follows from Lemma 2.
Lemma 4. ⋓ is the ⊆-greatest lower bound operation on P .
Proof. Let F and G, respectively, be propositions associated with
F and G. Then, because < F > ∩ < G > = < F ∩ G >, it follows
that
F ⋓ G = σ (< F ∩ G >)
and therefore F ⋓ G is a proposition. Thus,
F = σ (< F >) ⊇ σ (< F > ∩ < G >) = F ⋓ G
and
G = σ (< G >) ⊇ σ (< F > ∩ < G >) = F ⋓ G ,
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The above lemmas show that the description of the
experimental situation gives rise to an orthocomplemented
lattice. Aerts and Gabora [9] have a similar result for a different
psychological paradigm: They show that their empirical data is
representable as an orthocomplemented lattice that they imbed
in a Hilbert space.
Theorem 3 given later shows that P = hP , ⋒, ⋓, ⊥ , O , ∅ i also
satisfies the Orthomodular Law. The proof, which generalizes to
a wide class of paradigms with finitely many experiments, uses a
probability function that is defined on the set of P’s propositions.
The probability theory for this function is developed in the
following two sections.

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3.3. Probability Theory for P
Definition 8. Throughout the rest of this article, let P be the
following function on P : For each E in P ,
P(E) =

|<E>|
.
|<O>|

P is called (P)’s propositional probability function.
The following is the intended interpretation of P: For each
proposition E in P , P(E) is the probability that a randomly
chosen paradigm subject actually chose some outcome e in E. If
the subjects in < E > are known through data and theory, then
the value of P(E) completely computable from data.
Propositions E that span experiments are necessarily partially
based on counterfactuals. Because of this, they are theoretical in
nature. Nevertheless, as discussed at the end of Section 3, for
paradigm (P), P’s value for a proposition F is estimable from data
to a good approximation. As discussed at the end of Section 3
this is not generally true of other paradigms. However, for the
special case where a proposition comes from one of a paradigm’s
experiments it is generally true by an analog of the following
argument given for (P).
Because of the large numbers of subjects participating in (P)’s
experiments and the way they were randomly assigned in equal
numbers to each of (P)’s experiments, it follows that for each E in
℘(OA ),
⋆

| < E > | ≈ | < E > |,
where ≈ stands for “approximately” and | < E⋆ > | for
“the number of subjects in (B) that counterfactually chose some
outcome in E.” Thus, for E in ℘(OA ),
|<E>|
| < E > | + | < E⋆ > |
2| < E > |
P(E) =
=
≈
,
|<O>|
|<O>|
|<O>|
which is computable since | < E > | and | < O > | are known
from data.
Thus for each proposition E in ℘(OA ) and similarly for
each proposition F in ℘(OB ), P(E) and P(F) are estimable to
a good approximation from data. For G ∈ ℘(O) where G
spans experiments can be more complicated. For such spanning
propositions, theoretical assumptions as well as data are needed
to calculate P’s probabilities. As discussed at the end of Section 3,
this is possible for paradigm (P) but may not be possible for other
paradigms where the the theory may not complete enough to
estimate all spanning propositions.

3.4. Logical and Probabilistic Structure of
Orthomodular Event Lattices
Figure 1 is a Hasse diagram of the lattice P = hP , ⋒, ⋓, O, ∅i.
The set-theoretic boolean algebra generated by O has 26 = 64
elements. The elements at the bottom of Figure 1 but above ∅
are called atoms. They are lattice elements E such that there
there does not exist a lattice element F such that ∅ ⊂ F ⊂ E.
Figure 1 has 5 atoms, a, b, k, e, f . The set-theoretic boolean
algebra generated by these atoms has 25 = 32 elements. P has
12 elements—a substantial reduction from 64 or 32.

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FIGURE 1 | Hasse diagram for P = hP, ⋒, ⋓, ⊥ , O, ∅ i. ⊆ corresponds to
elements of P (nodes) being directly connected by edges to higher elements
(higher nodes).

In Figure 1, the lattice-theoretic intersection ⋓ of atoms, e.g.,
a ⋓ f, is the proposition ∅ . This is a consequence of assumption
(T3).
The following concepts are useful for the understanding of the
structure of orthomodular lattices.
Definition 9. X = hX , ⋒, ⋓,⊥ , X, ∅i is said to be an
ortholattice if and only if X is a complemented lattice event
algebra satisfying DeMorgan’s Laws.
Definition 10. Let X
=
hX , ⋒, ⋓,⊥ , X, ∅i be an
ortholattice. Then the following definitions hold.
1. Events C and D in X are said to be orthogonal, in symbols,
C ⊥ D if and only if C ⊆ D⊥ .
2. Q is said to be an orthoprobability function on X if and only
if
• Q is a function from X into the real interval [0,1];
• Q(X) = 1 and Q(∅) = 0; and
• for all C and D in X , if C ⊥ D then Q(C ⋒ D) = Q(C) +
C(D).
Q is said to be ⊂-monotonic if and only if for all C and D in

X , if C ⊂ D then Q(C) < Q(D).

3. An event lattice of the form Y = hY , ⋒, ⋓,⊥ , X, ∅i where
Y ⊆ X is said to be a subalgebra of X. Note that Y has
the same ⊆-maximal and minimal elements, X and ∅, as
X, and that the operations of Y are the restrictions of the
operations of X to Y .
4. A complemented lattice event algebra Z = hZ , ⋒, ⋓,⊥ , X, ∅i
is said to be an O6 subalgebra of X if and only there exist F
and G in Z such that the following two statements hold:

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|<F >| |<G>|
|<F⋒ G>|
=
+
|O |
|O |
|O |
= P(F) + P(G) ,

• Z = {∅, F, G, F ⊥ , G⊥ , X} and
• ∅ ⊂ F ⊂ G ⊂ X and ∅ ⊂ G⊥ ⊂ F ⊥ ⊂ X .

P(F ⋒ G) =

Figure 2 shows a Hasse diagram of an O6 subalgebra of X
when X is a set of propositions.
Lemma 7. Suppose X = hX , ⋒, ⋓,⊥ , X, ∅i is an ortholattice that
has no O6 subalgebra. Then X is orthomodular.
Proof. Theorem 2 (pp. 22–23) of Kalmbach [10]. (Also
Theorem 2.25 of [2].)
Lemma 8. P is a ⊂-monotonic orthoprobability function on P =
hP , ⋒, ⋓, ⊥ , O , ∅ i.
Proof. Suppose F and G are arbitrary elements of P such that
G ⊆ F⊥ . We first show
<F⋒G> = <F> ∪ <G> .

(5)

It is immediate that < F > ⊆ < F ⋒ G > and < G > ⊆ <
F ⋒ G >. Thus,
<F> ∪ <G>⊆<F⋒G> .

(6)

Suppose s is in

Then s is in σ (F) or s is in σ (G). Without loss of generality,
suppose s is in σ (F) = F. Then
(7)

and Equation (5) follows from Equations (6) and (7).
By the definitions of “proposition” and “⊥ ” and Equation (3),
< F⊥ > = < −F > = − < F >. Thus, < F > ∩ < F⊥ >
= ∅, and therefore, because G ⊆ F⊥ , < F > ∩ < G > = ∅.
Thus,
<F⋒G>=<F> ∪ <G>=<F>+<G> .
Therefore,

FIGURE 2 | O6 subalgebra.

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Theorem 3. P = hP , ⋒, ⋓, ⊥ , O , ∅ i is an orthomodular lattice
and P is an orthoprobability function on P.
Proof. By Lemma 8, P is a monotonic orthoprobability
function on P. Suppose P is not an orthomodular lattice. A
contradiction will be shown. Then by Lemma 7 there exists a
sublattice of P that has a Hasse diagram of the form displayed
in Figure 2. By the monotonicity of P,
P(F) < P(G),

(8)

and by the ortho-additivity of P,
P(F) + P(G⊥ ) = P(F ⋒ G⊥ ) = P(X) = 1 = P(F) + P(F⊥ ) . (9)
Equations (8) and (9) contradict one another, because, by the
monotonicity of P, P(G⊥ ) < P(F⊥ ) .

< F ⋒ G > = < σ (F ∪ G) > = < σ (F) ∪ σ (G) > .

< F ⋒ G > ⊆ < F > ∪ < G >,

showing ortho-additivity.
To show monotonicity suppose F and H are arbitrary elements
of P such that F ⊂ H. Then < F > ⊂ < H >. Then, by the
definition of P, P(F) < P(H).

The literature has studied orthomodular lattices as
generalizations of the logic underlying quantum mechanics.
Unfortunately, not all orthomodular lattices admit
orthoprobability functions [11]. This in itself is a clue that
for science something more than general orthomodular
lattices are needed. For P, the probability function P
was derived directly from (P)’s theory and empirical
considerations.

3.5. Generalizations and Properties of
Paradigm Probability Functions
Thus far, our analysis has focussed on the paradigm (P) and the
probabilistic structure P. Although the analysis sometimes used
special features of them, care was taken to present, when possible,
concepts and methods of proof that generalized to a wider
class experimental situations and a wider class of probabilistic
structures. There are, however, some conditions special to P and
P that do not apply to all between-subject paradigms involving
finitely many experiments. These are concerned with the use of
P’s atoms.
The boolean algebra B = h℘(O), ∪, ∩, −, O, ∅i spans (P)’s
experiments. Its set of atoms is O = {a, b, c, d, e, f }, which is
the set of outcomes of (P)’s experiments. (P)’s data consist of
records of the choices in O made by its subjects. In shifting the
analysis from B to P, it is desirable to keep the data intact.
(P)’s theoretical axioms and concepts does this by making the
propositions a, b, k, e, and f the atoms of P. (k results from
theoretical assumption (T2) that requires the identification of c
and d.) This allows the collected data about O to be transferred
to a, b, k, e, f. Concepts and theorems can exploit this transfer.
For example, this transfer is needed to implement the important

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concept of “actually determined” in obtaining consequences of
(P)’s theory and data.
Many of the previous results about P generalize to a paradigm
(Q) involving finitely many experiments, (A1 ), . . . , (An ), where
(Q) has disjoint experimental outcomes, disjoint subject
populations, and where for 1 ≤ i, j ≤ n, Ai ’s subject
population is randomly sampled from (Q)’s subject pool and
is the same size as Aj ’s subject population. In particular,
with appropriate generalizations of (P)’s theory, Lemmas 2 to
8 and Theorem 3 generalize to (Q)’s lattice of propositions
using the methods of proof similar to the those presented for
P and P.
(Q)’s theory consists of a set of statements describing
relationships among subjects’ responses across experiments.
Among these are statements that generalize (T1), (T2), and (T3)
of (P)’s theory in the following manner:
(T⋆ 1) Each of (Q)’s subjects paradigmatically choses an outcome
from each of (Q)’s experiments; and each of (Q)’s
outcomes is paradigmatically chosen by at least one of
(Q)’s subjects.
(T⋆ 2) Identifications of outcomes across experiments are made.
(T⋆ 3) The atomic propositions of (Q)’s propositional lattice
consist of propositions that correspond to the outcomes
that were not identified in (T⋆ 2) along with propositions
corresponding to each set of mutually identified outcomes
in (T⋆ 2).
“< >” and P have analogous definitions and results for (Q) to
those for (P).

3.6. Comparison with Quantum Probability
Many researchers of the formal foundations of quantum
mechanics have speculated that the underlying probability theory
for quantum mechanics is not interpretable in a physically
acceptable manner into a boolean probability theory (e.g.,
[1, 12–14]). Others have disagreed (e.g., [15]), producing
a long-running controversy that continues to the present
(e.g., 16).
Von Neumann was well aware of foundational difficulties
presented in his seminal 1932 book, Mathematische Grundlagen
der Quantenmechanik. It appears to me that such difficulties
are sharply increased and compounded by the importation
of formalisms involving probability from quantum mechanics
to cope with the difficult contextual issues presented in the
behavioral sciences.
Rédei [17] writes the following about the evolution of von
Neumann’s position about the nature of probability in quantum
mechanics.
What von Neumann aimed at in his quest
for quantum logic in the years 1935–1936 was
establishing the quantum analog of the classical
situation, where a Boolean algebra can be interpreted
as being both the Tarski-Lindenbaum algebra of
a classical propositional logic and the algebraic
structure representing the random events of a classical
probability theory, with probability being an additive

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Modeling of Context in Behavioral Science

normalized measure on the Boolean algebra satisfying
[monotonicity], and where the probabilities can also
be interpreted as relative frequencies. The problem
is that there exist no “properly non-commutative”
versions of this situation: The only (irreducible)
examples of non-commutative probability spaces
probabilities of which can be interpreted via relative
frequencies are the modular lattices of the finite
(factor) von Neumann algebras with the canonical
trace; however, the non-commutativity of these
examples is somewhat misleading because the noncommutativity is suppressed by the fact that the
trace is exactly the functional that insensitive for the
non-commutativity of the underlying algebra. So it
seems that while one can have both a non-classical
(quantum) logic and a mathematically impeccable
non-commutative measure theory, the conceptual
relation of these two structures cannot be the same
as in the classical commutative case—as long as one
views the measure as probability in the sense of relative
frequency. This must have been the main reason
why after 1936 von Neumann abandoned the relative
frequency view of probability in favor of what can be
called a “logical interpretation.” In this interpretation,
advocated by von Neumann explicitly in his address
to the 1954 Amsterdam Conference, (quantum) logic
determines the (quantum) probability, and vice versa,
i.e., von Neumann sees logic and probability emerging
simultaneously.
Von Neumann did not think, however, that this
rather abstract idea had been worked out by him as
fully as it should. Rather, he saw in the unified theory of
logic, probability, and quantum mechanics a problem
area that he thought should be further developed.
He finishes his address to the Amsterdam Conference
with these words [18]:
I think that it is quite important and will
probably shade a great deal of new light on logics
and probably alter the whole formal structure of
logics considerably, if one succeeds in deriving this
system from first principles, in other words for a
suitable set of axioms. All the existing axiomatizations
of this system are unsatisfactory in this sense, that
they bing in quite arbitrarily algebraical laws which
are not clearly related to anything that one believes
to be true or that one has observed in quantum
theory to be true. So, while one has very satisfactorily
formalistic foundations of projective geometry
of some infinite generalizations of it, including
orthogonality, including angles, none of them are
derived from intuitively plausible first principles
in the manner in which axiomatizations in other
areas are.
Now I think that at this point lies a very important
complex of open problems, about which one does
not know well of how to formulate them now, but

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which are likely to give logics and the whole dependent
system of probability a new slam.
Von Neumann’s concerns about probability theory in
quantum mechanics do not hold for the multi-experiment
behavioral paradigms presented in this article. The paradigms’
orthomodular lattice event structures follows directly from
their experimental designs and theories linking experiments.
This produces an orthoprobability probability function Q for a
paradigm’s lattice of propositions, Q. Because for propositions,
actual probabilities coincide paradigmatic probabilities, Q can
be estimated through a relative-frequency process for events
for which the underlying theory and collected data specify to
a good approximation which subjects paradigmatically chose
outcomes for those events. Paradigm (P) is an example where
such a relative frequency approach applies to all of its events: Its
event lattice has twelve elements. Of these, the probabilities of
two, O and ∅, are determined by definition. Five others, a, b,
k, e, f, are atoms and their actual probabilities are estimable by
collected data and thus, as described earlier, their paradigmatic
probabilities are estimable. The remaining five are complements
of the five atoms and these have as probabilities 1 minus the
probability of its atom, and thus they too are estimable. Now
consider the general case Q where F and G are lattice disjoint
propositions where it is known which subjects chose an element
of F and which chose an element of G. If it is the case that
F ∪ G ⊂ F ⋒ G then more information is required to estimate the
number of subjects who are in F ⋒ G. The additional information
has to come from the paradigm’s theory. For (P), its theory tells
us that a ⋒ f = {a, b, e, f }, which is the complement of k and thus
has number of subjects |O | − |k|. This number is known because
|O | and |k| are known.

4. CONCLUSIONS
Both the topological probability and the quantum-like
paradigm theories presented here are applicable to a variety
of psychological experimental situations where Kolmogorov
probability theory appears inadequate for modeling cognitive
processes. Although very different in how they handle
probabilities, they both can often offer explanations for puzzling
behavioral phenomena. From a modeling point of view, this
is not entirely surprising: After all, both are generalizations of
Kolmogorov probability, and, as such, both have greater freedom
to model behavioral data than the Kolmogorov theory. However,
because of their algebraic structural differences, they are likely
to suggest different cognitive mechanisms producing the data.
Topological probability functions are arguably “rational” in the
sense that they do not violate the key ideas of rationality inherent
in the Dutch Book Argument and the SEU model.
The probability theory of quantum mechanics and the
psychological paradigm probability theory developed here share
many formal characteristics, but at a fundamental level they are
about different kinds of uncertainty. The uncertainty in paradigm
probability theory is manufactured by the random assignment
of subjects to experiments by the scientist. It is not an inherent
part of the subjects, outcomes, or of the paradigm’s theory. The

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Modeling of Context in Behavioral Science

subjects in an experiment have actual and counterfactual choices.
These choices, as well as the theory connecting the paradigm’s
experiments, are modeled in deterministic manners. All of this is
very different than the probability theory of quantum mechanics,
where the uncertainty results from the randomness inherent an
ensemble of particles.
Systems satisfying the Kolmogorov axioms for probability
produce a probability theory founded on a σ -additive boolean
probability function. Such probability functions have come to
dominate the probability theories of mathematics, statistics, and
science. They are usually conceptualized as a single boolean
probability function defined over all relevant situations. They
are often interpreted as measuring the propensity of an event to
occur or a subjective degree of belief that an event will occur. Such
a propensity or degree are considered to be completely associated
with the event, and, as a consequence, does not depend on the
situation to which an event belongs. In this sense, propensity (or
degree of belief) is noncontextual.
Kolmogorov probability theory can be generalized to become
contextual by allowing events that belong to different situations
to have different propensities (or different degrees of belief).
These situations are characterized as having different probability
functions. This causes various challenging issues in behavioral
science, e.g., the identification of random variables across
situations, or descriptions of the relationships of random
variables across different probabilistic situations. Dzharafov and
Kujala [19] and Dzharafov et al. [20] have laid out a foundation
for such a generalization. It produces an alternative to the
single probability function interpretation of the Kolmogorov
theory that have many features in common with the probability
theory underlying quantum mechanics. There are several other
quantum-like probability theories in the literature that are not
discussed in this article (e.g., the probability theory of [21]). It is
beyond the scope of this article to go into their foundations or
relationships to the alternative probability theories described in
this article.
Context is an ill-understood concept in the behavioral
sciences. While there are many psychological experiments
illustrating its ubiquity and importance in psychological
phenomena, e.g., framing effects in cognitive psychology, there is
very little theory and experimentation describing the relationship
of contexts across different experiments. I believe part of the
reason for this has been the lack of mathematical theories
designed to model contextual relationships. For particle physics,
this kind of modeling was accomplished by von Neumann. His
method has been imported by Busemeyer and colleagues and
others into the behavioral science (e.g., [5]). This has produced
some interesting new phenomena (e.g., [22]) and has been
used as a unifying foundation for explaining many puzzling
psychological phenomena. Not surprisingly, this importation has
raised new, serious foundational and methodological issues.
Narens [2] interprets many results from lattice theory—
most known in the 1930s—as suggesting there are not many
alternatives to boolean algebras that are useful event spaces
for modeling probabilistic experimental phenomena, except
for those that are distributive (e.g., topological algebras) or
orthomodular (e.g., closed subspaces of a Hilbert space). This

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means that rich mathematical theories of probabilistic context are
likely very limited without giving up much more structure from
Kolmogorov probability theory, particularly, without greatly
reducing the parts of the event space displaying forms of
“probabilistic additivity.”

Modeling of Context in Behavioral Science

FUNDING
The research for this article was supported by grant SMA1416907 from NSF.

SUPPLEMENTARY MATERIAL
AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
approved it for publication.

REFERENCES
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(1936) 37:823–43. doi: 10.2307/1968621
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15. Sneed JD. Quantum mechanics and classical probability theory. Synthese
(1970) 21:34–64. doi: 10.1007/BF00414187

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The Supplementary Material for this article can be found
online at: http://journal.frontiersin.org/article/10.3389/fphy.
2017.00004/full#supplementary-material
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ArXiv:arxiv.org/pdf/1412.6987.
17. Rédei M. Von Neumann’s concept of quantum logic and quantum
probability. In: Rédei M, Stöltzner M, editors, John von Neumann
and the Foundations of Quantum Mechanics. Dordrecht: Kluwer
(2001).
18. von Neumann J. Unsolved problems in mathematics. In: Rédei M, Stöltzner
M, editors, John von Neumann and the Foundations of Quantum Physics.
Dordecht: Kluwer Academic (2001). p. 231–46.
19. Dzhafarov EN, Kujala JV. Conversations on contextuality. In: Dzhafarov EN,
Jordan JS, Zhang R, Cervantes VH, editors. Contextuality from Quantum
Physics to Psychology. World Scientific Press (2015). p. 1–22.
20. Dzhafarov EN, Kujala JV, Cervantes VH. Contextuality-by-default: a brief
overview of ideas, concepts, and terminology. In: Lecture Notes in Computer
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21. Foulis DJ, Randall CH. Operational statistics. I. Basic concepts. J Math Phys.
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22. Wang Z, Solloway T, Shiffrin R, Jerome R, Busemeyer JR. Context effects
produced by question orders reveal quantum nature of human judgments.
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756111
Conflict of Interest Statement: The author declares that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Narens. This is an open-access article distributed under the terms
of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) or licensor
are credited and that the original publication in this journal is cited, in accordance
with accepted academic practice. No use, distribution or reproduction is permitted
which does not comply with these terms.

February 2017 | Volume 5 | Article 4 | 126

ORIGINAL RESEARCH
published: 19 June 2017
doi: 10.3389/fphy.2017.00019

The Real and the Mathematical in
Quantum Modeling: From Principles
to Models and from Models to
Principles
Arkady Plotnitsky *
Theory and Cultural Studies Program, Purdue University, West Lafayette, IN, United States

Edited by:
Emmanuel E. Haven,
University of Leicester,
United Kingdom
Reviewed by:
Marco G. Mazza,
Max Planck Institute for Dynamics and
Self Organization (MPG), Germany
Gregg Jaeger,
Boston University, United States
*Correspondence:
Arkady Plotnitsky
plotnits@purdue.edu
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 16 November 2016
Accepted: 26 May 2017
Published: 19 June 2017
Citation:
Plotnitsky A (2017) The Real and the
Mathematical in Quantum Modeling:
From Principles to Models and from
Models to Principles.
Front. Phys. 5:19.
doi: 10.3389/fphy.2017.00019

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The history of mathematical modeling outside physics has been dominated by the
use of classical mathematical models, C-models, primarily those of a probabilistic or
statistical nature. More recently, however, quantum mathematical models, Q-models,
based in the mathematical formalism of quantum theory have become more prominent
in psychology, economics, and decision science. The use of Q-models in these fields
remains controversial, in part because it is not entirely clear whether Q-models are
necessary for dealing with the phenomena in question or whether C-models would still
suffice. My aim, however, is not to assess the necessity of Q-models in these fields,
but instead to reflect on what the possible applicability of Q-models may tell us about
the corresponding phenomena there, vis-à-vis quantum phenomena in physics. In order
to do so, I shall first discuss the key reasons for the use of Q-models in physics. In
particular, I shall examine the fundamental principles that led to the development of
quantum mechanics. Then I shall consider a possible role of similar principles in using
Q-models outside physics. Psychology, economics, and decision science borrow already
available Q-models from quantum theory, rather than derive them from their own internal
principles, while quantum mechanics was derived from such principles, because there
was no readily available mathematical model to handle quantum phenomena, although
the mathematics ultimately used in quantum did in fact exist then. I shall argue, however,
that the principle perspective on mathematical modeling outside physics might help us
to understand better the role of Q-models in these fields and possibly to envision new
models, conceptually analogous to but mathematically different from those of quantum
theory, that may be helpful or even necessary there or in physics itself. I shall, in closing,
suggest one possible type of such models, singularized probabilistic models, SP-models,
some of which are time-dependent, TDSP-models. The necessity of using such models
may change the nature of mathematical modeling in science and, thus, the nature of
science, as it happened in the case of Q-models, which not only led to a revolutionary
transformation of physics but also opened new possibilities for scientific thinking and
mathematical modeling beyond physics.
Keywords: principles, models, probability, statistics, reality, realism

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Plotnitsky

INTRODUCTION
The history of mathematical modeling outside physics has been
dominated by classical mathematical models, C-models, based on
mathematical models developed in classical physics, especially
probabilistic or statistical models, borrowed from classical
statistical physics or chaos and complexity theories. More
recently, however, models based in the mathematical formalism
of quantum theory, Q-models, primarily borrowed from
quantum mechanics but occasionally also quantum field theory,
became more current outside physics, specifically in psychology,
economics, and decision science, the fields (beyond physics)
with which I will be primarily concerned here [e.g., 1, 2]1 . My
abbreviations follows P. Dirac’s distinction between c-numbers
(classical numbers) and q-numbers (quantum numbers), because
the variables used in Q-models are in fact q-numbers. Quantum
mechanics and Q-models are based in the mathematics of
Hilbert spaces over complex numbers, C, with Hilbert-space
operators used as physical variables in the equations of quantum
mechanics, as against functions of real (mathematical) variables,
c-numbers, that serve as physical variables in classical physics.
The use of Q-models in these fields remains controversial,
because it is not entirely clear whether they are necessary
for dealing with the phenomena in question or whether Cmodels would suffice. It is true that debates and sometimes
controversies have also accompanied quantum mechanics since
its birth in 1925. These debates, initiated by the famous
confrontation between N. Bohr and A. Einstein on, in Bohr’s
phrase, “epistemological problems in atomic physics,” used
in the title of his account of this confrontation, have never
lost their intensity and appear to be interminable [3, v. 2,
pp. 32–66]. However, as Bohr’s phrase indicates, the reasons
for these controversies have been primarily philosophical. The
effectiveness of quantum mechanics or higher-level quantum
theories, such as quantum field theory, has not been in question:
they are among the best-confirmed theories in physics. The
situation is different in psychology, economics, and decision
science, where it is the scientific effectiveness or at least necessity
of Q-models that is doubted. My aim here, however, is not to
assess this effectiveness or necessity, but instead to reflect on
what the possible applicability of Q-models may tell us about
the corresponding phenomena in these fields vis-à-vis quantum
phenomena in physics. In order to do so, I shall first consider
the key reasons for the use of Q-models in physics. In particular,
I shall examine the fundamental principles that grounded and
indeed led to the development of quantum theory. Then I shall
consider a possible role of similar principles in using Q-models
beyond quantum theory. My emphases are due to the fact that
1 I shall only discuss the standard quantum mechanics or quantum field theory,
bypassing alternative theories of quantum phenomena, such as Bohmian theories,
which are sometimes used in mathematical modeling outside physics, but which
would require a separate consideration. By “quantum phenomena” I refer to
those physical phenomena in considering which Planck’s constant, h, must be
taken into account, and by “quantum objects” (thus different from quantum
phenomena) to those entities in nature that are responsible for the appearance of
quantum phenomena, manifested in measuring instruments involved in quantum
experiments or in certain natural phenomena.

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The Real and the Mathematical

psychology, economics, and decision science borrow already
available Q-models from quantum theory, rather than derive
them from their own fundamental principles, while quantum
mechanics and then quantum field theory were derived from such
principles. This is not surprising because there was at the time no
available mathematical model or (a more general concept, which
includes an interpretation of the model used) theory to effectively
handle quantum phenomena. The “old quantum theory” of M.
Planck, A. Einstein, N. Bohr, and A. Sommerfeld, which ushered
in the quantum revolution, became manifestly inadequate by
the time W. Heisenberg began his work on quantum mechanics
that he discovered in 1925 [4]. For the reasons explained below
(mostly a search for a more rigorous derivation of the formalism),
the research in quantum foundations is still concerned with
deriving quantum theory from such principles, a project in part
motivated by the rise of quantum information theory. That does
not appear to be a significant concern outside physics where
the use of Q-models is motivated primarily by their predictive
capacities, which is of course a crucial consideration in physics
as well. It may, however, be beneficial to consider the deeper
reasons for the possible use of Q-models in these fields, or, in
terms of my title, the real that gives rise to the mathematical
of Q-models there. The principle perspective on mathematical
modeling beyond physics might help us to do this and possibly
to envision new, post-quantum, models there or even in physics.
I shall, in closing, suggest one possible type of such models,
singularized probabilistic models, SP-models, some of which are
time-dependent, TDSP-models, and consider their implications
for mathematical modeling in science and for our understanding
of the nature of science2 .

PHYSICAL PRINCIPLES AND
MATHEMATICAL MODELS IN QUANTUM
MECHANICS
Theories, Principles, and Models in
Fundamental Physics
I would like to begin by outlining the key features of the standard
mathematical model of quantum mechanics, more customarily
used as a probabilistically or statistically predictive model in view
of the difficulties of in maintaining its representational capacities,
which continue to be debated:
(1) The Hilbert-space formalism over the field of complex
numbers, C, an abstract vector space of any dimension, finite
or infinite (in quantum mechanics, either finite or countably
infinite), possessing the structure of an inner product that
allows lengths and angles to be measured, analogously to an
n-dimensional Euclidean space (which is a Hilbert space over
real numbers R);
(2) The noncommutativity of the Hilbert-space operators, also
known as “observables,” which are mathematical entities
2 The

discussion to follow in part builds on two previous articles [5, 6], but only in
part: overall the present argument is different, especially (but not exclusively) by
virtue of considering SP-models.

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Plotnitsky

associated, in terms of probabilistic or statistical predictions,
with physically observable quantities;
(3) The nonadditive nature of the probabilities involved: the joint
probability of two or more mutually exclusive alternatives
in which an event might occur is, in general, not equal to
the sum of the probabilities for each alternative, and instead
obey the law of the addition of the so-called “quantum
amplitudes,” associated with complex Hilbert-space vectors,
for these alternatives (technically, these amplitudes are
linked to probability densities);
(4) Born’s rule or an analogous rule (such as von Neumann’s
projection postulate or Lüder’s postulate) added to the
formalism, which establishes the relation between
amplitudes as complex entities and probabilities as real
numbers (by using square moduli or, equivalently, the
multiplication of these quantities and their complex
conjugates) and (3) above3 .
In the development of quantum mechanics, discovered in 1925,
these features were not initially assumed, but were derived from
certain physical features of quantum phenomena and principles
arising from these features. The formalism was only given a
properly Hilbert-space form by J. von Neumann, in 1932, in The
Mathematical Foundations of Quantum Mechanics, a standard
text ever since [7]4 .
I shall now explain the concepts of theory, principle,
and model, as they will be understood here. By a theory,
I mean an organized assemblage of concepts, explanations,
principles, and models by means of which one is able to
relate, in one way or another, to the phenomena or (they
are not always the same) objects the theory considers. In
defining principles, I follow Einstein’s distinction between
“constructive” and “principle” theories, two contrasting, although
in practice often intermixed, types of theories [8, 9, pp.
35–50]. “Constructive theories” aim “to build up a picture
of the more complex phenomena out of the materials of
a relatively simple formal scheme from which they start
out” [8, p. 228]. Thus, according to Einstein, the kinetic
theory of gases, as a constructive theory in classical physics,
“seeks to reduce mechanical, thermal, and diffusional processes
to movements of molecules—i.e., to build them up out of
the hypothesis of molecular motion,” described by the laws
of classical mechanics [8, p. 228]. By contrast, principle
theories “employ the analytic, not the synthetic, method. The
elements which form their basis and starting point are not
hypothetically constructed but empirically discovered ones,
general characteristics of natural processes, principles that give
rise to mathematically formulated criteria which the separate
processes or the theoretical representations of them have
to satisfy” [8, p. 228]. Thus, thermodynamics, a classical
principle theory (parallel to the kinetic theory of gases as
a constructive theory), “seeks by analytical means to deduce

3 I bypass more technical definitions, found in standard texts and reference sources.
4 There

are alternative formalisms, such as those in terms of C∗ -algebras or more
recently category theory, thus far, all mathematically equivalent to the Hilbertspace formalism.

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The Real and the Mathematical

necessary conditions, which separate events have to satisfy,
from the universally experienced fact that perpetual motion is
impossible” [8, p. 228].
Principles, then, are “empirically discovered, general
characteristics of natural processes, ... that give rise to
mathematically formulated criteria which the separate
processes or the theoretical representations of them have to
satisfy.” I shall adopt this definition, but with the following
qualification, which is likely to have been accepted by Einstein.
Principles are not empirically discovered but formulated,
constructed, on the basis of empirically established evidence.
“The impossibility of perpetual motion” is hardly empirically
given; it is as a principle formulated on the basis of such
evidence.
Constructive theories are, more or less by definition, realist
theories, and conversely, many realist theories are constructive.
Realist theories represent, commonly causally, the phenomena
or objects they consider and their behavior, in science by
mathematical models, assumed to idealize how nature or reality
works, in the case of constructive theories at the simpler,
or deeper, level of reality constructed by a theory. In other
words, a constructive theory offer a representation of the
processes underlying and connecting the observable phenomena
considered, commonly by understanding the ultimate character
of these processes on the model of classical mechanics or
classical electrodynamics, as in the kinetic theory of gases,
as described above or other forms of classical statistical
physics. All such theories assume that the individual behavior
of the ultimate constituents of the systems they consider is
described by the laws of classical mechanics. A realist theory
may represent objects or phenomena it considers in a more
direct, if still idealized, manner, as classical mechanics (which
deals with individual or sufficiently small systems) or classical
electrodynamics do. I shall discuss the concepts of reality and
realism, which encompasses that of realist theory, in more
detail below. First, however, I shall define a mathematical
model.
By a “mathematical model” I refer to a mathematical structure
or set of mathematical structures that enables any type of
relation to the (observed) phenomena or objects considered.
(As I shall only deal with mathematical models here, the term
“model” hereafter refers to mathematical models.) All modern,
post-Galilean, physical theories are defined by their uses of
such models. The requirement of using mathematical models
may be seen as a principle, the mathematization principle,
“the M principle,” arguably the single defining principle of
all modern physics, from Galileo on. Such models may be
realist, representational, as in classical physics, specifically
classical mechanics, or predictive, as in classical statistical
physics (the models of which are, however, underlain by
representational models of classical mechanics), or in quantum
mechanics, without assuming realism and causality even in
considering elementary individual quantum processes, such
as those concerning elementary quantum objects, “elementary
particles.” This assumption is expressly abandoned or even
precluded in non-realist interpretations of quantum phenomena
and quantum mechanics, following Bohr and “the spirit of

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Copenhagen,” as Heisenberg called it [10, p. iv]5 . The M
principle is upheld in quantum mechanics, but, in non-realist
interpretations, in a way different from how it is used in realist
theories.
The probabilistic or statistical character of quantum
predictions must also be maintained by realist interpretations
of these theories or alternative theories (such as Bohmian
theories) of quantum phenomena, in conformity with quantum
experiments, in which only probabilistic or statistical predictions
are possible. The reasons for this is that the repetition of
identically prepared quantum experiments in general leads to
different outcomes, a difference that cannot be improved beyond
a certain limit (defined by Planck’s constant, h) by improving the
conditions of measurement, which is possible in classical physics.
This fact is also manifested in Heisenberg’s uncertainty relations,
which are statistical in character as well. This situation leads
to the quantum probability or (depending on interpretation)
quantum statistics principle, the QP/QS principle, arguably the
single defining principle in Q-models in physics and beyond,
keeping in mind that in psychology, economics, and decision
science, we do not have anything corresponding to elementary
individual physical processes, involving the ultimate elementary
constituents of nature, “elementary particles.” Nor do we have
anything analogous to h. The probabilities themselves necessary
for making correct predictions, in either quantum mechanics or
in using Q-models elsewhere, are, thus far, calculated by using
the Hilbert-space or mathematically equivalent formalisms and
the (non-additive) procedure described above that uses quantum
amplitudes and Born’s or a similar rule6 .
Realist models are, then, representational models, idealizing
the nature of objects or phenomena they consider. The term
“realism” will be primarily understood here as referring to
the possibility, at least, again, in principle, of such models,
and, in the first place, theories allowing for such models. One
could define another type of realism, which would refer to
theories that presuppose an independent architecture of reality
they consider, while allowing that this architecture cannot be
represented, either at a given moment in history or perhaps
ever, but if so, only due to practical human limitations [9, pp.
11–23]. In the first case, a theory that is strictly predictive may
be accepted, but with the hope that a future theory will do
better, by being a realist theory of the representational type.
Einstein adopted this attitude toward quantum mechanics, which
he expected to be eventually replaced by a (representational)
realist theory. Even in the second case, the ultimate nature of
reality is commonly deemed to be conceivable on realist models
of classical physics, possibly adjusting them to accommodate
new phenomena. However, this type of realism implies that
there is no representational theory or model of the ultimate
nature of the phenomena or objects considered. Either type of
realism is abandoned or even precluded in quantum mechanics,
5 The designation “the spirit of Copenhagen” is preferable to a more common “the
Copenhagen interpretation,” because there is no single Copenhagen interpretation.
6 That does not mean that an alternative way of doing so, for example, by bypassing
amplitudes or by using some an alternative formalism (not mathematically
equivalent to the standard one) is impossible.

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when interpreted in the spirit of Copenhagen. However, such
interpretations do assume the concept of reality, by which I refer
to what exists or is assumed to exist, without making any claim
upon the character of this existence, which type of claims defines
realist theories. By existence I refer to a capacity to have effects
on the world, ultimately, which also assume the existence of
the world by virtue of its capacity to have effects upon itself,
effects which establish by means of and thus in terms as effects
of our interactions with the world. In physics, the primary reality
considered is that of nature or matter. It is generally assumed to
exist independently of our interaction with it, which also assumes
that it has existed when we did not exist and will continue
to exist when we will no longer exist. This assumption is also
made in non-realist interpretations of quantum mechanics, in
the absence of a representation or even (as against the second,
non-representational type of realism defined above) conception
of the character of this existence. Thus, if realism presupposes a
representation or at least a conception of reality, this concept of
reality is that of “reality without realism” [9, 11]. The assumption
of this concept of reality is a principle, the RWR principle.
The existence or reality of quantum objects, a form of reality
beyond representation or even conception, is inferred from
effects they have on our world, specifically on experimental
technology. It has not been possible, at least thus far, to observe
a moving electron or photon, or for that matter even stationary
electrons (there are no stationary photons, which only exist
in motion before they are absorbed by other forms of matter,
such as electrons). It is only possible to observe traces of their
interactions with measuring instruments, traces that do not allow
us to reconstitute the independent behavior of quantum objects
movement, an impossibility reflected in Heisenberg’s uncertainty
relations. In non-realist, RWR-principle-based, interpretations,
quantum mechanics only predicts, in probabilistic or statistical
terms (no other predictions are, again, possible on experimental
grounds), effects manifested in measuring instruments impacted
by quantum objects.
While a principle theory, which, as I explained, need not be
constructive in Einstein’s sense, could be either realist or nonrealist, a constructive theory is by definition realist. Realist or,
it follows, constructive theories do involve principles, such as
the equivalence principle in general relativity, or the principle
of causality, which, to adopt Kant’s definition, commonly used
ever since, states that, if an event takes place, it has a cause of
which it is an effect [12, p. 305, 308]7 . Asymmetrically, however,
7 Causality

is, thus, an ontological category, characterizing the nature of reality. It
proceeds by connecting a cause (an event, phenomenon, a state of a system, or
force) to an effect, while the principle of causality connects an event to a cause.
Determinism is assumed here to be an epistemological category. It designates
our ability to predict the state of a system (ideally) exactly at any moment of
time once we know its state at a given moment of time. In classical mechanics
(which deals with a small number of objects), causality and determinism coincide.
Once a classical system is large, one can no longer predict its causal behavior
exactly. In other words, a system may be causal without our theory of its behavior
being deterministic, as is the case, for example, in classical statistical physics or
chaos theory. Causal influences are generally, although not always, assumed to
propagate from past or present towards future. Relativity theory further precludes
the propagation of physical influences faster than the speed of light in a vacuum,
c. Principle theories do not require causality, which is, again, difficult to assume

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a principle theory need not involve constructive aspects or
be realist. In non-realist, RWR-principle-based, interpretations,
quantum mechanics is a principle theory by definition, by virtue
of the RWR principle. It is not possible, in such interpretations, to
have a constructive theorization of the ultimate entities, quantum
objects, which are responsible for the observable quantum
phenomena, unless one sees quantum objects as constructed as
in principle unconstructible. According to Bohr, thus formulating
the RWR principle, “in quantum mechanics we are not dealing
with an arbitrary renunciation of a more detailed analysis of
atomic phenomena, but with a recognition that such an analysis
is in principle excluded,” beyond a certain point [3, v. 2, p. 62].
In this interpretation, quantum mechanics divorces itself from
the representation of the connections between observed quantum
phenomena, which it only relates in terms of predictions, in
general probabilistic or statistical in character, thus fulfilling the
M principle under the conditions of the RWR principle.
Finally, the present view does not assume a permanent,
Platonist, essence to any given principle, which can always be
abandoned under the pressure of new experimental findings
or new ways of theorizing previously available experimental
findings. Indeed, one might argue that the greatest form of
creative thinking in science or other theoretical fields is that
which lead to the invention of new principles, which implies
the transformation of principles, rather than any Platonist
permanence to them.

The Physical Principles of the Quantum
Theory
The RWR principle and the corresponding interpretation of
quantum mechanics emerged only in the 1930s. Heisenberg’s
discovery of quantum mechanics in 1925 and Bohr’s initial
interpretation of it, proposed in 1927, were based on the
following principles, with Bohr’s complementarity principle
added in 1927:
(1) the proto-RWR principle, according to which, “quantum
mechanics does not deal with a space–time description of the
motion of atomic particles” [3, v. 1, p. 48];
(2) the principle of discreteness or the QD principle, according
to which all observed quantum phenomena are individual
and discrete in relation to each other, which is fundamentally
different the atomic discreteness of quantum objects
themselves;
(3) the principle of the probabilistic or statistical nature of
quantum predictions, the QP/QS principle, even (in contrast
to classical statistical physics) in the case of primitive or
elementary quantum processes, in which nature also reflects
a special, non-additive, nature of quantum probabilities and
rules, such as Born’s rule, for deriving them, and
(4) the correspondence principle, which, as initially understood
by Bohr, required that the predictions of quantum theory
must coincide with those of classical mechanics in the
in quantum physics without, however, violating relativity or more generally the
principle of locality, which requires that all physical influences are local (still under
the assumption that they cannot, locally, propagate faster than c).

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classical limit, but was given by Heisenberg a new and
more rigorous form of “the mathematical correspondence
principle,” which required that the equations of quantum
mechanics convert into those of classical mechanics in
the classical limit, thus, in accordance with the M
principle.
I speak of the proto-RWR principle because Heisenberg saw the
project of describing the motion of electrons as unachievable at
the time, rather than “in principle excluded,” as Bohr assumed a
decade later [3, v. 2, p. 62]. This was, nevertheless, a radical move
on Heisenberg’s part, as Bohr was the first to realize: “In contrast
to ordinary [classical] mechanics, the new quantum mechanics
does not deal with a space–time description of the motion
of atomic particles. It operates with manifolds of quantities
[matrices] which replace the harmonic oscillating components
of the motion and symbolize the possibilities of transitions
between stationary states in conformity with the correspondence
principle. These quantities satisfy certain relations which take the
place of the mechanical equations of motion and the quantization
rules [of the old quantum theory]” [3, v. 1, p. 48].
Quantum discreteness was eventually (as part of Bohr’s
ultimate interpretation) recast by Bohr in terms of his concept
of “phenomenon,” defined in terms of what is observed in
measuring instruments under the impact of quantum objects, in
contradistinction to quantum objects themselves, which cannot
be observed or represented [3, v. 2, p. 64]. Quantum phenomena
are, in Bohr’s interpretation, irreducibly discrete in relation to
each other, and there is no continuous or any other conceivable
process that could be assumed to connect them. Probability has
a temporal structure by virtue of its futural and discrete nature:
one can only verifiably estimate future discrete events. Such
events may, however, be continuously and causally connected,
as they are in classical physics, even though we may not be
able to track these connections to make exact predictions, as
happens in classical statistical mechanics or chaos theory. By
contrast, in non-realist, RWR-principle-based, interpretations,
the nature of quantum phenomena and events precludes us
from causally (or otherwise) connecting them. This means that
only probabilistic or statistical predictions are possible, even
ideally and in principle, and even in dealing with elementary
individual quantum objects, such as those known as “elementary
particles,” and the processes and events they lead to, objects and
processes that cannot be decomposed into a smaller objects and
processes. This qualification distinguishes quantum mechanics
from classical probabilistic or statistical theories, or of course
classical mechanics where such predictions could, at least ideally,
be exact in dealing with individual classical objects or a small
number of classical objects. In quantum mechanics, in non-realist
interpretations, this type of idealization is not possible, a fact
reflected in the uncertainty relations. The theory only estimates
the probabilities or statistics of the outcomes of discrete future
events, on the basis of previous events, and tells us nothing
about what happens between events. Nor does it describe the
data observed in measuring instruments and hence quantum
phenomena. They are described by classical physics, which,
however, cannot predict them.

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The QP/QS principle was mathematically expressed in
Heisenberg’s scheme by matrices containing the necessary
probability amplitudes cum Born’s rule. Heisenberg only
formulated this rule in the case of electrons’ quantum jumps
in the hydrogen atom, rather than as universally applicable in
quantum mechanics, as Born did. Born’s rule is not inherent in
the formalism but is added to it—it is postulated.
The correspondence principle was central to Heisenberg’s
derivation of quantum mechanics. In its mathematical form,
introduced by Heisenberg, the principle required that both the
equations of quantum mechanics, which were formally those of
classical mechanics, and the variables used, which were different,
convert into those of classical mechanics in the classical limit, a
conversion automatic in the case of equations but not variables.
(The processes themselves, however, are still quantum even in
this limit.) Thus, the principle gave Heisenberg a half of the
mathematical architecture he needed.
An important qualification is in order. Heisenberg’s derivation
of quantum mechanics from principles cannot be considered a
strictly rigorous derivation, especially in a mathematical sense.
As he noted in The Physical Principles of the Quantum Theory
(from which title I borrow my title of this section): “The
deduction of the fundamental equation of quantum mechanics
is not a deduction in the mathematical sense of the word, since
the equations to be obtained form themselves the postulates
of the theory. Although made highly plausible, their ultimate
justification lies in the agreement of their predictions with the
experiment” [10, p. 108]. While Heisenberg, again, borrowed the
form of equations themselves from classical mechanics by the
mathematical correspondence principle, he virtually guessed the
variables he needed—one of the most extraordinary guesses in
the history of physics. A more rigorous derivation of quantum
mechanics from fundamental principles may, thus, be pursued.
More recent work in this direction has been in quantum
information theory in the case of discrete quantum variables,
such as spin, which require finite-dimensional Hilbert spaces, as
opposed to infinite-dimensional ones for continuous variables,
such as position and momentum (e.g., 13–15)8 . I shall comment
on this work below.
Bohr’s interpretation of quantum phenomena and quantum
mechanics added a new principle, the complementarity principle.
It arises from Bohr’s concept of complementarity and may
be defined as requiring: “(a) a mutual exclusivity of certain
phenomena, entities, or conceptions; and yet (b) the possibility of
considering each one of them separately at any given point, and (c)
the necessity of considering all of them at different moments for a
comprehensive account of the totality of phenomena that one must
consider in quantum physics” [9, p. 70].
In Bohr’s ultimate interpretation, this concept applies strictly
to what is observed in measuring instruments, quantum
phenomena, and not to quantum objects, placed beyond
representation or even conception. Complementarity is a
reflection of the fact that, in a radical departure from classical
physics or relativity, the behavior of quantum objects of
8 Among the

key earlier approaches are [16], Fuchs’s work, which “mutated” to the
program of quantum Bayesianism or QBism [17], and [18].

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the same type, say, electrons, is not governed by the same
physical law, especially a representational physical law, in all
possible contexts, specifically in complementary contexts. In
other words, the behavior of quantum objects has mutually
incompatible effects in complementary set-ups, although this
mutual incompatibility is, generally, manifested collectively, in
multiple identically prepared experiments. On the other hand,
the mathematical formalism of quantum mechanics offers correct
probabilistic or statistical predictions of quantum phenomena in
all contexts, in non-realist interpretations, under the assumption,
that quantum objects and processes are beyond representation or
even conception, by the RWR principle.
In some non-realist interpretations, such as the one the
present author would favor, following W. Pauli, individual
quantum events are not subject even to the probabilistic laws of
quantum mechanics. This makes these laws collective, statistical
[9, pp. 173–186; 11]. The QP/QS principle, accordingly, becomes
strictly the QS principle. According to Pauli:
As this indeterminacy is an unavoidable element of every initial
state of a system that is at all possible according to the [quantummechanical] laws of nature, the development of the system can
never be determined as was the case in classical mechanics.
The theory predicts only the statistics of the results of an
experiment, when it is repeated under a given condition. Like
the ultimate fact without any cause, the individual outcome of a
measurement is, however, in general not comprehended by laws.
This must necessarily be the case, if quantum or wave mechanics is
interpreted as a rational generalization of classical physics, which
take into account the finiteness of the quantum of action [h]. The
probabilities occurring in the new laws have then to be considered
to be primary, which means not deducible from deterministic
laws. [19, p. 32]

Thus, in Pauli or the present view, this “beyond the law” includes
the probabilistic or, in this view, statistical laws of quantum
mechanics, laws that, thus, only apply to statistical multiplicities
of repeated quantum events. Individual quantum events are
not subject to laws, even to the probabilistic or statistical laws
of quantum mechanics. Their outcomes cannot, in general,
be assigned a probability: they are strictly random9 . Only the
statistics of multiple (identically prepared) experiments could
be predicted and repeated, which repeatability appears to have
been, thus far, necessary for scientific practice. Whether, however,
one interprets quantum mechanics on such statistical lines or
on the Bayesian lines, by assigning probability to individual
events, we are compelled to rethink the concept of physical law
as unavoidably contextual. This is “an entirely new situation as
regards the description of physical phenomena that, the notion
of complementarity aims at characterizing” [20, p. 700].
There are other important features of quantum phenomena,
mathematically expressed in the quantum-mechanical
formalism, in particular, the so-called “quantum non-locality,”
which refers to the existence of the statistical correlations
9 Randomness may be defined by this impossibility. This concept of randomness is

not ontological, because one cannot ascertain the reality of this randomness, but
epistemological. It is ultimately a matter of assumption or belief, practically justified
in a given interpretation.

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between spatially separated quantum events, and “quantum
entanglement,” which reflects these correlations in the formalism.
These features were discovered later and played no role in the
initial derivation of quantum mechanics by either Heisenberg
or Schrödinger. They do figure significantly in quantum
information theory and recent attempts, mentioned above, to
derive quantum mechanics from the principles of quantum
information. Their analysis would require a treatment beyond
my scope10 . A few key points may, however, be mentioned.
First, while quantum entanglement is a clearly defined feature of
the formalism, the situation is different in the case of quantum
non-locality. Although originating in the experimentally
well-confirmed fact that certain spatially separated quantum
phenomena or events exhibit statistical correlations (not found
in classical physics), quantum non-locality is a complex and
much debated issue. The problematic was in effect introduced
in 1935 in the famous article by Einstein et al. [22]. I qualify
because neither EPR’s article nor Bohr’s equally famous reply to
it [20] used the language of correlations or entanglement. The
latter term was introduced, in both German [Verschränkung]
and English, by Schrödinger in his response to EPR’s article,
known as “the cat-paradox paper,” after the paradox found there
[23]. The subject remained dormant until the 1960s, when it
was rekindled by the Bell and Kochen-Specker theorems, even
to the point of nearly defining the current debate concerning
quantum foundations. The theoretical and experimental research
on the subject during the last decades has been massive and
literature concerning it is immense. The term “non-locality” is
not uniformly used in referring to quantum correlations, because
it may suggest some sort of instantaneous physical connections
between distant events, a “spooky action at a distance,” as
Einstein called it. Such connections are incompatible with
relativity, although the principle of locality, which prohibits
such connections, is independent of relativity. This type of
physical non-locality, which is found, for example, in Bohmian
mechanics, is commonly viewed as undesirable. The absence of
realism allows one to avoid physical non-locality, as Bohr argued
in his reply to EPR’s article, which contended that quantum
mechanics is either incomplete or physically nonlocal [20, 22].

FROM MODELS TO PRINCIPLES IN
Q-MODELING OUTSIDE PHYSICS
Q-Models, Fundamental Principles, and
Reality without Realism Outside Physics
In addressing Q-models in physics in preceding discussion, my
main question, arising from the history of quantum theory,
was: Given certain fundamental physical principles, established
on the basis experimental evidence, in particular the QD and
QP/QS principles, and perhaps adopting additional principles,
such as the correspondence principle or the RWR (or protoRWR) principle, what are the mathematical models that would
10 I have discussed the subject, also in relation to complementarity, in Plotnitsky
(9, pp. 136–54). These connections also bring in a related (EPR-correlation)
concept, “contextuality.” This concept plays a significant role in Q-modeling
beyond physics [1, pp. 363–5, 21].

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enable us to handle this evidence? In turning now to the Qmodels beyond physics, my main question is reverse: Assuming
that mathematical Q-models apply in psychology, economics,
and decision science, which features and which fundamental
principles are behind such models, and how they accord with
the fundamental principles of quantum mechanics? There are
two sets of principles I have in mind. The first contains the
principles that led to the emergence of quantum mechanics; and
the second the principles of quantum information theory, which
are, however, in accord with most principles of the first set. I
shall be primarily concerned with this first set (apart from the
correspondence principle, unique to quantum theory), but will
also comment on the second11 .
But why is this question important in the first place? As
noted from the outset, if there are phenomena outside physics
that appear to require Q-models, one need, unlike at the time
of the introduction of quantum mechanics, not invent such
models at this point. One can borrow them, “ready-made,” from
quantum theory, which is what happed in the case of Q-modeling
outside physics. Nevertheless, establishing, now inferentially,
fundamental principles behind Q-models might allow us to
make important conclusions about the nature of the phenomena
handled by these models. To put it in stronger terms, finding
the fundamental principles behind a given model, even if this
model is already available, is important because otherwise we
don’t have a rigorous theory or a rigorous model, which is true
even if a constructive theory is available, but is all the more
important if it is not. Otherwise, we don’t really know what
our models are models of, especially, again, in the absence of
a constructive theory and realism, which absence is likely if Qmodels apply and is my main interest here. These considerations
are also relevant in pursuing projects of more rigorous derivation
of quantum mechanics from principles in physics, for example
on lines of quantum information theory, even though the theory
itself is already established. Part of the reason is, again, that
doing so can give us a deeper understanding of quantum
phenomena and quantum theory. More, however, is at stake.
The main value of such projects lies in solving outstanding
problems of fundamental physics, as in quantum field theory
(which still has unresolved problems, its extraordinary successes
notwithstanding) or quantum gravity, which has no model as
yet [24, 25]. The same argument applies to Q-modeling beyond
physics. The future of mathematical modeling there is at stake as
well.
Before addressing the relationships between fundamental
principles and Q-models in psychology, economics, and decision
science, it may be helpful to summarize the non-realist, the RWRprinciple-based, interpretation of quantum phenomena and
quantum mechanics outlined in Section Physical Principles and
Mathematical Models in Quantum Mechanics. While quantum
objects are assumed to exist, the character of this existence
or reality is, by the RWR principle, assumed to be beyond
representation and even conception. As such, this reality is
different from the reality of quantum phenomena, which are
11 I

have discussed the role of principles of quantum information theory beyond
physics in Plotnitsky [6].

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defined by what is observed in measuring instruments under the
impact of quantum objects and, thus, can be represented. There
are no mathematically expressed physical laws corresponding
to the behavior of quantum objects. There are, however,
mathematical laws that, expressing the QP/QS principle, enable
correct probabilistic or statistical predictions of the outcomes of
quantum experiments, manifested in measuring instruments, in
all contexts. In addition, there are two interpretations of these
mathematical laws. The first is probabilistic, along Bayesian lines,
in which case these laws are seen as allowing one to assign
probabilities to the outcomes of individual quantum events in
accordance with one or the other law of the available set of laws,
specifically those applicable in complementary situations. The
second is statistical, when no such probabilities could be assigned
because the outcomes of individual quantum experiments are
not comprehended even by these laws and are seen as random,
while these laws are assumed to predict the statistics of multiple
identically prepared experiments in the corresponding contexts.
It is clear, however, that this conceptual architecture, in either
the Bayesian or statistical interpretation, cannot apply unaltered
in considering, along non-realist lines, human phenomena found
in psychology, economics, or decision science and the possible
Q-models there. This is because, while there are individual
objects or, the case may be, (human) subjects and processes
to consider, there are no elementary objects of the type found
in quantum physics. There is nothing analogous to elementary
particles, such as electrons or photons, and there is rarely a
completely random individual behavior. When one deals in
these fields with large multiplicities one can, either in using Cor Q-models, average the individual behavior and statistically
disregard the differences in this behavior, differences defined by
psychological or other human and social factors, in which case
one could apply either a Bayesian or statistical interpretation of
the Q-model used. While, however, this averaging is sometimes
possible in psychology, economics, and decision science, there
are often significant obstacles in using it. Each sequence of events
considered in such situations is singular, unique. Accordingly, if
a Q-model applies in a given class of such cases, it would have
to be interpreted on Bayesian lines, if one can establish such a
class. If not, then, as discussed below, another type of models may
be possible, the singularized probabilistic (SP) models, some of
which are time-dependent (TDSP). Each such model is unique
to the individual situation considered, rather than applicable to
a class of individual situations; and this uniqueness may pose
difficulties for scientific use of such models.

The QP/QS Principle and the
Complementarity Principle
Beginning with Tversky and Kahneman’s work in the 1970–80’s
[e.g., 26], it has been primarily the presence of probabilistic data
akin to those encountered in quantum physics that suggested
using Q-models in cognitive psychology, decision science, and
economics [e.g., 1, 2]12 . Economic behavior may also involve
psychological factors of the type analyzed by Tversky and
Kahneman. (Kahneman was eventually awarded a Nobel Prize
12 I also refer to these works for more detailed discussions of the ways in which
Q-models are used in these fields.

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in economics.) The recourse to Q-models is motivated by the
fact that one could not effectively use the classical (additive)
rules but could use the quantum-mechanical-like (non-additive)
rules for predicting the probabilities of the outcomes of
certain psychological experiments, such as those involving
responses to certain specific questions, asked sequentially. These
responses were found to be statistically dependent on the
order in which they were asked, which, again, in parallel
with quantum mechanics, suggested that a non-commutative
model and, in combination with the non-additive rules for
calculating the probabilities involved, a Q-model could be more
effective13 . To clarify this parallel, in quantum mechanics,
simultaneously measuring, or simultaneously asking questions
concerning, two or more complementary variables, such as
the position and the momentum of a given quantum object,
are mutually exclusive or incompatible. Correlatively, changing
the order of measuring (of asking the question concerning)
the position and then the momentum of a quantum object,
in general, changes the outcomes and hence our predictions
concerning them. This circumstance is reflected, experimentally,
in the uncertainty relations, and mathematically, in the noncommutativity of the multiplication of the corresponding
Hilbert-space operators in the formalism, and epistemologically,
in the complementarity of these two measurements. One can,
analogously, consider psychologically incompatible and, thus,
complementary questions in psychology and attempt to handle
the corresponding events statistically by a Q-model [e.g., 1, pp.
259–260]. The situation involves further complexities in and
outside quantum physics, which I put aside here. I would like,
however, to mention R. Spekkens’s article, which introduced
“a toy theory,” based on the following principle, linked to
complementarity: “the number of questions about the physical
state of a system that are answered must always be equal
to the number that are unanswered in a state of maximal
knowledge. Many quantum phenomena are found to have
analogs within this toy theory.” Many but not all! For the theory
expressly fails to reproduce some among the crucial features
of quantum theory, specifically and intriguingly some of those
related to correlations and entanglement, such as “violations of
Bell inequalities and the existence of a Kochen-Specker theorem”
[27, p. 032110]. This failure reminds us that models based on the
existence of incompatible questions, in and outside physics, may
mathematically differ from quantum mechanics.
Q-models are, then, used to predict probabilities and
correlations found in such experiments, without being expressly
concerned with the principles characterizing the situations
considered, but only assuming certain mathematical principles
inherent in the quantum-mechanical formalism. Some among
the principles of the first kind are, nevertheless, implicitly
at work, specifically the QP/QS principle or the principle of
incompatibility, in effect complementarity14 . Whether these Qmodels are required or C-models, models derived from the
13 As noted earlier, this does not mean that such probabilities could not be predicted

by means of alternative models even in quantum physics.
14 Complementarity has received some attention outside physics, beginning with
Bohr’s own (tentative) suggestions. Inspired by Bohr and others did propose using
the concept in philosophy, biology, and psychology. See Plotnitsky [28, pp. 158–66]
and [29].
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mathematics of classical physics, suffice remains, again, an open
question, although it is difficult to assume that C-models could
provide the non-additive probabilities necessary in such cases.
A model alternative to that of quantum mechanics, possibly
also free of quantum amplitudes and dealing directly with
probabilities, is, in principle, possible even, as noted earlier, in
quantum physics, but such a model is unlikely to be akin to
those of classical physics. Thus, while they are both realist and
causal, Bohmian models are mathematically different from those
of classical physics. It may also be possible to construct a realist
and causal mathematical model that would represent a deeper
level of reality and that would have quantum mechanics as its
limit, and then extend this model beyond physics [e.g., 30].
In any event, one can see the QP/QS principle, in part
in conjunction with complementarity, as the main principle
behind the use of Q-models beyond physics, accompanied, as
in quantum mechanics, by the specific (non-additive) calculus
of probability. Indeed, the QP/QS principle, along with the
QD principle, was the starting principle for Heisenberg. The
role of complementarity, only implicit initially by virtue of
the non-commutative nature of Heisenberg’s scheme, became
apparent shortly thereafter, helped by Heisenberg’s discovery of
the uncertainty relations in 1927. It became clear that noncommutativity, the uncertainty relations, and complementarity
were correlative, representing, respectively, the mathematical,
physical, and epistemological aspects of the quantum-mechanical
situation, defined by quantum discreteness (the QD principle).
As noted earlier, quantum discreteness was eventually rethought
by Bohr in terms of quantum phenomena, defined by what
is observed in measuring instruments impacted by quantum
objects, as opposed to the nature of quantum objects and
processes, which are beyond conception and, hence, cannot be
thought of as either discrete or continuous.
The psychological, economic, and decision-making
phenomena treated by means of Q-models do not exhibit
this type of irreducible discreteness or individuality. The
processes that connect these phenomena are more akin to
processes considered in classical physics, especially in chaos or
complexity theory, again, often providing mathematical models,
C-models, used in these fields. Now, assuming the defining
role of, jointly, the QP/QS principle and the complementarity
principle in considering these phenomena, could some form of
the QD principle, correlative to the QP/QS principle in quantum
mechanics, find its place in considering or even in order to derive
Q-models in these fields? And if so, or in the first place, would
the RWR principle, or a proto-RWR principle of the type used
by Heisenberg, also be applicable? There are reasons to believe
that such might be the case.

The RWR and QD Principles
Bohr thought that, along with the complementarity principle,
the RWR principle might apply in biology and psychology. In
considering biology, he argued as follows:
The existence of life must be considered as an elementary fact
that cannot be explained, but must be taken as a starting point
in biology, in a similar way as the quantum of action, which

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The Real and the Mathematical

appears as an irrational element from the point of view of the
classical mechanical physics, taken together with the existence of
elementary particles, forms the foundation of atomic physics. The
asserted impossibility of a physical or chemical explanation of the
function peculiar to life would in this sense be analogous to the
insufficiency of the mechanical analysis for the understanding of
the stability of atoms. [31, p. 458; emphasis added]

The ultimate character of biological processes may, thus, be
beyond representation or even conception, in accord with
the RWR principle. Once the theory suspends accounting for
the connections between the phenomena considered, these
phenomena are unavoidably discrete, leading to the QD
principle, and our predictions concerning them are unavoidably
probabilistic, leading to the QP/QS principle. Our predictions
concerning them are likely to follow a (non-additive) probability
calculus of the type used in quantum probability, and thus are
likely to require a Q-model. This is because, by the RWR or
proto-RWR principle, it would be difficult or even impossible
to treat the processes connecting the phenomena considered
as either continuous or causal. Bohr’s appeal to “an irrational
element” is noteworthy, and I shall comment on it below. It is
important that, as Bohr clearly implies here, this approach is
possible even if the nature of biological processes is not physically
quantum in the sense of being able to have physically quantum
effects. (The ultimate constitution of all matter is quantum, but
this constitution does not manifest itself apart from quantum
experiments.) If they were quantum, such processes would
be unrepresentable or inconceivable in Bohr’s interpretation.
At stake here, however, are parallel, rather than physically
connected, situations that may require using the same type of
mathematical models, Q-models, without possible connections
between the systems defining these situations15 .
A recent article by Haven and Khrennikov provides an
instructive example for possible roles of both the RWR and
QD principle in market economics in their Q-modeling of
market phenomena involving arbitrage as analogous to quantum
tunneling [33]. The term “quantum tunneling” refers to a
quantum object’s capacity to “tunnel” through an energy barrier
that it would not be able to surmount if it behaved classically. It
is a quantum phenomenon par excellence. The quantum process
itself behind any given case of quantum tunneling cannot be
observed. One only ascertains that a particle can be found
beyond the barrier, which is to say, that the corresponding
measurement will register an impact of this particle on the
measuring instrument beyond the barrier. Thus, in accord with
the general situation that obtains in quantum mechanics, one
deals with two discrete phenomena, connected by probabilistic
or (in which case, we need multiple trials) statistical predictions
concerning the second event on the basis of the first. “Arbitrage”
is the practice of taking advantage of a price difference between
two or more markets: striking a combination of matching deals
that capitalize on the imbalance, the profit being the difference
15 There

are several recent arguments for such connections, most prominent of
which is arguably that by Penrose [32] and developed in several subsequent
studies. The model itself that Penrose has in mind is, thus far, only mathematically
conjectured, following certain approaches to quantum gravity.

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This view at least allows for an interpretation of the phenomenon
of arbitrage in terms of the QD and the RWR principles,
even if it does not require it. Haven and Khrennikov, while,
again, allowing for the applicability of the QD principle, do not
appear to subscribe to the RWR principle, or even to the protoRWR principle16 . In effect, however, they follow the proto-RWR
principle, insofar as they are not concerned with representing
how arbitrage actually occurs, any more than Heisenberg was
concerned with representing the behavior of the electron in
the hydrogen atom in deriving his formalism. They are only
concerned with predicting the probabilities or statistics of future
events of arbitrage.
Thus, situations governed the QD, QP/QS, and RWR (or
proto-RWR) principles are possible in economics, psychology,
and decision science, and just as in quantum mechanics, they
may allow for either a statistical or Bayesian view of the Qmodel used. When finite-dimensional Q-models (dealing with
discrete variables, such a spin) are used, as they often are in these
fields, one can also consider the application of the principles of
quantum information theory. While I cannot address the subject
in detail, the operational framework, used in this field, merits a
brief detour. This framework allows one to arrive at Q-models
in a more rigorous and first-principle-like way, by using the

rules governing the structure of operational devices, “circuits,” via
recent work on monoidal categories and linear logic [13–15, 34].
According to Chiribella et al.: “The operational-probabilistic
framework combines the operational language of circuits with the
toolbox of probability theory: on the one hand experiments are
described by circuits resulting from the connection of physical
devices, on the other hand each device in the circuit can
have classical outcomes and the theory provides the probability
distribution of outcomes when the devices are connected to
form closed circuits (that is, circuits that start with a preparation
and end with a measurement)” [13, p. 3]. A circuit is an
arrangement of measuring instruments capable of quantum
measurements and predictions, which are, again, probabilistic or
statistical, and sometimes, as in the EPR type of experiments,
are correlated, which gives a circuit a very specific architecture,
corresponding only to quantum but not classical experiments.
A realist representation of a circuit is possible because a circuit
is described by classical physics, even though it interacts with
quantum objects, and thus has a quantum stratum, enabling
this interaction. Hence, the information obtained by means of a
circuit is physically classical, too, but the architecture and mode
of transmission of this information is quantum: they cannot be
generated by a classical process.
As discussed earlier, Heisenberg found the formalism of
quantum mechanics by adopting, in addition to the QD, QP/QS,
and proto-RWR principles, the mathematical correspondence
principle and, by the latter principle, using the equations
of classical mechanics, while changing the variables in these
equations. This principle was not exactly the first principle. In
particular, it depended on formally adopting the equations of
classical mechanics, while one might prefer these equations to be
a consequence of fundamental quantum principles. Heisenberg’s
variables were new, which was his great discovery. But they were
new more of a guess, a logical guess, fitting the probabilities
of transitions between the energy levels of the electron in the
hydrogen atom he worked with. In the operational framework,
one derives finite-dimensional quantum theory in a more
first-principle-like way, in particular, independently of classical
mechanics (which does not exist for discrete variables, such as
spin). This derivation is made possible by applying the rules
that define the operational language of circuits, as the language
of monoidal categories and linear logic, and thus giving a
mathematical structure to operational circuits themselves and
thus, in effect, to measuring instruments [13, p. 4, 33]. These rules
are more empirical, but they are not completely empirical (which
no rules may ever be), because circuits are given a mathematical
structure, from which the mathematical architecture of the theory
emerges17 . The resulting formalism is equivalent to the standard
Hilbert-space formalism. As in Heisenberg, one only deals with
“mathematical representations” providing the probabilities or
statistics of the outcomes of discrete quantum experiments, in
accord with the QD and QP/QS principles, without providing a
representation of quantum processes themselves, in accord with
the RWR principle.

16 As indicated earlier, elsewhere Khrennikov argued for a classical-like model at
the ultimate level of the constitution of nature in physics [30].

17 See also Plotnitsky

between the market prices. An arbitrage is a transaction that
involves no negative cash flow at any probabilistic or temporal
state and a positive cash flow in at least one state; in simple
terms, it is the possibility, ideally, of a risk-free profit at zero
cost. In practice, there are always risks in arbitrage, sometimes
minor (such as fluctuation of prices decreasing profit margins)
and sometimes major (such as devaluation of a currency or
derivative). In most ideal models, an arbitrage involves taking
advantage of differences in price of a single asset or identical
cash-flows.
Now, if arbitrage can be modeled analogously to quantum
tunneling in physics, one might expect features analogous to
those found in quantum tunneling, which dramatically exhibits
the character of quantum phenomena. Haven and Khrennikov
are primarily concerned with the use of Q-models in predicting
the probabilities involved, by QP/QS principle (accompanied by
the non-additive calculus of probabilities), rather than with the
QD and the RWR, or proto-RWR, principles. They do, however,
offer some considerations concerning discreteness:
We believe that the equivalent of quantum discreteness in this
paper corresponds to the idea that each act of arbitrage is a
discrete event corresponding to the detection of a quantum
system after it passed ... the barrier. In reality arbitrage
opportunities do not occur on a continuous time scale. They
appear at discrete time spots and often experience very short
lives. We would like to argue that it is the tunneling effect
which is closely associated to the occurrence of arbitrage. ...
We also mentioned the wave function in the discussion above,
and quantum discreteness is narrowly linked with quantum
probabilities. [33, p. 4095]

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[9, pp. 248–58] and Hardy [15].

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In the areas of social science, which concerns human subjects,
establishing the mathematical architecture for such “circuits” is a
formidable task. However, given important recent work along the
lines of category theory beyond physics [e.g., 35], this approach
may prove to be viable in enabling a principle approach in
Q-modeling outside physics18 .

Q-Theories as Rational Theories of the
Irrational
As indicated earlier, while the main reasons for using Q-models
in psychology, economics, and decision science are due to the
quantum-like nature or calculus of the probabilities associated
with predicting certain phenomena, the underlying dynamics
of the cognitive or psychological processes leading to each
such phenomenon individually might, in principle, be causal
or partially causal. This dynamics might also not be causal,
especially given the quantum (non-additive) character of the
probabilities involved. If it is causal or partially causal, then,
unlike quantum processes, in non-realist interpretations, an
analysis of these psychological processes may be possible, rather
than “in principle excluded” [3, v. 2, p. 62]. This is because one
might expect psychological, social, or economic reasons shaping
these situations, and one of the tasks of analyzing them to explain
these reasons, an imperative that is hard to avoid, as is clearly
apparent in Tversky and Kahneman’s articles [26, 37] or in Pothos
and Buseymeyer’s survey [1].
Psychological, social, or economic research using Q-models
may renounce this task, especially in statistical analysis, thus in
effect assuming a form of proto-RWR principle, akin to that used
by Heisenberg. Even in this case, however, the question would
still arise to what degree the QP/QS, QD, and (strictly) RWR
principles, or the principles of quantum information theory,
could apply in these fields, in particular in considering individual
situations. As explained earlier, in quantum mechanics, in
non-realist interpretations, the latter could either be treated
on Bayesian lines or, in statistical interpretations, assumed to
be random, which assumption would, again, be difficult in
the fields in question at the moment. Some considerations of
discreteness are unavoidable because, as noted, probability has
an irreducibly futural and discrete character by dealing with
estimates concerning discrete future events.
It is a more complex question whether one can renounce, as
one does in quantum mechanics, in non-realist interpretations,
considering or even assuming the existence of continuous
processes connecting these events. I would surmise that such
may be the case and that our brains may work, at least
sometimes, in accordance with the QD, the QP/QS, and the
RWR principles. This means they would not be relying on
and calculating hidden causality connecting events but would
instead functions by relying on the quantum-like workings of
probabilities and correlations. This type of brain functioning
would define what may be called a Bayesian Q-brain, which
would require the corresponding Bayesian models. Importantly,

The Real and the Mathematical

however, this kind of Bayesian brain is fundamentally different
from rational Bayesian agents, associated with the term Bayesian
in cognitive psychology. Indeed, Q-models there are in part
advanced in these fields against this concept of human agency.
A Bayesian Q-brain need not always function “rationally,” at
least, not in accordance with any single concept of rationality.
A corresponding Bayesian Q-model, if possible, would allow
one to predict the outcomes of decisions governed by the brain
processes of the individual subjects involved without having,
even conjecturally, a full access to these processes, by the RWR
principle. Nor do those who make these decisions have this
access: these processes are unconscious, and, if one assumes the
RWR principle, this part of the unconscious is not causal or
“rational” (in its own way), as S. Freud, for example, saw it [38].
Freud’s thinking on this point was, however, ultimately more
complex, even if against his own grain.
It is instructive to return, in this context, to Bohr’s invocation
of “an irrational element,” in the passage cited above and repeated
elsewhere in his writings. The idea and even the language of
irrationality have often been seen as problematic by Bohr’s critics
and even by some of his advocates. I would argue this assessment
to be a result of misunderstanding Bohr’s meaning. This
“irrationality” is not any “irrationality” of quantum mechanics,
which Bohr saw as a rational theory, a “rational quantum
mechanics,” and argued for its rational character throughout his
writing (e.g., 3, v. 1, p. 48; 3, v. 2, p. 63). However, he did see it as a
rational theory of something—the nature of quantum objects and
processes—that is inaccessible to rational thinking, or at least to
a rational representation. If, as he says, “the quantum of action
[h], which appears as an irrational element from the point of view
of the classical mechanical physics,” it only means that cannot be
rationally incorporated into the latter [31, p. 458].
Tversky and Kahneman’s and related arguments are, too,
sometimes seen as related to “irrational” elements in decisionmaking. This decision-making replaces purportedly “rational”
Bayesian agents with at least partially “irrational” Bayesian
agents. The “rational” Bayesian agents, as explained above, use
probabilistic reasoning subject to updating their estimates on the
basis of new information (which defines the Bayesian approach
to probability). The irrationality of “irrational” Bayesian agents
may be divided into three main, sometimes overlapping, types.
The first type is in effect a form of rationality. This rationality
is, however, different from rationality presumed to be dominant
in the class of situations considered, say, the rationality of
maximizing one’s monetary benefits. In addition, this alternative
rationality may be unconscious. The second type of irrationality
refers to something that could be explained. However, it defies
explaining it as anything assumed to be rational, say, as a form
of rational behavior, beforehand. This irrationality may, upon
further analysis, reveal itself to be the irrationality of the first type,
but it may also be an alternative form of rationality19 . Finally,
the third type of irrationality is that invoked by Bohr: a realist
19 Some

18 See

also a recent approach to representing sensation-perception dynamics
in terms of quantum-like mental instruments, which are akin to “circuits,” in
Khrennikov [36].

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might still see, as Freud did, this “irrationality” as a form of unconscious
“rationality.” Once again, however, Freud, against his own grain, could not
ultimately avoid giving the unconscious a stratum that is beyond representation,
if not conception.

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theory cannot incorporate it in its handling of the corresponding
phenomena, while a non-realist Q-model or theory can make it
part of its probabilistically predictive scheme without explaining
it. In this way, QD, QP (or, if averaging is possible QS), and RWR
principles can be brought together in this domain.
There is yet another possibility, which leads to a different
type of models or theories, conforming to the QD, QP (but not
QS), and RWR principles. I shall call such models or theories
singularized probabilistic (SP) models or theories, keeping in
mind their non-realist, RWR-principle-based, character. Realist
SP models are possible, but I shall not be concerned with them.
SP-models may also be time-dependent (TDSP). Such models
can only be briefly sketched here in conceptual and somewhat
abstract terms, but their possibility is intriguing. SP- or TDSPmodels need not be mathematically related to Q-models, but they
might be, given the shared principles in which they are based.

Singularized Probabilistic (SP) Theories
and Models
Let us recall that, as reflected in the complementarity principle,
in quantum mechanics there is no single, uniform physical
law applicable to quantum behavior in all contexts, while the
same mathematical formalism or model can be used in all
contexts. Depending on whether an interpretation is statistical
or (Bayesian) probabilistic, the individual quantum behavior
is either assumed to be random or to be subject to the
probabilistic law, the application of which is defined by the
context. By contrast, in the case an SP-model or theory, the
following situation obtains. While, as in quantum physics,
there is no single uniform physics law, realist or not, each
individual behavior obeys its own singular law, defined by its own
mathematical model, rather than conforms to one or another
contextual probabilistic or statistical law, from a (determinable)
set of such laws determined by the theory, using a single
mathematical model. Under the RWR principle, assumed here
for SP-models, such a model still does not represent the reality
of the ultimate processes considered, which makes the absence
of not only determinism but also causality automatic, just as
in quantum mechanics under the RWR principle. One cannot,
however, any longer adopt a statistical view, which assumed
multiplicities of events that could be averaged (in quantum
mechanics, contextually). In each case, only a Bayesian view of
the corresponding (unique) model is possible. Such individual
laws and accompanying mathematical models may also be
changing in time, a change observed each time a new observation
occurs. If so, the corresponding model or theory becomes time
dependent, TDSP.
The concept of an SP and especially a TDSP model or theory
is a radical idea, to my knowledge, rarely, if ever, entertained,
at least in science20 . Indeed, it is not clear whether such
theories and, especially, the mathematical models defined by
them are scientifically viable, particularly if the corresponding
mathematical laws are assumed to be changing in time, possibly
20 Something akin to this possibility has been suggested in physics in Ungar and
Smolin [39], but in a different context and based it on a very different set of
principles than those adopted here, most especially because, as against the present
argument, they assume realism and causality.

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The Real and the Mathematical

on small scales. For an effective scientific practice to be possible,
one might need regularities beyond those found in each singular
situation, for which a mathematical model, unique to it, would
be introduced, say, in order to predict the outcome of events.
Such changes of laws and models could, in principle, be governed
mathematically, have an overall mathematical model. Thus, one
could have a set of models mathematically parameterized so as
to allow one to use them for different individual situations and to
adjust them to make effective predictions in all of these situations.
If not, then each case would require its own mathematical model.
Would mathematical-experimental sciences, as they are practiced
now, still be possible, then?
Furthermore, there might, in a given domain, be individual
cases the character of which will defeat our attempt to treat them
by mathematical means. Indeed, this is already so in the case
individual quantum processes if one adopts a statistical view,
according to which each individual process is random, beyond
the law. Now, however, there would not be statistical regularities,
of the type found in quantum physics, applicable to multiplicities
of repeatable cases (handled, moreover, by the same model,
even if contextually), because there would be no repeatable
cases in any meaningful sense. There would be neither statistical
averaging, nor individual mathematical probabilistic treatment.
This situation may be more familiar in literature, which is
concerned with the particular or the singular, for example, with a
unique life history of a novel’s protagonist. One also encounters
this singularity or uniqueness in life itself. Such histories resist
and even preclude statistical averaging, again, allowed by,
otherwise equally unique, histories (which cannot be thought of
as classical trajectories of motion) of individual quantum objects,
as well as mathematical handling. But they may become, at least
outside physics, perhaps especially, in psychology (which often
deals with the same human conditions as literature), part of
science, a science that will combine science and non-science, or
at least mathematical, both of the more standard or the SP/TDSP
type, and nonmathematical modeling. Indeed, as just indicated,
the SD/TDSP-modeling already poses complexities for scientific
practice. Could this situation also emerge in physics, for example,
in dealing with quantum gravity? This is not inconceivable. If
it does, it will not end mathematical modeling in physics or,
again, beyond, or the mathematical-experimental character of
modern science, which has defined it beginning with Galileo.
It might, however, change both, just as it happened in the
case of quantum theory, which not only led to a revolutionary
transformation—physical, mathematical, and philosophical—of
physics itself but also opened new possibilities for scientific
thinking and mathematical modeling beyond physics.

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
approved it for publication.

FUNDING
This work was funded by The Purdue Distinguished
Professorship Research Fund.
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Plotnitsky

ACKNOWLEDGMENTS
I would like to thank Mauro G. D’Ariano, Emmanuel Haven,
Gregg Jaeger, and Andrei Khrennikov for helpful discussions
concerning the subjects considered in this article. I am grateful

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Conflict of Interest Statement: The author declares that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2017 Plotnitsky. This is an open-access article distributed under the
terms of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) or licensor
are credited and that the original publication in this journal is cited, in accordance
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which does not comply with these terms.

June 2017 | Volume 5 | Article 19 | 139

REVIEW
published: 04 July 2017
doi: 10.3389/fphy.2017.00018

Quantization, Frobenius and Bi
Algebras from the Categorical
Framework of Quantum Mechanics
to Natural Language Semantics
Mehrnoosh Sadrzadeh *
Theory Group and Computational Linguistics Lab, School of Electronic Engineering and Computer Science, Queen Mary
University, London, United Kingdom

Edited by:
Emmanuel E. Haven,
University of Leicester,
United Kingdom
Reviewed by:
Yousef Azizi,
Institute for Advanced Studies in Basic
Sciences, Iran
Jan Sladkowski,
University of Silesia in Katowice,
Poland
Alexander Vladimirovich Bogdanov,
Saint Petersburg State University,
Russia
*Correspondence:
Mehrnoosh Sadrzadeh
mehrnoosh.sadrzadeh@qmul.ac.uk

Compact Closed categories and Frobenius and Bi algebras have been applied to
model and reason about Quantum protocols. The same constructions have also been
applied to reason about natural language semantics under the name: “categorical
distributional compositional” semantics, or in short, the “DisCoCat” model. This model
combines the statistical vector models of word meaning with the compositional models
of grammatical structure. It has been applied to natural language tasks such as
disambiguation, paraphrasing and entailment of phrases and sentences. The passage
from the grammatical structure to vectors is provided by a functor, similar to the
Quantization functor of Quantum Field Theory. The original DisCoCat model only used
compact closed categories. Later, Frobenius algebras were added to it to model long
distance dependancies such as relative pronouns. Recently, bialgebras have been added
to the pack to reason about quantifiers. This paper reviews these constructions and their
application to natural language semantics. We go over the theory and present some of
the core experimental results.
Keywords: compact closed categories, frobenius algebras, bialgebras, quantization functor, categorical quantum
mechanics, compositional distributional semantics, pregroup grammars, natural language processing

1. INTRODUCTION
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 12 December 2016
Accepted: 23 May 2017
Published: 04 July 2017
Citation:
Sadrzadeh M (2017) Quantization,
Frobenius and Bi Algebras from the
Categorical Framework of Quantum
Mechanics to Natural Language
Semantics. Front. Phys. 5:18.
doi: 10.3389/fphy.2017.00018

Frontiers in Physics | www.frontiersin.org

Categorical compositional distributional semantics is a model of natural language that combines
the statistical vector models of word meanings with the compositional models of grammar. The
grammatical structures are modeled as morphisms of a compact closed category of grammatical
types, the vector representations of word meanings are modeled as morphisms of the category
of finite dimensional vector spaces, which is also a compact closed category. The passage from
grammatical structure to vectorial meaning is by connecting the two categories with a structure
preserving map, in categorial words, a functor.
F : Grammar H⇒ Meaning
This passage allows us to build vector representations for meanings of phrases and sentences, by
using the vectors of the words and the grammatical structure of the phrase or sentence. Formally,
this procedure is the application of the image of the functor on the grammatical structure to the

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CQM and Natural Language

meaning vectors of the words. Still more formally, given a string
of words w1 w2 · · · wn , one first formalizes their grammatical
structure as a morphism α in the compact closed category of
grammar, introduced by Lambek and Lambek and Preller, as
the categorical semantics of pregroup type-logical grammars,
see Lambek [1, 2]. We denote these below by Preg. The vector
meanings of words live the category of finite dimensional vector
spaces, denoted below by FVect. The more concrete version of
the above functor is thus as follows:
F : Preg H⇒ FVect
The functor F transforms α to a linear map in FVect. This linear
map is applied to the vectors of the words within the phrase or
sentence. The whole procedure is formalized below:
−−−−−→
−
→
−
→
−
→
(∗) −
w−
1 w2 · · · wn = F( w 1 ⊗ w 2 ⊗ · · · ⊗ w n )
→
Vectors of words, i.e., the −
w i ’s are represented by morphims
I → V of the category of finite dimensional vector spaces, for
V the vector space in which the meaning of the word lives. The
tensor product ⊗ between these morphisms is the categorical way
of packing them together. This model, referred to by DisCoCat,
for Categorical Compositional Distributional, was the first model
that put together the vector meanings of words by taking into
account their grammatical structure, in order to build a vector
for the phrase or sentence containing the words.
DisCoCat relates to the categorical models of Quantum
phenomena in two ways. One is through the function F; quoting
from Coecke et al. [3]:
“A structure preserving passage to the category of vector spaces is
not a one-off development especially tailored for our purposes.
It is an example of a more general construction, namely,
a passage long-known in Topological Quantum Field Theory
(TQFT). This general passage was first developed by Atiyah [4]
in the context of TQFT and was given the name “Quantization,”
as it adjoins “quantum structure” (in terms of vectors) to a
purely topological entity, namely the cobordisms representing
the topology of manifolds. Later, this passage was generalized to
abstract mathematical structures and recast in terms of functors
whose co-domain was FVect by Baez and Dolan [5] and Kock
[6]. This is exactly what is happening in our [DisCoCat] semantic
framework: the sentence formation rules are formalized using
type-logics and assigned quantitative values in terms of vector
composition operations. This procedure makes our passage from
grammatical structure to vector space meaning a “Quantization”
functor. Hence, one can say that what we are developing here is a
grammatical quantum field theory for Lambek pregroups. ”

The other connection

is that the DisCoCat model, i.e., the
tuple Preg, FVect , F , was originally inspired by the categorial
model of Quantum Mechanics, as developed by Abramsky and
Coecke [7]. CQM, for Categorical Quantum Mechanics, models
Quantum protocols using compact closed categories and their
vector space instantiations (more specifically they use dagger
compact closed categories and category of Hilbert spaces, which
have also been used in DisoCat, e.g., see [8]). The aim of this

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review is to briefly describe the DisCoCat model and its recent
extensions with Frobenius and Bi algebras. These extensions were
inspired by extensions to categorical Quantum Mechanics: the
work of Coecke et al. [9, 10] for the use of Frobenius Algebras and
Coecke and Duncan [11] for Bialgebras. These extensions have
enabled us to reason, in a structured way, about logical words
of language such as relative pronouns “who, whom, what, that,
which, etc.” and quantifiers “all, some, at least, at most, none, etc.”
In what follows, we will first review the advances made in the
DisCoCat model in a chronological order; then go through the
the core of theoretical underpinnings of the model and finally
present some of the main experiments performed to validate the
theoretical predictions.

2. A CHRONOLOGICAL OVERVIEW OF
DISCOCAT
The origins of the DisCoCat model goes back to the work of Clark
and Pulman [12], presented in the AAAI Spring Symposium on
Quantum Interaction (QI) in 2007. The paper discussed vector
models of word meaning, otherwise known as distributional
semantics, and outlined an open problem they faced. The open
problem is how to extend distributional models so that they can
assign vector meanings to phrases and sentences of language. In
their proposed extended model, Clark and Pulman, inspired by
the Harmonic Grammars of Smolensky [13], argue for the use of
tensors. The vector meaning of a string of words w1 w2 · · · wn , as
defined by them, is as follows:
−
−−−
−−→
w−! w
1 · · · wn =

X
i

−
→
→
wi⊗−
ri

→
where −
r i is a vector representing the grammatical role played
by word wi in the string. The problem with this model is
that firstly, vector meanings of sentences grow as the sentence
becomes larger and building tensor models for them becomes
costly. Secondly, sentences that have different grammatical
meanings live in different spaces and thus their meanings
cannot be compared to each other. Further, Clark and Pulman
do not provide any experimental support for their models.
Subsequently, in a paper presented in QI in 2008, Stephen et al.
[14], addressed the former two of these problems by presenting
the first DisCoCat model. An extended version of this paper later
appeared in Lambek’s 90’th Festschrift [15] in 2010. The original
DisCoCat model presented there worked along side the following
triangle:

where the space between the FVect and Preg was interpreted
as pairing. That is, instead of working with a functor between
the two categories of grammar and meaning, there is only one
category: the category whose objects are pairs (V, p) of V a vector

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CQM and Natural Language

space and p its grammatical type (i.e., the grammatical type of
the words living in that category), and whose morphisms are also
pairs (f : V → W, p ≤ q), for f : V → W a linear map and
≤ :p → q the partial ordering of the pregroup grammar. In this
model, pregroups where treated as partial order categories. The
functorial form of DisCoCat, described in the introduction, was
fist introduced by Preller and Sadrzadeh [16] in 2010 and later
connected to the Quantization of TQFT by Coecke et al. [3] in
2013. Similar connections, albeit not in a functorial form and not
to TQFT but to CQM in general, were also drawn in a paper by
Lambek [17]. It should be noted that the main contribution of
Coecke et al. [3] was however, to extend the functorial passage
and thus the DisCoCat from Lambek Pregroups to the original
monoidal calculus of Lambek [18].

F : Monoidal Closed Cats H⇒ FVect
Although grammar-aware vector space models of meaning
existed for adjective noun phrases via the work of Baroni and
Zamparelli [19], but the above DisCoCat model was the first one
where this grammar-awareness was theoretically defined for all
language constructs. Later, in 2013, in the paper by Grefenstette
et al. [20], it was shown how the concrete constructions of
Baroni and Zamparelli [19] can be used in the DisCoCat to
build matrices and tensors for intransitive and transitive verbs.
But extending these concrete models to words such as relative
pronouns, quantifiers and prepositions proved to be problematic,
due to data sparsity, as also shown in Baroni et al. [21].
The theoretical predictions of DisCoCat were first
experimentally verified in a paper by Grefenstette and Sadrzadeh
[22]. They presented an algorithm to build the matrices/tensors
of the model and implemented it on intransitive and transitive
verbs and further applied these to a disambiguation task. The
intransitive version of the task was originally developed by
Mitchell and Lapata [23], the extension to transitive was a novel
contribution, leading to a dataset that was later used by the
community in many occasions. In a short paper in the same
year [24] presented a few extensions to the constructions of the
latter model. The full set of results, together with extensions to
adjective-noun phrases and sentences containing them, appeared
in the journal article by Grefenstette and Sadrzadeh [25].
The model described above had one flaw, namely that
sentences of different types acquired vectors that lived in different
vectors spaces. This made it impossible to fully benefit from
DisCoCat, one of whose original promises was an extension of the
model of Clark and Pulman to one that one can actually compare
sentences. This shortcoming was later overcome by Kartsaklis et
al. [26], where it was shown that two possible applications of
Frobenius algebras on the concrete model of Grefenstette and
Sadrzadeh [22] solves the problem and leads us to a uniform
sentence space.
After the above, DisCoCat has been extended to cover
larger fragments of language and also it has been implemented
on different vector models with different implementation
parameters and experimented with in different settings. These
latter contributions include the following:

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• Prior disambiguation of tensors by sense-clustering and
separating different meanings, based on the contexts in
which the words appear. For the case of a verb, we first
disambiguate its subject and objects by clustering, then build
verb tensors using disambiguated versions of these vectors.
The experimental support we provided for this work showed
that the disambiguated models indeed perform better than
their ambiguous versions, these results were presented in
Kartsaklis et al. [27] and Kartsaklis and Sadrzadeh [28].
• Experiments in support of entangled tensors in the linguistics
applications listed above. Our so called tensors are elements
of tensor spaces, which are in turn built from spaces of the
types of the corresponding words (e.g., a verb or an adjective).
Some elements of such spaces are separable and some are not.
In the separable case, for t ∈ A⊗B we have two other elements
a ∈ A, b ∈ B such that t = a ⊗ b. In the non-separable case,
this factorization will not be possible. In Quantum Mechanics,
the non-separable elements correspond to entangled states of
a physical system. A question arises that whether our linguistic
tensors are separable or not. We answered this question by
measuring the degree of entanglement of our tensors and
showing that the ones with a higher degree led to better results,
as presented in Kartsaklis and Sadrzadeh [29].
• Application of neural word embeddings in the tensor models.
Recent work on deep neural nets has led to creation of large
sets of vectors for words, referred to by word embeddings,
as presented in Mikolov et al. [30, 31]. These vectors are
now hosted by Google in their Tensor Flow platform1 . The
popularity and good performance of the vectors in various
tasks and models makes one ask whether or not they would
work well in a tensor framework as well. The work in Milajevs
et al. [32] showed that this is indeed the case.
• Developing a theory of entailment for the tensor models.
Distributional semantics supports word-level entailment via a
distributional inclusion hypothesis, where inclusion relations
between features of words is put forward as a signal for
the entailment relations between words. We showed how
this model can be extended from word to sentence-level in
a compositional fashion. We worked out different feature
inclusion relations for features of sentences that were built
using different compositional operations. We developed new
datasets and measured performances of our feature inclusion
relations, the results appeared in Kartsaklis and Sadrzadeh
[33, 34].
We have also extended the model theoretically, where the
major contributions are as follows:
• The use of Frobenius and Bi algebras to model linguistic
phenomena that involve certain types of rearranging of
information within phrases and sentences. An example of such
a phenomena is relative and quantified clauses, such as “all
men ate some cookies,” “the man who ate the cookies” and “the
man whose dogs ate the cookies.” We extended the DisCoCat
model from compact closed categories to ones with Frobenius
and Bi algebras over them and showed how the copying
1 https://www.tensorflow.org/versions/r0.11/tutorials/word2vec/index.html

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Sadrzadeh

and merging operations of these algebras allow us to model
meanings of quantified and relative clauses and sentences. The
work done here includes that of Clark et al. [35], Sadrzadeh et
al. [36, 37], Hedges and Sadrzadeh [38] and Sadrzadeh [39].
• Density matrices are the core of vector space models of
Quantum Mechanics and indeed it has been shown that the
category of density matrices and completely positive maps
is also a compact closed category. The question arises so
as whether and how these matrices have applications in
linguistics. The work of Piedeleu et al. [8], Balkir et al. [40, 41]
and Bankova et al. [42] showed that density matrices can
model different meanings of ambiguous words and that they
can also model a hierarchy of these meanings and thus be
applied to entailment.

3. OVERVIEW OF THEORY
This section reviews the theoretical framework of a DisCoCat.
Its structure is as follows: in Section 3.1, we will review the
distributional semantic models. We show how the motivating
idea of these models are formalized in terms of vector
representations and describe some theoretical and experimental
parameters of the model and some of the major applications
thereof. In Section 3.2. we review the grammatical model that
was first used as a basis for compositional distributional models,
namely the pregroup grammars of Lambek. We review the
theory of pregroup algebras and exemplify its applications to
reasoning about grammatical structures in natural language. In
Section 3.3, we show how a functorial passage can be developed
between a pregroup grammar, seen as a compact closed category,
and the category of finite dimensional vector spaces and linear
maps. We describe how this passage allows one to assign
compositional vector semantics to words and sentences of
language. This passage is similar to the one used in TQFT, where
the grammatical part is replaced by the category of manifolds
and cobordisms. Section 3.4, describes the theory of Frobenius
and Bi algebras over compact closed categories. In Section 3.5,
we show how these algebras can model meanings of relative
and quantified clauses and sentences. In Section 3.6, we go
through the graphical calculus of compact closed categories and
Frobenius and Bi algebras over them. We exemplify how these
are used in linguistics, where they depict flows of information
between words of a sentence.

CQM and Natural Language

Given an m × n co-occurrence matrix, every target word t can
be represented by a row vector of length n. For each feature c,
the entries of this vector are a function of the raw co-occurrence
counts, are computed as follows:
rawf (t) =

P

c N(f , t)

k

for N(f , t) the number of times the t and f have co-occurred in the
window. Based on L, the total number of times that t has occurred
in the corpus, the raw count is turned into various normalized
degrees. Some common examples are probability, conditional
probability, likelihood ratio and its logarithm: The lengths of the
corpus, window, and normalization scheme are parameters of the
model, as are the sizes of the feature and target sets, there has been
a plentiful of papers who study these parameters, for example see
Lapesa and Evert [45], Bullinaria and Levy [46], and Turney [47].
The distance between the meaning vectors, for instance the
cosine of their angle, provides an experimentally successful
measure of similarity of their meanings. For example, in the
vector space of Figure 1, cited from Coecke et al. [3], the angle
between meaning vectors of “cat” and “dog” is small and so is
the angle between meaning vectors of “kill” and “murder.” Such
similarity measures have been implemented on large scale data
(up to a billion words) to build high dimensional vector spaces
(tens of thousands of basis vectors). These have been successfully
applied to automatic generation of thesauri and other natural
language tasks such as automatic indexing, meaning induction
from text, and entailment, for example see Curran [48], Lin [49],
Landauer and Dumais [50], Geffet and Dagan [51], and Weeds et
al. [52].

3.2. Pregroup Grammars
A pregroup algebra, as defined by Lambek [1], is a partially
ordered monoid (P, ≤, ·, 1) where every element has a left and a
right adjoint, which means that for every element p ∈ P we have
a pr ∈ P and a pl ∈ P such that the following four inequalities
hold:
p · pr ≤ 1 ≤ pr · p

pl · p ≤ 1 ≤ p · pl

3.1. Vector Models of Natural Language
Given a corpus of text, a set of contexts and a set of target words,
the vector models of words work with a so called co-occurrence
matrix. This has at each of its entries “the degree of co-occurrence
between the target word and the context,” developed amongst
other by Salton et al. [43] and Rubenstein and Goodenough [44].
This degree is determined using the notion of a window: a span
of words or grammatical relations that slides across the corpus
and records the co-occurrences that happen within it. A context
can be a word, a lemma, or a feature. A lemma is the canonical
form of a word; it represents the set of different forms a word can
take when used in a corpus. A feature represents a set of words
that together express a pertinent linguistic property of a word.

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FIGURE 1 | A subspace of a vector space model of meaning, built from real
data, cited from Coecke et al. [3].

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CQM and Natural Language

An example of a pregroup in arithmetics is the set of
unbounded monotone functions on integers, where the monoid
multiplication is function composition with the identity function
its unit, and the left and right adjoints defined using min and max
of integers. For reasons of space, we will not give these definitions
here and refer the reader to Lambek [1, 2].
Pregroup algebras are applied to natural language via the
notion of a pregroup grammar, defined to be a pair hD, Si, where
D is a pregroup lexicon and S ⊂ B is a set of designated types,
containing types such as that of a declarative sentence s, and a
question q. A pregroup lexicon is a binary relation D, defined as
D ⊆ 6 × T(B)
where T(B) is the free pregroup generated over B (for the free
construction see [1]).
Given a pregroup grammar, as specified in Lambek [1], one
says that a string of words w1 w2 · · · wn of language is grammatical
iff for 1 ≤ i ≤ n, there exists a (wi , ti ) ∈ D, such that we have a
type t ∈ T(B) ∩ D[6] such that the following partial order holds
in T(B):
t1 · t2 · · · · · tn ≤ t
An example of a pregroup lexicon is presented in Table 1:
The pregroup reductions corresponding sentences (1) “men
kill dogs,” (2) “men kill cute dogs,” and (3) “men do not kill dogs”
are as follows (all cited from [3]):
(1) n · nr · s · nl ≤ 1 · s · 1 = s

(2) n · nr · s · nl · n · nl · n ≤ 1 · s · 1 · 1 = s

(3) n · nr · s · jl · σ · σ r · j · jl · σ · σ r · j · nl · n

≤ 1 · s · jl · 1 · j · jl · 1 · j · 1 = s · jl · j · jl · j ≤ s

3.3. Quantization
In order to formalize the structure preserving passage between
syntax: pregroup grammars and semantics: vector models,
we formalize both of these in the language of compact
closed categories [53]. For this reason, we very briefly recall
some definitions. A compact closed category has objects A, B;
morphisms f : A → B; a monoidal tensor A ⊗ B that has a unit
I; and for each object A two objects Ar and Al together with the
following morphisms:
ǫAr

r
ηA

A ⊗ Ar −→ I −→ Ar ⊗ A

ǫAl

l
ηA

Al ⊗ A −→ I −→ A ⊗ Al

These morphisms have to satisfy certain other conditions, among
which are the four yanking equations, which for reasons of space

TABLE 1 | Type assignments for a toy language in a Lambek pregroup; table from
Coecke et al. [3].
Men

Dogs

Cute

Kill

To kill

Do

Not

n

n

n · nl

nr · s · nl

σ r · j · nl

nr · s · jl · σ

σ r · j · jl · σ

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we will not give here. It is evident (and has also been proven,
see for example [17, 54]), that pregroup algebras are compact
closed categories. This is by taking the above ǫ and η maps
to be the four adjoint inequalities of a pregroup algebra. Finite
dimensional vector spaces with linear maps as morphisms are
also compact closed categories, this has been shown by Kelly and
Laplaza [53]. This category is symmetric, thus the left and right
adjoints collapse to one, that is for V a finite dimensional vector
space, we have V l = V r = V ∗ , where V ∗ is the dual space of V. In
the presence of a fixed basis, however, one obtains the equivalence
V ≡ V ∗ . Assuming so, the ǫ and η maps are defined as follows,
→
for {−
r i }i a fixed basis:
ǫ = ǫl = ǫr : V ⊗ V → R
X −
→
::
cij →
r i⊗−
r j 7→
ij

l

r

η = η =η : R→V ⊗V

∗

X
ij

→
→
cij h−
r i|−
r ji

: :1 7→

X−
→
→
r i⊗−
ri
i

Now we can define the structure preserving map via the following
Quantization functor:
F : Preg H⇒ FVect
explicitly defined as follows:
• For n, s ∈ B and two atomic vector spaces W and S, we have
F(n) = W and F(s) = S,
• For p, q ∈ T(B) \ (B ∪ {1}), we have F(p · q) = F(p) ⊗ F(q),
• For 1 ∈ T(B), we have F(1) = R,
• For adjoints we have, F(pr ) = F(pl ) = F(p),
• For morphisms, we have F(p ≤ q) = F(p) → F(q).
We have now formally defined a DisCoCat: the tuple
(Preg, FVect, F), as defined above. It is in this setting that
one obtains vector representations for sentences by applying
the definition (∗) of introduction. For example, the vector
representations of two of our example sentences above become
as follows:

→ −−→
−−−−−−−−→
−→ ⊗ −
men kill dogs = (ǫW ⊗ 1S ⊗ ǫW ) −
men
kill ⊗ dogs
X
−−→ →
−→ | −
→
→
=
cijk h−
men
w i ih−
w k | dogsi −
sj
ijk

−−−−−−−−−−−−→
men kill cute dogs =

→ −−→ −−→
−→ ⊗ −
men
kill ⊗ cute ⊗ dogs)
(ǫW ⊗ 1S ⊗ ǫW ⊗ ǫW ) (−
XX
−−→ →
−−→ | −
→
→
→
→
=
cijk clm hmen
w i ih−
w k |−
w l ih−
w m | dogsi−
sj
ijk lm

An important observation is that in this setting one obtains
that, vector representations of words that have atomic types, e.g.,
−→
−→ −
men and dogs with type n are vectors −
men,
dogs ∈ W. The
representations of other
e.g., cute and kill with types nr s
P words,
−
→
→
→
r
l
and n sn are matrices ij cij w i ⊗ −
w j ∈ W ⊗ W for {−
wi }i a basis
P
→
→
→
for W and tensors ijk cijk −
wi ⊗−
s j⊗−
w k ∈ W ⊗ S ⊗ W, for
→
{−
s j }j a basis in S.
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3.4. Frobenius and Bi Algebras
Both Frobenius and Bi algebras are defined over a symmetric
monoidal category C. Frobenius Algebras were developed in
their current from by Kock [6] and McCurdy [55], bialgebras
by McCurdy [56] and Bonchi et al. [57]. Formally, they are both
denoted by tuples (X, δ, ι, µ, ζ ) where, for X an object of C, the
triple (X, δ, ι) is an internal comonoid and the triple (X, µ, ζ ) an
internal monoid; i.e., the following are coassociative and counital,
respectively associative and unital morphisms of C:
δX : X → X ⊗ X

ιX : X → I

µX : X ⊗ X → X

ζX : I → X

One difference between these two is that the Frobenius algebra
satisfies the following so-called Frobenius condition (due to [58]
who originally introduced the algebraic form of these in the
context of representation theorems for group theory):
(µX ⊗ 1X ) ◦ (1X ⊗ δX ) = δX ◦ µX = (1X ⊗ µX ) ◦ (δX ⊗ 1X )

The bialgebras satisfy a weaker version of this condition, referred
to by Q3 in McCurdy [56]
δX ◦ µX = (µX ⊗ µX ) ◦ (1X ⊗ σX,X ⊗ 1X ) ◦ (δX ⊗ δX )
for σX,X the symmetry morphism of the category C. Both these
algebras do satisfy other conditions, which we will not give here.
→
In FVect, any vector space V with a fixed basis {−
vi }i has a
Frobenius algebra over it, explicitly given by:
→
→
→
δV : :−
vi 7→ −
vi ⊗ −
vi
−
→
−
→
→
µ : : v ⊗ v 7→ δ −
v
V

i

j

ij i

→
ιV : :−
vi 7→ 1
X−
→
ζV : :1 7→
vi
i

where δij is the Kronecker delta. These definitions were
introduced in Coecke et al. [9, 10] to characterize vector space
bases.
Bialgebras over vectors spaces were introduced in Coecke
and Duncan [11] to characterize phases. For linguistic purposes,
however, we use a different definition, first introduced in Hedges
and Sadrzadeh [38]. Let V be a vector space with basis P (U),
where U is an arbitrary set. We give V a bialgebra structure as
follows:
ιP (U) |A = 1
ζP (U) = |U

δP (U) |A = |A ⊗ |A

µP (U) (|A ⊗ |B) = |A ∩ B

The Frobenius and the bialgebra δ act similarly here: they both
→
copy their input, that is given a vector −
υ , the produce two
−
→
−
→
copies of it υ ⊗ υ . The slight difference in this special natural
language instantiation is that the inputs to the bialgebra δ’s are
vectors whose basis are subsets of a universal set U, whereas
the inputs to Frobenius algebra δ’s can be any vector. The main
difference between these two algebras are in their µ maps. The
Frobenius µ, when inputted with two same vectors, returns one
of them, the bialgebra µ acts on any two input vectors, but of
course these both have to have as basis subsets of P (U), and

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returns the “intersection” of these two vectors. By “intersection
of vectors” we mean (as defined above), a vectors whose basis is
the intersection of the basis of the input vectors.
The reason for working with the above bialgebras is that they
are there to model generalized quantifiers of Barwise and Cooper
[59]. These quantifiers are defined as maps with the type P (U) →
PP (U). In order to see why, consider the following definition for
the logical quantifiers “all” and “some”:
[[some]] (A) = {X ⊆ U | X ∩ A 6= ∅}


every (A) = {X ⊆ U | A ⊆ X}
A similar method is used to define non-logical quantifiers, for
example “most A” is defined to be the set of subsets of U that
has “most” elements of A, “few A” is the set of subsets of U
that contain “few” elements of A, and similarly for “several”
and “many.” These functions can be formalized as relations over
P (U), where they will thus obtain the type P (U) → P (U).
These relations can be formalized in the category of sets and
relations, which is also compact closed. The above definitions
are vector space generalizations of the bialgebras defined for
relations. They enable us to work with intersection of these
relations. This is an operation that allows Barwise and Cooper
to formalize an important property of generalized quantifiers
of natural language, i.e., that they are conservative. For details
and the from-relation-to-vector embedding, see Hedges and
Sadrzadeh [38].

3.5. Relative Pronouns and Quantifiers
In order to model relative pronouns and quantifiers, according to
the developments of Clark et al. [35], Sadrzadeh et al. [36, 37], and
Hedges and Sadrzadeh [38], one adds to the pregroup lexicon, the
following assignments:
To subject relative pronouns “who, that, which,” we assign type
nr nsl n
To object relative pronouns “whom, that, which,” we assign type
nr nnll sl
To determiners of any role “a, the, all, every, some, none, at most,
many, · · · ,” we assign type nnl

The vectorial representations of the subject and object relative
pronouns are as follows, respectively for each case:
−−−→
Sbj Rel := (1W ⊗ µW ⊗ ζS ⊗ 1W ) ◦ (ηW ⊗ ηW )
−−−−→
Obj Rel := (1W ⊗ µW ⊗ 1W ⊗ ζS ) ◦ (ηW ⊗ ηW )
where the µW and ζS are maps of the Frobenius algebras defined
over the W and S spaces. The vectorial representation of the
determiners are as follows:
−−−−−−−→
determiner : = (1W ⊗ ǫW ) ◦ (1W ⊗ µW ⊗ ǫW ⊗ 1W )
 
◦(1W ⊗ d ⊗ δW ⊗ 1W⊗W )
◦(1W ⊗ ηW ⊗ 1W⊗W ) ◦ (ηW ⊗ 1W )

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computations thereof. For example see Figure 2 for the diagram
for teleportation:
These diagrams depict the flow of information between the
parties involved in the protocol and also simplify the tensor
contraction computations. In the setting of language, every
language construct can be seen as a protocol, with words as
the involved parties. The same diagrammatic calculus has been
widely used to show how information flows amongst the words
of a phrase or sentence and to depict the meaning of the language
unit resulting from it. In the interest of space, we will not
introduce this diagrammatic calculus here, but provide examples,
−−−−−−−−−−−→
men who eat cake : = (1W ⊗ ǫW )◦(ǫW ⊗1W ⊗ S ⊗ ǫW ⊗1S ⊗ ǫW ) via the following Figures 3–5.
−→ −
→ −−→
−→ ⊗ −
(−
men
who ⊗ eat ⊗ cake)

where the µW and δW maps are bialgebraic maps defined over the
space W = VP (U) , which is notation for a vector space spanned
 
by the subsets of the set U. The d map has type W → W, it
is a linear map that directly encodes the relational graph of the
generalized quantifier d.
By applying definition (∗) from the introduction, one obtains
vectorial representations for relative clauses and quantified
sentences. An example of the former is “men who eat cake,” which
acquires the following vectorial representation:

4. OVERVIEW OF EXPERIMENTS

This simplifies as follows, after opening up the meaning of “who”
using the “Obj Rel” definition above:


→ −−→
−→ ⊗ −
(µW ⊗ ǫW ) −
men
eat ⊗ cake


X−
X
X
→
→
→
−
→
= (µW ⊗ ǫW )
w k ⊗(
αij −
w i ⊗−
w j )⊗
w
l
ij

k∈K

=
=

l∈L

X

→
→
→
→
αij µW (−
wk⊗−
w i ) ⊗ ǫ W (−
wj⊗−
w l)

X

→
αij δki −
w i δjl =

ij,k∈K,l∈L

ij,k∈K,l∈L

X

→
αkl −
wk

k∈K,l∈L

An example of a quantified sentence is “most cats snore,” which
acquires the following vectorial representation,
−−→ −→
−−→
(ǫW ⊗ 1S ) ◦ (1W ⊗ ǫW ⊗ 1W⊗S )(most ⊗ cats ⊗ −
snore)
The above simplifies to the following
−−−−→
−→ ⊗ −
−−→
men
snore)
(ǫW ) ◦ ([[most]] ⊗ µW ⊗ 1S ) ◦ (ǫW ⊗ 1S )(−

Our first set of experiments was on two datasets, both consisting
of pairs of transitive sentences with ambiguous verbs and their
two eminent meanings. One of the datasets had adjectives
modifying the subjects and objects, the other contained bare
subjects and objects. The goal was to disambiguate verbs, based
on the sentences in which they occurred. A non-compositional
distributional approach to this task would be to build vector
representations for verbs and their different meanings (in this
case the two most eminent ones); then measure the cosine of the
angle between the vector of the verb and those of the meanings
and use this as a measure of disambiguation. In other words, if
the vector of the verb was closer to one of the meaning vectors,
that meaning would be chosen as the right meaning for the verb.
This non-compositional method, however, does not take into
account the specific sentence in which the verb has occurred. In
our compositional version, we build a vector representation for
each sentence of the dataset; specifically, we build a vector for
the sentence with the verb in it, and two for the two sentences
where the verb is replaced with one of its two eminent meanings.
Then we compare the distances between these sentence vectors.
The sentence vectors were built using different composition
operators, and the non-compositional verb vector was taken as

Different instantiations for U and S are provided in Hedges and
Sadrzadeh [38], as an example consider U to be the set of words of
language, in which case P (U) represents the set of, what is called
“lemmas” of language, i.e., the set of canonical forms of words.
One takes S = VP (S′ ) ,for S′ an abstract sentence space, denoted
by the symbol S in our previous examples. In this case, the above
categorical definition takes a concrete instantiation as follows:
X

X

ijk B∈[[most]]([[cat]])

snore most
cicat cjk
cB hB | Ai ∩ Aj i|sk

FIGURE 2 | Diagram of information Flow in the teleportation protocol, cited
from Abramsky et al. [7].

P cat
−→
−−−→
where
we have cats: =
i ci |Ai for Ai ⊆ U and snore: =
P snore
jk cjk |Aj ⊗ Ak , for Aj ⊆ U and |Ak a basis vector of S.

3.6. Diagrams
The compact closed categorical setting of Abramsky and
Coecke comes equipped with a diagrammatic calculus, originally
developed in Joyal and Street [60] and referred to by string
diagrams. This calculus allows one to draw diagrams that
depict the protocols of Quantum mechanics and simplify the

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FIGURE 3 | Diagram of information flow in the negative transitive sentence,
cited from Preller and Sadrzadeh [16].

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FIGURE 4 | Diagram of information flow in a relative clause with an object relative pronoun, cited from Clark et al. [35].

TABLE 3 | Example entries from the adjective-transitive dataset, cited from
Grefenstette and Sadrzadeh [25].

FIGURE 5 | Diagram of Information Flow in a Sentence with a Quantified
Subject, cited from Hedges and Sadrzadeh [38].

TABLE 2 | Example entries from the transitive dataset, cited from Grefenstette
and Sadrzadeh [25].
Sentence 1

Sentence 2

HIGH-LOW Tag

Man draw sword

Man attract sword

LOW

Report draw attention

Report attract attention

HIGH

Man draw sword

Man depict sword

HIGH

Report draw attention

Report depict attention

LOW

a comparison base line. The results show that one of the tensor
composition methods, namely our Kronecker model, performed
best. This was the first time 3 and 4 word sentences were used to
disambiguate a single word. A precursor to this experiment, was
that of Mitchell and Lapata [23], where ambiguous verbs were
disambiguated using 2-word “Sbj Verb” or “Verb Obj” phrases.
A snapshot of two of these datasets are presented in
Tables 2, 3. The first two entries are the two sentences in question
and the last entry is a tag we gave to the sentences based on how
similar the meanings of the sentences in the pair were. As you
can see, the first dataset consists of “Sbj Verb Obj” sentences, the
second dataset consists of sentences of the form “Adj Sbj Verb
Adj Obj” where the subject and object are moreover modified by
adjectives:
We asked human annotators (on Amazon Turk) to assign
a degree of similarity to each pair of the dataset, using a
number from 1 to 7, ranking the degree of similarity of the
sentences therein. If the sentence “Sbj Verb1 Obj” was ranked

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Sentence 1

Sentence 2

Statistical table show good result

Statistical table express good result

Statistical table show good result

Statistical table depict good result

to have an average high similarity with the sentence “Sbj Verb2
Obj,” then we concluded that “Verb1” had the same meaning
as “Verb2,” thus disambiguating it. We implemented different
models to compute vectors for sentences and used the cosine
of the angle between them as a measure of similarity. The
results are presented in Table 4. In the “Sbj Verb Obj” dataset,
the vectors built via the Kronecker model achieve the highest
correlation with the annotators’ judgments. This model is one of
the DisCoCat models, a model that has consistently performed
very well. In the “Adj Sbj Verb Adj Sbj” dataset, the model
referred to by Categorical Adj has consistently performed the
best. This model builds a matrix for the adjective and matrix
multiplies it with the vector of the noun to obtain a vector for
the adjective noun phrase. The exact results denote the degree
of correlation (computed by using Spearman’s ρ) between the
human judgments and the judgments predicted by the models.
These seem quite low, but so is the inter annotators agreement,
presented in the last line of each table. This agreement is an
upper bound for the experiment, denoting the degree to which
the human annotators agreed amongst themselves about their
similarity judgments. Having this upper bound in mind, we see
that the “Adj Sbj Verb Adj Sbj” dataset performed better than
the “Sbj Verb Obj” dataset (since it had larger compositional
contexts), as it aligns to human judgment in about 60% of the
time.
A criticism to this first set of experiments was that they relied
on human judgments and that these were not done according to
clear guidelines. One argument against lack of such a judgment
was that human annotators were asked to judge the degree of
similarity between sentences and that is a hard task. It was
argued that similarity has different interpretations in different
contexts and annotators might have had different interpretations
(different to ours) in mind when judging the dataset. In a second
task, we avoided this weakness by working on term-definition
pairs mined from a junior dictionary. The terms were words and

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TABLE 5 | Sample of the dataset for the term/definition comparison task, cited
from Kartsaklis et al. [26].

TABLE 4 | Model correlation coefficients with human judgments, cited from
Grefenstette and Sadrzadeh [25].
Model

ρ

Term

Main definition

Def. 2

Def. 3

Verb Baseline

0.13

Blaze

Large strong fire

Huge potent flame

Substantial heat

Bigram Baseline

0.16

Husband

Married man

Male spouse

Trigram Baseline

0.15

Partner of a
woman

Horror

Great fear

Intense fright

Add

0.10

Disturbing
feeling

Multiply

0.16

Categorical

0.16

Apologize

Say sorry

Kronecker

0.26

Express regret or
sadness

Acknowledge
shortcoming or
failing

UpperBound

0.62

Embark

Get on a ship

Enter boat or
vessel

Commence trip

Model

ρ

Vandalize

Break things

Cause damage

Produce
destruction

Verb Baseline

0.20

Bigram Baseline

0.14

Trigram Baseline

0.16

Additive

0.10

TABLE 6 | Accuracy results for the term/definition comparison task, Kartsaklis et
al. [26].
CpObj

Multp

Addt

Cont

Multiplicative
Nouns

0.24

0.22

0.17

0.09

0.20

Verbs

0.28

0.30

0.25

0.07

AdjNoun

0.05

Combined

0.19

0.20

0.12

0.06

CategoricalAdj

0.20

AdjMult

The best performing models are highlighted in boldface.

Categorical
AdjMult

0.14

AdjNoun

0.16

CategoricalAdj

0.19

TABLE 7 | Examples from a Term-Description dataset, cited from Sadrzadeh et al.
[37].
Term

Description

1

Emperor

Person who rule empire

2

Queen

Woman who reign country

0.26

3

Mammal

Animal which give birth

AdjNoun

0.17

4

Plug

Plastic which stop water

CategoricalAdj

0.27

5

Carnivore

Animal who eat meat

6

Vegetarian

Person who prefer vegetable

Kronecker
AdjMult

Upperbound

0.48

The best performing models are highlighted in boldface.

the definitions were short descriptions given by the dictionary
as the meaning of the word. The goal was to distinguish which
definition was describing which word. We built word vectors
for the terms and phrase vectors for the definitions and used
the cosine of the angle between these vector as a classifier. We
collected five definitions whose vectors were closest to the vector
of the verb and then verified whether the correct dictionary
definition was amongst these five. The model that classified more
terms to their correct definitions was considered to be the better
model.
A snapshot of the dataset is presented in Table 5. The accuracy
results are presented in Table 6, where the DisCoCat CopyObj
model achieves the highest accuracy for the terms that are nouns,
and the second best for terms that are verbs (28% vs. the accuracy
of 30% reached by multiplying the word vectors).

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Based on this experiment, we formed a toy experiment and did
a preliminary evaluation of the application of Frobenius algebras
to modeling relative clauses. Similar to the above experiment, we
mined term-description pairs from a dictionary, but this time
the terms were chosen such that their descriptions had a relative
pronoun in them. We then proceeded as before: built vectors for
the term and for the description. A snapshot of the dataset is
presented in Table 7.
The latter used three different composition operators: simple
addition and point wise multiplication, i.e., we just added and
point wise multiplied the word vectors within the descriptions
to obtain a vector representation for the whole relative clause.
We also built vector representations using the Frobenius model
presented above and an extension of it to possessive relative
pronoun “whose.” In the first two models, we had a choice of
either building a vector for the relative pronoun or dropping
it and thus treating it as noise. We presented results for both

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TABLE 8 | Results for the Term-Description dataset, cited from Sadrzadeh et al.
[37].
Frob
’s = Id ’s =

Frob
P −−−−→
i (noun1 )i

Mult

Mult

without
Rel. Pr.

with Rel.
Pr.

Add
/with without
Rel. Pr.

MRR

0.82

0.71

0.78

0.76

0.75

Acc

0.75

0.56

0.62

0.62

0.62

The best performing models are highlighted in boldface.

TABLE 9 | Models correlation with human judgments for the disambiguation task
with normal and neural (NWE) vectors, cited from Milajevs et al. [32].
Method

GS11

KS14

NWE

Verb only

0.212

0.325

0.107

Addition

0.103

0.275

0.149

Multiplication

0.348

0.041

0.095

Kronecker

0.304

0.176

0.117

Relational

0.285

0.341

0.362

Copy subject

0.089

0.317

0.131

Copy object

0.334

0.331

0.456

The best performing models are highlighted in boldface.

of these options. For the possessive Frobenius case, we had the
choice of either building a vector for ’s, or treating it as the
unit vector. Again, we presented the results of both cases. We
tested which model achieved a better accuracy (Acc) and a mean
reciprocal rank (MRR). The results are presented in the Table 8,
where both of the Frobenius models achieve the highest accuracy
and MRR.
The differences between the two Frobenius models only
applies to the possessive relative clauses, which were not reviewed
in this article, in the benefit of space. In these models, one has
to build a linear map for the “ ’s ” phoneme. In one model we
summed all the nouns that were modified by this morpheme,
in the other, we simply took it to be the identity map, i.e., the
unit map. In either case, the Frobenius model performed better
than our other implemented models, e.g., the other two models
in which we added the vectors of the words, taking into account
the vector of the relative pronoun or ignoring it, and two other
similar models where we multiplied them.
As another set of experiments, we used the neural word
embeddings of Mikolov et al. [30]. The motivation behind
this task was the popularity and success of the word
embeddings. Often, when vector representations are built from
scratch using count-based methods and on a given corpus,
many parameters have to be taken into account (e.g., size
of the window, the normalization scheme for the counts,
the dimensions of the vector spaces and its size). The
preprepared word embeddings provides a platform wherein
new vectors need not be built for each task and parameters
need not be individually tuned by each experimenter and for
each experiment. The word embeddings provide a standard

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framework (to some extent) for all experimenters to do
experiments and compare their results in a more unified
manner. It also relieves us from the task of building the vectors
ourselves.
We used the neural noun vectors of Mikolov et al. [30] and
built adjective matrices and verb tensors and re-experimented
with the disambiguation task presented above. The results are
presented in Table 9. GS11 and KS14 denote the “Sbj Verb Obj”
and “Adj Sbj Verb Adj Obj” datasets with count-based vectors,
and NWE denotes both of these datasets together with the word
embedding vectors.
Our hope was that one of the tensor-based models would do
better here; this would indicate that the tensor models worked
better regardless of their underlying vectors, count-baed or
neural. This was shown to be indeed the case, as the DisCoCat
CopyObj model achieves the highest correlation with human
judgments.

5. BRIEF SUMMARY AND FUTURE WORK
In this paper, we reviewed the general field of categorical
compositional distributional semantics, to which we referred
as DisCoCat. This field introduces grammar awareness into
vectors models of language, otherwise known as distributional
semantics, thus enabling these models to build vectors for
phrases and sentences, using the vectors of the words therein
and their corresponding grammatical relations. The setting
of a DisCoCat is that of a compact closed category, to
which later Frobenius and Bi algebras were added to reason
about relative pronouns and quantifiers. Compact closed
categories, Frobenius and Bi algebras are also the building
blocks of the categorical approach to Quantum Mechanics,
known under the acronym CQM. Another connection to
Quantum formalisms, is the structure preserving passage
from grammatical structure to vectorial meaning, which is
through a functor similar to the Quantization functor of
Topological Quantum Field Theory. In this paper, we presented
a chronology of the developments of the DisCoCat, briefly
went through its theoretical underpinnings and its experimental
validations.
What remains to be done is to relate the setting of DisCoCat
to the Quantum logical approaches to language, such as the work
done by Preller [61], by Widdows [62], and the original seminal
work of Van Rijsbergen [63].

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
approved it for publication.

FUNDING
Support by EPSRC for Career Acceleration Fellowship
EP/J002607/1 and AFOSR International Scientific Collaboration
grant
FA9550-14-1-0079
is
gratefully
acknowledged
by MS.

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Sadrzadeh

CQM and Natural Language

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Conflict of Interest Statement: The author declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
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author(s) or licensor are credited and that the original publication in this
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July 2017 | Volume 5 | Article 18 | 151

ORIGINAL RESEARCH
published: 14 April 2016
doi: 10.3389/fphy.2016.00012

Inconclusive Quantum
Measurements and Decisions under
Uncertainty
Vyacheslav I. Yukalov 1, 2* and Didier Sornette 1, 3
1

Department of Management, Technology and Economics, ETH Zürich, Zürich, Switzerland, 2 Bogolubov Laboratory of
Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia, 3 Swiss Finance Institute, University of Geneva,
Geneva, Switzerland

Edited by:
Emmanuel E. Haven,
University of Leicester, UK
Reviewed by:
Jan Sladkowski,
The University of Silesia, Poland
Salvatore Micciche’,
Universitá Degli Studi di Palermo, Italy
*Correspondence:
Vyacheslav I. Yukalov
yukalov@theor.jinr.ru
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 25 January 2016
Accepted: 24 March 2016
Published: 14 April 2016
Citation:
Yukalov VI and Sornette D (2016)
Inconclusive Quantum Measurements
and Decisions under Uncertainty.
Front. Phys. 4:12.
doi: 10.3389/fphy.2016.00012

Frontiers in Physics | www.frontiersin.org

We give a mathematical definition for the notion of inconclusive quantum measurements.
In physics, such measurements occur at intermediate stages of a complex measurement
procedure, with the final measurement result being operationally testable. Since the
mathematical structure of Quantum Decision Theory (QDT) has been developed
in analogy with the theory of quantum measurements, the inconclusive quantum
measurements correspond, in QDT, to intermediate stages of decision making in
the process of taking decisions under uncertainty. The general form of the quantum
probability for a composite event is the sum of a utility factor, describing a rational
evaluation of the considered prospect, and of an attraction factor, characterizing
irrational, subconscious attitudes of the decision maker. Despite the involved irrationality,
the probability of prospects can be evaluated. This is equivalent to the possibility of
calculating quantum probabilities without specifying hidden variables. We formulate a
general way of evaluation, based on the use of non-informative priors. As an example,
we suggest the explanation of the decoy effect. Our quantitative predictions are in very
good agreement with experimental data.
Keywords: quantum measurements, decision theory, inconclusive events, quantum probability, non-informative
priors, decoy effect

1. INTRODUCTION
The standard theory of quantum measurements [1] is based on the projection operator
measure corresponding to operationally testable events. Simple measurements really have to
be operationally testable in order to possess physical meaning. However, if a measurement is
composite, consisting of several parts, the intermediate stages do not have to necessarily be
operationally testable, but can be inconclusive.
As a typical example, we can recall the known double-slit experiment, when particles pass
through a screen with two slits and then are registered by particle detectors some distance away
from the screen. This experiment can be treated as a composite event consisting of two parts, one
being the passage through one of the slits and second, registration by detectors. The registration of
a particle by a detector is an operationally testable event, since the particle is either detected or not,
with the result being evident for the observer. But the passage of the particle through one of the
slits is not directly observed, and the experimentalist does not know which of the slits the particle
has passed through. In that sense, the passage of the particle through a slit is an inconclusive event.
The existence of this inconclusive event, occurring at the intermediate stage of the experiment,

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Yukalov and Sornette

is intimately associated with an interference effect. Otherwise, if
the experimentalist would precisely determine the slit through
which the particle has passed, the interference pattern registered
by the particle detectors would be destroyed. The existence of
interference is precisely due to the presence of the inconclusive
event that happened at the intermediate stage.
The occurrence of inconclusive events in decision making
is even more frequent and important. Practically any decision,
before it is explicitly formulated, passes through a stage of
deliberation and hesitation accompanying the choice. That is,
any decision is actually a composite event consisting of an
intermediate stage of deliberation and of the final stage of taking
a decision. The final stage of decision making is equivalent
to an operationally testable event in quantum measurements.
While the intermediate stage of deliberation is analogous to an
inconclusive event.
The analogy between the theory of quantum measurements
and decision theory has been mentioned by von Neumann
[1]. Following this analogy, Quantum Decision Theory (QDT)
has been advanced [2–7], with the mathematical structure that
is applicable to both decision making as well as to quantum
measurements. The generality of our framework, being equally
suitable for quantum measurements and decision making, is its
principal difference from all other attempts that employ quantum
techniques in psychological sciences. An extensive literature on
various quantum models in psychology and cognitive science can
be found in the books [8–11] and review articles [12–15].
Any approach, applying quantum techniques to decision
theory, is naturally based on the notion of probability.
This is because quantum theory is intrinsically probabilistic.
Respectively, the intrinsically probabilistic nature of QDT is
what makes it principally different from stochastic decision
theory, where the choice is treated as always being deterministic,
while in the process of choosing the decision maker acts
with errors [16–20]. Such stochastic decision theories can be
termed as “deterministic theories embedded into an environment
with stochastic noise.” The standard way of using a stochastic
approach is to assume a probability distribution over the values
characterizing the errors made by the subjects in the process of
decision making. Then the parameters entering the distribution
are fitted to a posteriori empirical data by maximizing the loglikelihood function. Such a procedure allows one to better fit
the given set of data to the assumed basic deterministic decision
theory, in particular due to the introduction of additional fitting
parameters. However, it does not essentially change the structure
of the underlying deterministic theory, although improving it
slightly. And, being descriptive, the classical stochastic approach
does not provide quantitative predictions.
Contrary to classical stochastic theory, in the quantum
approach, we do not assume that the choice of a decision maker
is deterministic, with just some weak disturbance by errors.
Following the general quantum interpretation, we consider
the choice process, including deliberations, hesitations, and
subconscious estimation of competing outcomes, as intrinsically
random. The probabilistic decision, in the quantum case, is not
just a stochastic decoration of a deterministic process, but it
is an unavoidable random part of any choice. The existence

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Inconclusive Quantum Measurements and Decisions under Uncertainty

of the hidden, often irrational subconscious feelings and
deliberations, results in the appearance of quantum interference
and entanglement. The difference between classical stochastic
decision theory and QDT is similar to the difference between
classical statistical physics and quantum theory. In the former,
all processes are assumed to be deterministic, with statistics
coming into play because of measurement uncertainties, such as
no precise knowledge of initial conditions and the impossibility
of measuring exactly the locations and velocities of all particles.
In contrast, quantum theory is principally probabilistic, which
becomes especially important for composite measurements.
A detailed mathematical theory of quantum measurements
in the case of composite events has been developed in our
previous papers [21–23]. In the present paper, we concentrate our
attention on composite measurements including intermediate
inconclusive events and on the application of this notion for
characterizing decision making under risk and uncertainty.
The importance of composite events, including intermediate
inconclusive events, in decision theory makes it necessary to pay
a special attention to the correct mathematical formulation of
such events and to the description of their properties allowing
for the quantitative evaluation of the corresponding quantum
probabilities. We show that, despite uncertainty accompanying
inconclusive events, it is possible to give quantitative evaluations
for quantum probabilities in decision theory, based on noninformative priors. Considering, as an illustration, the decoy
effect, we demonstrate that even the simple non-informative
priors provide predictions in very good agreement with
experimental data.

2. COMPOSITE QUANTUM
MEASUREMENTS AND EVENTS
In this section, we give a brief summary of the general scheme
for defining quantum probabilities for composite events. As we
have stressed above, in our approach, the mathematics is the same
for describing either quantum measurements or decision making.
Therefore, when referring to an event, we can keep in mind either
a fact of measurement or a decision action.
Let An be a conclusive operationally testable event labeled by
an index n. And let B = {Bα } be a set of inconclusive events
labeled by α. Defining the space of events as a Hilbert space H,
we associate with an event An a state |ni in this Hilbert space and
an event operator P̂n ,
An → |ni → P̂n = |nihn| .

(1)

The event operator for an operationally testable event is a
projector.
The set of inconclusive events B generates in the Hilbert space
H the state |Bi and the event operator P̂B ,
B → |Bi → P̂B = |BihB| ,

(2)

where the state reads
|Bi =

X
α

bα |αi ,

(3)

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Inconclusive Quantum Measurements and Decisions under Uncertainty

with coefficients bα being random complex numbers. The event
operator for an inconclusive event is not necessarily a projector,
but a member of a positive operator-valued measure [7, 21–23].
The space of events, in the quantum approach, is the Hilbert
space
O
H = HA
HB
(4)
that is a tensor product of the spaces
HA = span{|ni} ,

HB = span{|αi} .

Each decision maker is characterized by an operator ρ̂ that can be
termed the strategic state of a decision maker, which, in quantum
theory, corresponds to a statistical operator. The pair {H, ρ̂}, in
physics, is named a statistical ensemble, and in decision theory, it
is a decision ensemble.
A composite event is called a prospect and is denoted as
πn = An

O

B.

(5)

A prospect πn generates a state |πn i in the Hilbert space of events
H and a prospect operator P̂(πn ),
πn → |πn i → P̂(πn ) = |πn ihπn | ,

(6)

with the prospect state
|πn i = |ni

O

|Bi =

X
α

bα |nαi .

(7)

The prospect operator is a member of a positive operator-valued
measure, which implies that these operators satisfy the resolution
of unity [21, 23]. Since they contain random quantities bα , the
corresponding random resolution has to be understood not as
a direct equality between numbers, but, e.g., as the equality in
mean [24].
The prospect probability is
p(πn ) = Tr ρ̂ P̂(πn ) ,

(8)

with the trace over the space H. To form a probability measure,
the prospect probabilities are to be normalized:
X
n

p(πn ) = 1 ,

0 ≤ p(πn ) ≤ 1 .

(9)

Taking explicitly the trace in expression (Equation 8) and
separating diagonal and off-diagonal terms, we see that the
prospect probability
p(πn ) = f (πn ) + q(πn )

(10)

is represented as a sum of a positive-definite term
f (πn ) =

X
α

|bα |2 hnα|ρ̂|nαi

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(11)

and a sign-undefined term
q(πn ) =

X

α6=β

b∗α bα hnα|ρ̂|nβi .

(12)

The appearance of the sign-undefined term is a purely quantum
effect responsible, in quantum measurements, for interference
patterns. The attenuation of this quantum term is called
decoherence. In quantum theory, decoherence can be due to
external as well as to internal perturbations and the influence of
measurements [25–27]. And in QDT, decoherence can occur due
to the accumulation of information [28].
The disappearance of the quantum term implies the transition
to classical theory. This is formulated as the quantum-classical
correspondence principle [29], which in our case reads as
p(πn ) → f (πn ) ,

q(πn ) → 0 .

(13)

This principle tells us that the term f (πn ) plays the role of classical
probability, hence is to be normalized:
X
n

f (πn ) = 1 ,

0 ≤ f (πn ) ≤ 1 .

(14)

When decisions concern a choice between lotteries, the classical
term f (πn ) has to be defined according to classical decision theory
based either on expected utility theory or on some non-expected
value functional. This suggests to call f (πn ) a utility factor, since
it is defined on rational grounds and reflects the utility of a
choice. The quantum term is caused by the interference and
entanglement effects in quantum theory, which correspond, in
decision making, to irrational effects describing the attractiveness
of choice. Therefore, it can be called the attraction factor. From
Equations (9) and (14), it follows the alternation law
X
n

q(πn ) = 0 ,

−1 ≤ q(πn ) ≤ 1 .

(15)

Note that, in quantum theory, the definition of the composite
event in the form of prospect (Equation 5) is valid for any
type of operators, since they are defined on different spaces.
No problem with non-commutativity of operators defined on
a common Hilbert space arises. This way makes it possible to
introduce joint quantum probabilities for several measurements
[21, 23]. Contrary to this, considering operators on the same
Hilbert space does not allow one to define joint probabilities
for non-commuting operators. Sometimes, one treats the Lüders
probability of consecutive measurements as a conditional
probability. This, however, is not justified from the theoretical
point of view [21, 23, 30] and also contradicts experimental data
[31, 32]. But defining the quantum joint probability according to
expression (Equation 8) contains no contradictions.
In the present section, the general scheme of QDT is
presented. Being limited by the length of this paper, we cannot
go into all mathematical details that have been thoroughly
described in our previous publications. However, we would
like to stress that for the purpose of practical applications, it
is not necessary to study all these mathematical peculiarities,

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Inconclusive Quantum Measurements and Decisions under Uncertainty

but it is sufficient to employ the final formulas following the
prescribed rules. One can use the formulated rules as given
prescriptions, without studying their justification. The main
formulas of this section, which are necessary for the following
application, are: the form of the quantum probability (Equation
10), the normalization conditions (Equations 9 and 14), and the
alternation law (Equation 15). More details required for practical
application will be given in the sections below.

3. NON-INFORMATIVE PRIOR FOR UTILITY
FACTORS
To make the above scheme applicable to decision theory, it is
necessary to specify how one should quantify the values of the
utility factors and attraction factors. Here we show how these
values can be defined as non-informative priors.
Let us consider a set of lotteries Ln = {xi , pn (xi ) : i =
1, 2, . . . , Nn }, enumerated by the index n = 1, 2, . . . , NL , with
payoffs xi and their
P probabilities pn (xi ). The related expected
utilities U(Ln ) = i u(xi )pn (xi ) can be defined according to the
expected utility theory [33]. For the present consideration, the
utility functions u(x) do not need to be specified. For instance,
they can be taken as linear functions, since this choice has the
advantage of making the utility factors independent from the
units measuring the payoffs.
In QDT, the act of choosing a lottery Ln , denoted as An ,
together with the accompanying set of inconclusive events B,
including the decision-maker hesitations [6, 30], compose the
prospect (Equation 5). Depending on whether the expected
utilities are positive on negative, there can be two cases.
If the expected utilities of the considered set of lotteries are all
positive (non-negative), such that
U(Ln ) ≥ 0

(n = 1, 2, . . . , NL ) ,

(16)

then it is reasonable to require that zero utility corresponds to
zero utility factor:
f (πn ) → 0 ,

U(Ln ) → 0 .

(17)

The case where the utility factor is simply proportional to the
related expected utility trivially obeys this condition (Equation
17). Taking into account the normalization condition (Equation
14) gives the utility factor
U(Ln )
f (πn ) = P
.
n U(Ln )

(18)

(n = 1, 2, . . . , NL ) ,

(19)

the required condition is that an infinite loss corresponds to zero
utility factor:
f (πn ) → 0 ,
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|U(Ln )| → ∞ .

|U(Ln )|−1
f (πn ) = P
.
−1
n |U(Ln )|

(21)

The utility-factor forms (Equations 18 and 21) coincide with the
choice probabilities in the Luce choice axiom [34]. It is possible
to show that generalized forms for the utility factors can be
derived by maximizing a conditional Shannon entropy or from
the principle of minimal information [12, 35, 36].
In the case of positive expected utilities, we consider the
information functional, taking into account the normalization
condition (Equation 14) and the expected log-likelihood 3. This
functional reads as
"
#
X
X
I[ f (πn ) ] =
f (πn ) ln f (πn ) + λ
f (πn ) − 1
n

+α

n

"
X
n

#

f (πn )3(πn ) − 3

(22)

,

where
3(πn ) = − ln U(Ln ) ,

U(Ln ) ≥ 0 .

Minimizing functional (Equation 22) results in the utility factor
U α (Ln )
f (πn ) = P α
n U (Ln )

(α > 0) ,

(23)

in which the positive sign of α is prescribed by the condition that
the larger utility implies the larger factor.
In the case of negative expected utilities, the information
functional takes the form
"
#
X
X
I[ f (πn ) ] =
f (πn ) ln f (πn ) + λ
f (πn ) − 1
n

n

"

+γ 3−

X

f (πn )3(πn )

#

, (24)

n

where
3(πn ) = − ln |U(Ln )| ,

U(Ln ) < 0 .

Then its minimization yields the utility factor

When the expected utilities are negative, which happens in the
domain of losses, such that
U(Ln ) < 0

The simplest way to satisfy this condition (Equation 20) is that
the utility factor is inversely proportional to the related expected
utility. Taking into account the normalization condition, we get

(20)

|U(Ln )|−γ
f (πn ) = P
−γ
n |U(Ln )|

(γ > 0) ,

(25)

with the positive sign of γ prescribed by the requirement that the
larger cost implies the smaller factor.
The utility factors (Equations 23 and 25) are the examples
of power-law distributions that are known in many applications
[35–37].

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4. NON-INFORMATIVE PRIOR FOR
ATTRACTION FACTORS
Although the attraction factor characterizes irrational features
of decision making, it can be estimated by invoking noninformative prior assumptions. An important consequence of the
latter is the quarter law derived earlier [4, 5, 12]. Here we first
give the new, probably the simplest, derivation of the quarter law
and, second, we show how this law can be used for estimating the
attraction factors in the case of an arbitrary number of prospects.
Let us consider the sum
1
NL

NL
X
n=1

|q(πn )| =

Z

1

ϕ(x)x dx

(26)

0

NL
1 X
[ δ(x − q(πn )) + δ(x + q(πn )) ]
NL

(27)

n=1

plays the role of the attraction-factor distribution. The latter is
normalized as
Z 1
ϕ(x) dx = 1 ,
(28)
−1

since the attraction factors, in view of condition (Equation 15),
vary in the interval [−1, 1]. If q(πn ) does not equal zero, then
normalization (Equation 28) is evident. And when q(πn ) = 0,
then one should use the identity
1

Z

0

δ(x) dx =

0

x1

1
.
2

q(πn ) = qmax − (n − 1)1 .

qmax =

(30)

NL − 1
1.
2

(36)

And the quarter law (Equation 30) leads to the gap

2

P

NL
.
|N
+ 1 − 2n|
L
n

(37)

If NL is even, then
NL
X
n=1

|NL + 1 − 2n| =

NL2
2

(NL even) ,

while when NL is odd, then

n=1

NL
1 X
1
|q(πn )| = .
NL
4

(35)

With notations (Equations 32 and 34), the alternation condition
(Equation 15) yields

NL
X

In that case, integral (Equation 26) results in the quarter law

(independent of n) . (34)

Taking their average values as determining their typical values,
we omit the symbol h.i representing the average operator and use
Equation (34) to represent the n-th attraction factor as

1=

(29)

xNL −1

where the series η1 ≤ η2 ≤ ... ≤ ηNL of inequalities ensure
the ordering. It is then straightforward to show that the average
values of the q(πn ) are equidistant, i.e., the difference between any
two neighboring factors is on average

1
2

for the semi-integral of the Dirac function.
The use of a non-informative prior implies that the values of
the attraction factor are not known. A full ignorance is captured
by a uniform distribution, which, according to normalization
(Equation 28), gives
ϕ(x) =

Pr[q(π1 ) < η1 , q(π2 ) < η2 , ..., q(πNL ) < ηNL |η1 ≤ η2 ≤ ...
Z ηN
Z η1
Z η2
L
dx1
dx2 ....
dxNL , (33)
≤ ηNL ] =

1 ≡ hq(πn )i−hq(πn+1 )i = const

of the attraction factor moduli, where
ϕ(x) ≡

Given the unknown values of the attraction factors, the noninformative prior assumes that they are uniformly distributed
and at the same time they must obey the ordering constraint
(Equation 31). Then, the joint cumulative distribution of the
attraction factors is given by

|NL + 1 − 2n| =

NL2 − 1
2

(NL odd) .

This allows us to represent gap (Equation 37) as

n=1

If the prospect lattice L = {πn } consists of NL prospects, we
can always arrange the corresponding attraction factors in the
ascending order, such that
q(πn ) > q(πn+1 )

(n = 1, 2, . . . , NL − 1) .

(31)

1=

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(32)




1
NL

(NL even)

NL
NL2 −1

(NL odd) .

(38)

And for the largest attraction factor, we find

We denote the largest attraction factor as
qmax ≡ q(π1 ) > 0 .





qmax =







NL −1
2NL

(NL even)

NL
2(NL +1)

(NL odd) .

(39)

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The above expressions make it possible to evaluate, on the basis
of the non-informative prior, the whole set
QNL ≡ {q(πn ) : n = 1, 2, . . . , NL }
of the attraction factors:
 1

 2NL (NL − 2n + 1)
q(πn ) =

 N2L
(NL − 2n + 1)
2(N −1)

(40)

(NL even)
(41)
(NL odd) .

L

For example, in the case of two prospects, we have
1=

1
,
2

qmax =

1
4

(NL = 2) ,

which yields the attraction set


1
1
Q2 =
, −
.
4
4
For three prospects, we get
1=

3
,
8

qmax =

3
8

(NL = 3) ,

hence
Q3 =



3
3
, 0, −
8
8



.

Similarly, for four prospects, we find
1=

1
,
4

qmax =

3
8

(NL = 4) ,

with the attraction set
Q4 =



3 1
1
3
, , −
, −
8 8
8
8



.

When there are five prospects, then
1=

5
,
24

qmax =

5
12

(NL = 5) ,

from where
Q5 =



5
5
5
5
,
, 0, −
, −
12 24
24
12



.

Thus, we can evaluate the attraction factors for any number of
prospects, obtaining a kind of a quantized attraction set. In the
case of an asymptotically large number NL of prospects, we have
1≃

1
,
NL

qmax ≃

1
2

(NL ≫ 1) ,

(42)

and
q(πn ) ≃

1 2n − 1
−
.
2
2NL

(43)

The non-informative priors can be employed for predicting
the results of decision making. This makes the principal
difference compared with the introduction into expected utility
of adjustment parameters that are fitted post-hoc to the given
experimental data [38].

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5. QUANTITATIVE EXPLANATION OF
DECOY EFFECT
We now show how the non-informative priors of the attraction
factors can be employed to explain the decoy effect and for
quantitative prediction in decision-making. Throughout this
section, we denote, for simplicity, the objects of choice, say A,
as well as the act of choosing an object A, by the same letter A. As
has been emphasized above, the act of choice under uncertainty
is a composite prospect. But, again for simplicity, we employ the
same letter for denoting the action A and the related prospect
(Equation 5).
The decoy effect was first studied by Huber et al. [39], who
called it the effect of asymmetrically dominated alternatives. Later
this effect has been confirmed in a number of experimental
investigations [40–43]. The meaning of the decoy effect can be
illustrated by the following example. Suppose a buyer is choosing
between two objects, A and B. The object A is of better quality, but
of higher price, while the object B is of slightly lower quality, while
less expensive. As far as the functional properties of both objects
are not drastically different, but B is cheaper, the majority of
buyers value the object B higher. At this moment, the salesperson
mentions that there is a third object C, which is of about the
same quality as A, but of a much higher price than A. This causes
the buyer to reconsider the choice between the objects A and
B, while the object C, having the same quality as A but being
much more expensive, is of no interest. Choosing now between A
and B, the majority of buyers prefer the higher quality but more
expensive object A. The object C, being not a choice alternative,
plays the role of a decoy. Experimental studies confirmed the
decoy effect for a variety of objects: cars, microwave ovens,
shoes, computers, bicycles, beer, apartments, mouthwash, etc.
The same decoy effect also exists in the choice of human mates
distinguished by attractiveness and sense of humor [44]. It is
common as well for animals, for instance, in the choice of female
frogs of males with different attraction calls characterized either
by low-frequency and longer duration or by faster call rates
[45].
The decoy effect contradicts the regularity axiom in decision
making telling that if B is preferred to A in the absence of C, then
this preference has to remain in the presence of C.
In the frame of QDT, the decoy effect is explained as follows.
Assume buyers consider an object A, which is of higher quality
but more expensive, and an object B, which is of moderate quality
but cheaper. Suppose the buyers have evaluated these objects A
and B, which implies that the initial values of the objects are
described by the utility factors f (A) and f (B). In experiments, the
latter correspond to the fractions of buyers evaluating higher the
related object. When the decoy C, of high quality but essentially
more expensive, is presented, it attracts the attention of buyers to
the quality characteristic. The role of the decoy is well understood
as attracting the attention of buyers to a particular feature of
the considered objects, because of which the decoy effect is
sometimes named the attraction effect [40]. In the present case,
the decoy attracts the buyer attention to quality. The attraction,
induced by the decoy, is described by the attraction factors
q(A) and q(B). Hence the probabilities of the related choices

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Inconclusive Quantum Measurements and Decisions under Uncertainty

are now
p(A) = f (A) + q(A) ,

p(B) = f (B) + q(B) .

Since the quality feature becomes more attractive, q(A) > q(B).
According to the non-informative prior, we can estimate the
attraction factors as q(A) = 1/4 and q(B) = −1/4.
To be more precise, let us take numerical values from the
experiment of Ariely and Wallsten [43], where the objects under
sale are microwave ovens. The evaluation without a decoy results
in f (A) = 0.4 and f (B) = 0.6. In the presence of the decoy, we
predict that the choice probabilities can be evaluated as
p(A) = f (A) + 0.25 ,

p(B) = f (B) − 0.25 .

This gives p(A) = 0.65 and p(B) = 0.35. The experimental values
for the choice between A and B, in the presence of but excluding
C, correspond to the fractions pexp (A) = 0.61 and pexp (B) = 0.39,
which is close to the predicted probabilities.
Another example can be taken from the studies of the frog
mate choice [45], where frog males have attraction calls differing
in either low-frequency sound or call rate. The males with lower
frequency calls are denoted as A, while those with high call rate,
as B. In an experiment with 80 frog females, without a decoy, it
was found that females evaluate higher the fastest call rate, so that
f (A) = 0.35 and f (B) = 0.65. In the presence of an inferior
decoy, attracting attention to the low-frequency characteristic,
the non-informative prior predicts the probabilities
p(A) = 0.35 + 0.25 = 0.6 ,

p(B) = 0.65 − 0.25 = 0.4 .

The empirically observed fractions are found to be pexp (A) =
0.6 and pexp (B) = 0.4, in remarquable agreement with our
predictions.
To make it clear how the decoy effect fits the title of
the paper “Inconclusive quantum measurements and decisions
under uncertainty,” it is worth extending the comments that have
been mentioned in the Introduction.
Our principal point of view is that decision making, generally,
almost always deals with composite events, since any choice
is accompanied by subconscious feelings and irrational biases.
The latter are often difficult to formalize and, even more, their
weights usually are not known and are practically unmeasurable.
This is why these subconscious irrational factors can be treated
as what is called inconclusive events. When choosing between
several possibilities, say An , one actually considers composite
prospects, as defined in Equation (5). And the composite nature
of choices requires the use of quantum techniques, as has been
explained in our previous paper [30]. Otherwise, the probabilities
of simple events could be characterized by classical theory. It is
the composite nature of the considered prospects that yields the
appearance of the quantum term q(πn ) related to interference
and coherence effects. In that way, the choice between the objects
in the decoy effect is also a composite prospect, being composed
of the choice as such and accompanying subconscious feelings
forming an inconclusive set. This is why the use of QDT here is
necessary and why it gives so good results.

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It is admissible to give a schematic picture of the choice in
the decoy effect by analogy with the double-slit experiment in
physics, which is mentioned in the Introduction. Thus, making
a concrete selection of either an object A or B is the analog of
the registration of the particle by a detector. But before such a
selection is done, there exists the uncertainty of deciding which
of the object features are actually more important. These not
precisely defined acts of hesitation play the role of the slits,
with the uncertainty associated with which of them the particle
has passed through. When it is known which of the slits the
particle has passed through, then the interference effects in
physics disappear. Similarly, in decision theory, if the values
of each object are clearly defined, there are no hesitations, no
interference, and the selection can be based on classical rules.
Such an objective evaluation in the decoy effect happens in the
absence of any decoy, when a decision maker rationally evaluates
the features of the given objects, say quality and price. The
appearance of a decoy induces hesitations concerning which of
the features are actually more important. These hesitations before
the choice are the analogs of the uncertainty of which slits will
be visited by the traveling particle. The uncertainty results in
the interference and the arising quantum term, whether in the
registration of a particle or in the final choice of a decision maker.

6. DISCUSSION
We have presented a mathematical formulation for the concept
of inconclusive quantum measurements and events. This type
of measurements in physics happens at intermediate stages of
composite measuring procedures, while the final measurement
stage is operationally testable. In decision making, inconclusive
events correspond to the intermediate stage of deliberations.
Invoking non-informative priors, it is possible to estimate the
prospect probabilities, thus, predicting the results of decision
making.
Generally, invoking more information on the properties of the
attraction factor, it is possible to define its form more accurately
than the value given by non-informative prior. For example, from
condition (Equation 9) it follows that
−f (πn ) ≤ q(πn ) ≤ 1 − f (πn ) .
Hence, for a positive q(πn ), we have
0 ≤ q(πn ) ≤ 1 − f (πn ) .
While for a negative q(πn ), we get
−f (πn ) ≤ q(πn ) ≤ 0 .
Therefore, the attraction factor has to satisfy the limits
q(πn ) → +0 ,

f (πn ) → 1 ,

q(πn ) → −0 ,

f (πn ) → 0 .
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Inconclusive Quantum Measurements and Decisions under Uncertainty

This suggests that the absolute value of the attraction factor can
be modeled by an expression proportional to
q(πn ) ∝ f µ (πn )[1 − f (πn )]ν ,
with µ and ν being positive parameters and the sign defined by
the ambiguity and risk aversion principle [4–6, 12]. More detailed
study of such a form will be given in a separate paper.
But it turns out that even the simple non-informative prior
provides us a rather good estimate allowing for quantitative
predictions in decision making. And we have illustrated the
approach by the decoy effect for which the non-informative
priors yield quantitative predictions in very good agreement with
empirical data.
In this paper, decision making by separate subjects is
considered. We think that the theory can be generalized by
considering societies of decision makers. The exchange of
information in a society should certainly influence the decisions
of the society members. To develop a theory of many agents, it
is necessary to generalize the apporach by treating a dynamical
model of agents exchanging information. Then, we think, it
would be feasible to describe the behavior of the agents operating

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AUTHOR CONTRIBUTIONS
All authors listed, have made substantial, direct and intellectual
contribution to the work, and approved it for publication.

ACKNOWLEDGMENTS
Financial support from the ETH Zürich Risk Center is
appreciated. We are greateful for discussions to E.P. Yukalova.

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Conflict of Interest Statement: The authors declare that the research was
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be construed as a potential conflict of interest.
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