Review

Quantum Operation of Affective Artificial Intelligence

V.I. Yukalov

Bogolubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, Dubna 141980, Russia

and

Instituto de Fisica de São Carlos, Universidade de São Paulo,
CP 369, São Carlos 13560-970, São Paulo, Brazil

e-mail<EMAIL_ADDRESS>

###### Abstract

The review analyzes the fundamental principles which Artificial Intelligence
should be based on in order to imitate the realistic process of taking
decisions by humans experiencing emotions. Two approaches are considered, one
based on quantum theory and the other employing classical terms. Both these
approaches have a number of similarities, being principally probabilistic. The
analogies between quantum measurements under intrinsic noise and affective
decision making are elucidated. It is shown that cognitive processes have many
features that are formally similar to quantum measurements. This, however, in
no way means that for the imitation of human decision making Affective
Artificial Intelligence has necessarily to rely on the functioning of quantum
systems. The analogies between human decision making and quantum measurements
merely demonstrate formal common properties in their functioning. It is in
this sense that one has to understand quantum operation of Artificial
Intelligence. Appreciating the common features between quantum measurements
and decision making helps for the formulation of an axiomatic approach
employing only classical notions. Artificial Intelligence, following this
approach, operates similarly to humans, by taking into account the utility of
the considered alternatives as well as their emotional attractiveness.
Affective Artificial Intelligence, whose operation takes account of the
cognition-emotion duality, avoids numerous behavioural paradoxes of
traditional decision making. A society of intelligent agents, interacting
through the repeated multistep exchange of information, forms a network
accomplishing dynamic decision making based on the evaluation of utility and
affected by the emotional attractiveness of alternatives. The considered
intelligent networks can characterize the operation of either a human society
of affective decision makers, or the brain composed of neurons, or a typical
probabilistic network of an artificial intelligence.

Keywords: artificial intelligence, quantum measurements, quantum intrinsic
noise, affective decision making, cognition-emotion duality, behavioural
paradoxes, dynamic decision making, collective decision making, probabilistic
networks

Contents

1. Introduction

2. Measurements under intrinsic noise

2.1. Quantum algebra of events

2.2. Operationally testable events

2.3. Modes of intrinsic noise

2.4. Noise-decorated alternatives

2.5. Quantum probability space

2.6. Quantum-classical correspondence

2.7. Probability of superposition states

2.8. Alternative-noise entanglement

2.9. Entanglement production by measurements

2.10. Time dependence of probability

2.11. Quantum state reduction

2.12. Consecutive measurements of alternatives

2.13. Immediate consecutive measurements

2.14. Synchronous noiseless measurements

2.15. Synchronous measurements under noise

2.16. Swap order relations

2.17. Quantum versus classical probabilities

2.18. Quantum decision theory

3. Affective decision making

3.1. Evolutionary origin of emotions

3.2. Problems in decision making

3.3. Behavioural probabilities of alternatives

3.4. Quantification of utility factor

3.5. Magnitude of attraction factor

3.6. Multiple attraction factors

3.7. Problems in classifying attractiveness

3.8. Explicit attraction factors

3.9. Buridan’s donkey problem

3.10. Kahneman-Tversky lotteries

3.11. Verification of quarter law

3.12. Contextuality of attraction factors

3.13. Choice between bundled alternatives

3.14. Quantum versus classical consciousness

4. Resolution of behavioural paradoxes

4.1. St. Petersburg paradox

4.2. Martingale illusion

4.3. Allais paradox

4.4. Independence paradox

4.5. Ellsberg paradox

4.6. Prisoner dilemma

4.7. Disjunction effect

4.8. Conjunction fallacy

4.9. Disposition effect

4.10. Ariely paradox

4.11. Decoy effect

4.12. Planning paradox

4.13. Preference reversal

4.14. Preference intransitivity

4.15. Order effects

5. Networks of intelligent agents

5.1. Multistep decision making

5.2. Types of interactions and memory

5.3. Networks with uniform memory

5.4. Network with mixed memory

5.5. Dynamic regimes of preferences

5.6. Attenuation of emotion influence

5.7. Continuous decision making

5.8. Discrete versus continuous processes

5.9. Time discounting of utility

5.10. Collective network operation

6. Conclusion

##  1 Introduction

Artificial Intelligence is understood as intelligence demonstrated by
machines, as opposed to natural intelligence displayed by animals including
humans. The main Artificial Intelligence textbooks define the field as the
study of artificial intelligent systems perceiving the information obtained
from the environment and taking decisions and actions for the goal attainment
[1, 2, 3, 4, 5, 6]. There is wide agreement among artificial intelligence
researchers that to be called intelligence, it is required to be able to use
logical strategy and make judgments under uncertainty.

A system possessing intelligence is termed an intelligent agent. That system,
evaluating the available information, is able to take autonomous actions and
decisions directed to the achievement of the desired goals and may improve its
performance with learning or using obtained knowledge [1, 2, 3, 4, 5, 6].
Often, the term intelligent agent is applied to systems possessing artificial
intelligence. However the intelligent agent paradigm is closely related and
employed with respect to agents in economics, in cognitive science, ethics,
philosophy, as well as in many interdisciplinary socio-cognitive modeling and
simulations. Generally, from the technical or mathematical point of view, the
notion of intelligent agent can be associated with either real or artificial
intelligence. An intelligent agent could be anything that makes decisions, as
a person, firm, machine, or software.

In this review, we concentrate on one of the most difficult and important
problems of Artificial Intelligence, that is on the mechanism of taking
decisions similarly to this process in humans, whose decisions practically
always are accompanied by emotions. The achievement of human-level machine
intelligence has been a principal goal from the beginning of works on
Artificial Intelligence [1, 2, 3, 4, 5, 6]. The key-point of the present
review is the description of how affective decision making could be
mathematically formalized to the level sufficient for the functioning of
Artificial Intelligence imitating human decision processes in which emotions
are an inevitable part. Below, talking about Artificial Intelligence we keep
in mind Affective Artificial Intelligence.

In order to formulate the basic operational algorithms of Affective Artificial
Intelligence, it is necessary to develop a mathematical description of human
affective decision making. The problem of emotion quantification consists of
two sides. One side is the assessment of emotions experienced by a subject as
reactions on external events, e.g., hearing voice or looking at pictures. The
arising emotions can include happiness, anger, pleasure, disgust, fear,
sadness, astonishment, pain, and so on. The severity or intensity of such
emotions can be estimated by studying the expressive forms manifesting
themselves in motor reactions, such as facial expressions, pantomime, and
general motor activity, and by measuring physiological reactions, such as the
activity of the sympathetic and parasympathetic parts of the autonomic nervous
system, as well as the activity of the endocrine glands. Vegetative
manifestations of emotions can be noticed by studying changes in the
electrical resistance of the skin, the frequency and strength of heart
contractions, blood pressure, skin temperature, hormonal and chemical
composition of the blood, and like that. There exists a vast literature on the
methods of emotion detection and appraisal in speech, facial expressions, and
body gestures [7, 8]. The study and development of systems and devices that
can recognize, interpret, process, and simulate human affects is named
Affective Computing [9, 10]. These problems will not be touched in the review.

The other side of the story is the challenge of characterizing how emotions
influence decision making. To formulate the principles of functioning of
Affective Artificial Intelligence in the process of taking decisions, it is
necessary to be able to quantify the role of emotions in this process. It is
this objective that is in the center of the present review.

This goal confronts the basic problem of how emotions, arising in the process
of decision making, could be defined and quantified. It seems to be too
difficult, if possible at all, to develop a formalized quantification of
emotions allowing for the selection, in the presence of emotions, of an
optimal alternative in the cognitive process of making decisions. The
mathematical description of emotion influence in the process of decision
making is the hard problem that has not found yet a comprehensive solution
[11].

Difficulties start with the fact that there is no a unique generally accepted
definition of what is emotion as compared to cognition. It is possible to
mention the long-standing dispute about whether emotion is primary and
independent of cognition [12, 13], or secondary and always dependent upon
cognition [14, 15], although there is the point of view that this dispute is
largely semantic, being induced by dissimilar definitions [16].

The studies on brain organization often support the assumption that there is a
considerable degree of functional specialization and that many regions of
brain can be conceptualized as either affective or cognitive. Popular examples
are the amygdala in the domain of emotion and the lateral prefrontal cortex in
the case of cognition. However, there are arguments [17, 18] that complex
cognitive-emotional behaviours have their basis in dynamic coalitions of
networks of brain areas, none of which should be conceptualized as
specifically affective or cognitive. Different brain areas exhibit a high
degree of connectivity for regulating the flow and integration of information
between brain regions, which results in the intense cognitive-emotional
interactions. Usually, under “emotions” one understands just a placeholder for
something much broader than emotions in a narrow sense, including affective
processes in general [19]. There are arguments that the notions of emotion,
cognition, and the related phenomena can be more precisely defined in a
functional framework, for example in terms of behavioural principles [20],
with respect to emotion taxonomy [21], to emotion regulation [22], or studying
the emotion appraisal during the dynamics of the emotion process [23, 24, 7].
More references on the definition of emotions and their relation to cognition
can be found in the surveys [25, 26, 27].

The functional framework keeps in mind the operational separation of cognition
and emotion as the notions related to the process of decision making that
comprises two sides, reasoning and affective [11, 28]. Under the reasoning
side one means the ability of formulating explicit rules allowing for a
normative choice. And the affective side implies the possibility of making a
choice being influenced by emotions that not always allow for explicit formal
prescriptions. The reasoning-affective dichotomy in decision making is often
called rational-irrational duality [29]. As is explained above, there is no
strictly speaking uniquely defined and absolutely separated notions of
cognitive and affective, as well as of rational and irrational. However, our
goal is not to plunge into semantic debates, but to describe an approach
taking into account two aspects of decision making, normative allowing for the
explicit evaluation of utility and affective that seems to avoid the
characterization by prescribed formal rules. The kaleidoscope of emotions can
be quite ramified and not allowing for sharp categorical definitions, because
of which it is labeled [23, 24, 7] as idiosyncratic and fuzzy. This fuzziness
is the main obstacle in the attempts of quantifying the influence of emotions
on decision making.

Thus the principal difference between a standard programmed robot or computer
and human-type intelligence is the cognition-emotion duality of human
consciousness in the process of taking decisions. For clarity, one can talk
about human intelligence, although the same duality in decision making is
typical of practically all alive beings, as numerous empirical studies prove.
The animals likely feel a full range of emotions, including fear, joy,
happiness, shame, embarrassment, resentment, jealousy, rage, anger, love,
pleasure, compassion, respect, relief, disgust, sadness, despair, and grief
[30].

The cognition-emotion duality of human consciousness, exhibited when taking
decisions, combines rational conscious evaluation of utility of the intended
actions with irrational subconscious emotions. The latter are especially
noticeable in decisions under risk and uncertainty. This duality is the cause
of a number of behavioural paradoxes in classical decision making, when human
actions contradict expected utility theory. So, in order to formulate explicit
algorithms for the operation of Affective Artificial Intelligence, comprising
cognition-emotion duality, it is necessary to develop an adequate theory of
affective decision making that could give realistic predictions under
uncertainty.

The existence of the cognition-emotion duality in decision making hints on the
possibility of its description by resorting to the techniques of quantum
theory, in which there also exists duality, the so-called particle-wave
duality [31]. Although the nature of these notions in physics and decision
theory is rather different, but, probably, the mathematical techniques of
quantum theory could hint on the similar description of both phenomena. Bohr
[32, 33] was the first to assume that the functioning of the human brain could
be described by the techniques of quantum theory. Since then, there have
appeared numerous publications discussing the possibility of directly applying
quantum techniques for characterizing the process of human decision making.
These discussions, assuming that consciousness is quantum or quantum-like have
been summarized in many review works, e.g. [34, 35, 36, 37, 38, 39], where
numerous references on different attempts of applying quantum techniques to
the description of consciousness are cited.

It is necessary to accept that many researchers are rather sceptical with
regard to the parallelism between quantum physics and cognitive processes
because of the following reasons:

(i) First of all, according to the current neurophysiological knowledge, the
brain is in no way a quantum system, hence, it has nothing to do with quantum
consciousness. The assumption that the brain’s neurons act as miniature
quantum devices, thus that the brain functions similarly to a quantum computer
[40, 41] has been justly criticized [42] by showing that decoherence effects
do not allow for neurons to act as quantum objects. This does not exclude that
some quantum processes do exist in the brain, which are studied in quantum
biophysics [43, 44]. Nevertheless, the brain as a whole and its functioning
seem to have nothing to do with quantum theory.

(ii) The above objection is usually refuted by saying that the possibility of
describing human thinking processes by means of quantum theory does not
require the assumption that human brains are some quantum systems. Instead, it
holds that, although the brain is not a quantum object, but cognition and the
process of human thinking can be mathematically formalized into the language
of quantum theory. This is similar to the situation presented by the theory of
differential equations, which was initially developed for describing the
motion of planets. But now the theory of differential equations is employed
everywhere, being just an efficient mathematical tool not necessarily related
to planet motion. In the same way, quantum theory may provide a convenient
framework for the mathematical description of thinking processes. The critics,
however, insist that the analogies are superficial, do not prescribe practical
recipes, and sometimes even contradict empirical data qualitatively [45, 46].

(iii) Moreover, the simple logic teaches us that, if the brain is a classical
object, then its functioning should be described by classical equations, since
it is exactly its properties, including functioning, that classify an object
as classical or quantum. If the properties of an object cannot in principle be
described by a classical theory, but allow for only quantum description, then
this object is quantum, which contradicts our current knowledge on the brain.

(iv) The direct use of quantum theory for describing decision making
introduces a large number of unknown parameters and ambiguous notions that
cannot be characterized on the level of observable quantities associated with
decision making. For instance, what is a Hamiltonian in psychological
processes? How to define and measure numerous coefficients entering wave
functions describing the brain states? What is the evolution equation for
statistical operators characterizing the brain? And a lot of other ambiguously
defined notions appear [47].

(v) The most important goal of any theory is the ability of predicting
quantitative results that could be verified in experiment. However, none of
the purely quantum variants of decision making has ever predicted some
numerical data. The maximum what can be done is the consideration of
particular cases and fitting parameters for the assumed interpretation of
these cases. In order to extract quantitative information from the derived
quantum relations, it is necessary to complement them by a number of
assumptions not related to quantum techniques. In that sense the complicated
quantum substructure becomes excessive, similarly to the excessiveness of
nonlocal hidden variables for explaining quantum phenomena [48].

(vi) The fact that some events in decision making can qualitatively be
interpreted as due to quantum processes does not exclude the possibility of
other interpretations in terms of classical language. According to the Occam’s
razor principle, the simplest of competing theories has to be preferred to the
more complex, so that explanations of unknown phenomena should be sought first
in terms of known quantities. Hence quite complicated theories based on
quantum formulas are to be disregarded in favor of much simpler explanations
based on classical notions, provided these exist. Entities should not be
multiplied beyond necessity. The simplest theory is the best [49].

To understand whether the functioning of consciousness is described by quantum
or classical rules is important, since, depending on the involved formalism,
the operation of artificial intelligence has to be characterized in the same
language. Examining the above objections to the use of quantum techniques for
the formalization of decision making, it is possible to say the following:
First, although at the present time, the influence of quantum effects on the
functioning of the brain has not been convincingly argued, it cannot be
absolutely excluded. Second, even if actual quantum effects play no role in
the brain operation and consciousness does not need quantum description, the
investigation of the analogies between decision making and quantum processes
can enrich both of them suggesting their more profound comprehension. The
peculiarities of quantum phenomena, that are better conceived, can give hints
on the ways of characterizing consciousness functioning.

The point of view advocated in this review can be summarized as follows: The
brain is a classical object, hence its basic property, that is consciousness,
by definition, has to be classical. Otherwise it would be a meaninglessness to
say that a classical object has quantum properties. Nevertheless, there exists
a number of formal analogies in the description of quantum measurements and
decision making. These analogies need to be carefully investigated for two
reasons:

(i) Although being formal, the analogies between different phenomena very
often suggest concrete practical ways for describing these phenomena.

(ii) Borrowing some ideas from the nominal analogies between two different
approaches helps to compare these approaches and to choose the more efficient
and simple theory.

The formal analogy between quantum and conscious phenomena has been noticed
long time ago by von Neumann, who mentioned that the quantum theory of
measurements can be interpreted as decision theory [50]. This concept has been
developed by other researchers, for instance by Benioff [51, 52]. Thus quantum
measurement is analogous to decision making, hence the measurement of an
observable is similar to the choice of an alternative in decision making.
Accepting these analogies, we can go further. Keeping in mind that emotions
appear subconsciously during the process of decision making, they can be
associated with intrinsic noise produced by a measuring device during the
measurement procedure. In that way, the observable-noise duality is equivalent
to the cognition-emotion duality. In the same way as in physical measurements
the detection of signals can be either hindered by noise or the addition of
the appropriate amount of noise can boost a signal and hence facilitate its
detection [53, 54], in decision processes, emotions can either hinder or
facilitate decision making.

In quantum measurements, there can exist observable-noise entanglement, which
in decision making corresponds to correlations mimicking cognition-emotion
entanglement. If the intrinsic noise is presented as a superposition of
several modes, then there appears noise interference, hence there can arise
the emotion interference. In that way, it is possible to expect different
similarities between quantum measurements and decision making. So, even if
consciousness does not function exactly by the same rules as quantum
measurements, but anyway the many found similarities can provide useful hints
for formalizing the operation of decision procedures, hence for the creation
of artificial intelligence.

Concluding, in order to avoid confusion, it is necessary to stress what the
review is about and what are not the review aims. This is in no way a survey
of the general field of quantum techniques applied to the characterization of
consciousness, because of which thousands of articles on such applications are
not discussed, but only the main books are cited, where the vast number of
references can be found. Concentrating on the ideas and methods for emotion
quantification, citations are given only to those works where the role of
emotions in decision making is studied and especially where practical methods
of their description are discussed, but we do not plunge into the ocean of
papers where no these problems are touched upon. While in the majority of
works discussing the applications of quantum theory to consciousness neither
the role of emotions is considered, nor their quantification is touched at
all.

The very first requirement in the way of creating human-like artificial
intelligence is the formulation of explicit mathematical rules of its
operation. This paper does not pretend to describe all technical stages of
actual artificial intelligence functioning, but it aims at formulating
explicit mathematical algorithms for the operation of a human-like artificial
intelligence in the process of taking decisions. Without a mathematical
description of such rules and algorithms, no device can be modeled. But in
order to mathematically formulate the process of choice in human-like decision
making to be implemented by an artificial intelligence, it is compulsory to
understand and mathematically describe the process of choice by humans, whose
actions an artificial intelligence is planned to mimic. Therefore the pivotal
aim of the paper is to analyze the combination of the following problems,
whose solution is necessary for the mathematical formulation of decision
making by an intelligence, whether human-like artificial or human:

(1) The analysis of the role of emotions in decision making and survey of the
related literature, whether it employs quantum or classical language. This is
necessary for understanding the basic qualitative principles of Affective
Intelligence Processing

(2) The exposition of a practical way for emotion quantification in the
process of taking decisions. This is a prerequisite for the formation of an
Affective Artificial Intelligence requiring, for its functioning, the
existence of explicit quantitative algorithms.

(3) The comparison of two ways, quantum and classical, for the formulation of
the practical principles of Affective Decision Making. This is compulsory for
selecting the most appropriate method that would be self-consistent, simple,
and providing quantitative recipes for its operation.

(4) The comprehension of how the classical approach has to be modified in
order to provide the same practical results as with the use of quantum
techniques. Again, this is impossible to understand without a comparison of
both approaches, quantum and classical. Otherwise, the reader would constantly
exclaim: Why this or that assumption has been made? Where this or that formula
has appeared from?

These goals are realized in the review. An exhaustive survey of literature
discussing the role of emotions in decision making is given. The attempts of
emotion quantification are described being based on the available literature.
As is evident from numerous given citations, there are plenty of papers
discussing the role of emotions in classical terms. The detailed comparison of
quantum and classical techniques is given. It is shown that the classical
approach can be modified by taking account of emotions, in such a way that to
give the same results as in the language of quantum decision theory. For
example, all paradoxes of classical decision making can be quantitatively
explained without any use of quantum theory.

However, without comparing the two different approaches for taking into
account emotions, it would be impossible: First, to make a conclusion which of
them is preferable, and second, how it would be necessary to modify the
classical theory so that it would give the same results as the quantum
approach. Therefore all parts of the review are of equal importance and would
lose sense being separated. Thus it is impossible to justify one of the
approaches without comparing it with the other. On the other side, after
different approaches are formulated, they can be employed independently and
their effectiveness compared.

The layout of the review is as follows. In Sec. $2$, the general theory of
quantum measurements in the presence of intrinsic noise is introduced. The
analogies with decision making are emphasized. Assuming that the functioning
of noisy quantum measurements is similar to that of affective decision making
suggests the general framework for the latter. The comparison of a quantum and
a modified classical approaches does not merely provide interesting analogies,
but it allows for the formulation of the most simple and effective theory of
Affective Decision Making.

Quantum techniques, of course, are not a common knowledge and can strongly
hinder the use of quantum theory for practical applications. Therefore, if the
same phenomena can be described in quantum language and also in classical
terms, it is reasonable to resort to the simpler classical approach, but not
to play science specially complicating the consideration with fashionable
terminology. The theory has to be as simple as possible, in order that it
could be straightforwardly employed by anyone, including those who may not
know quantum techniques. This concerns decision theory as well, which can be
developed as a branch of quantum theory or can be reformulated into an
axiomatic form that, from one side mimics some quantum operations and
structures, but, from the other side, does not require the knowledge of
quantum terminology. Section 3 accomplishes this goal showing that the theory
of affective decision making can be formulated in an axiomatic way that does
not need to resort to quantum theory. Being formulated in mathematical terms,
affective decision theory can be implemented for the operation of artificial
intelligence. Section 4 considers the famous behavioural paradoxes in decision
making and shows that, on the aggregate level, these paradoxes do not arise in
the frame of the affective decision theory. In that sense, an artificial
intelligence, obeying the rules of this theory, will act as a typical human
decision maker. In Section 5, the structure of networks composed of
intelligent agents taking decisions in the presence of emotions is described.
Section 6 concludes.

##  2 Measurements under intrinsic noise

One of the main points advocated in the present review is the existence of an
analogy between human emotions in decision making and intrinsic noise in
quantum measurements. This obliges us to investigate the structure of quantum
probability for measurements under intrinsic noise in order to find out the
answers to two principal questions:

1\. Can this analogy be employed for developing affective decision theory that
could be sufficiently formalized for being useful for describing the operation
of human-level artificial intelligence?

2\. Whether this analogy is merely nominal or it goes deeper than that,
requiring the use of quantum techniques for adequately representing
behavioural decision making?

In physics, noise is modeled by introducing concrete noise terms into
Hamiltonians or evolution equations and prescribing the corresponding
distributions [53, 54, 55, 56]. Our aim here is not the analysis of concrete
models, but the study of the general structure of probabilities for quantum
events, decorated by the presence of intrinsic noise [57]. This is because we
wish to compare these probabilities with those arising in decision theory.
However the explicit nature of intrinsic noise, mimicking emotions, appearing
in the process of taking decisions, is not known. Thus only general structures
can be compared. In parallel with the physics terminology, we shall mention
decision-making analogies of the considered notions. The mentioned analogies
do not imply that the process of taking decisions by humans has to be
necessarily treated as a quantum procedure, but conversely, this rather means
that quantum measurements can be handled as formally similar to decision
making [51, 52, 58, 59, 60]. The most important analogy is between intrinsic
noise in quantum measurements and emotions in decision making [60, 61].

###  2.1 Quantum algebra of events

Let us consider quantum events $A_{n}$, enumerated with the index
$n=1,2,\ldots$. For concreteness, we consider a discrete index $n$, while in
general it could be continuous. Events can be the results of measurements for
observable quantities. In decision theory, an event can be the choice of a
particular alternative from the given set of alternatives. The collection of
quantum events forms a ring [62, 63],

$\mathbb{A}=\\{A_{n}:~{}n=1,2,\ldots\\}$ (2.1)

possessing two binary operations, addition and conjunction. Addition, or
union, or disjunction, implies that for any $A,B\in\mathbb{A}$ there is the
union $A\cup B\in\mathbb{A}$, meaning either $A$ or $B$ and enjoying the
properties

$A\bigcup B=B\bigcup A\qquad({\rm commutativity})\;,$
$A\;\bigcup\;\left(B\;\bigcup\;C\right)=\left(A\;\bigcup\;B\right)\;\bigcup\;C\qquad({\rm
associativity})\;,$ $A\;\bigcup\;A=A\qquad({\rm idempotency})\;.$

Conjunction, or multiplication, means that for any $A,B\in\mathbb{A}$ there
exists $A\cap B\in\mathbb{A}$ implying both $A$ and $B$ and having the
properties

$A\;\bigcap\;\left(B\;\bigcap\;C\right)=\left(A\;\bigcap\;B\right)\;\bigcap
C\;\qquad({\rm associativity})\;,$ $A\;\bigcap\;A=A\qquad({\rm
idempotency})\;.$

In general, conjunction is not commutative,

$A\;\bigcap\;B\neq B\;\bigcap\;A\qquad({\rm no\;commutativity})\;,$

and not distributive,

$A\;\bigcap\;\left(B\;\bigcup\;C\right)\neq\left(A\;\bigcap\;B\right)\;\bigcup\;\left(A\;\bigcap\;C\right)\qquad({\rm
no\;distributivity})\;.$

The ring $\mathbb{A}$ includes the identical event $1\in\mathbb{A}$, which is
an event that is identically true. For this event,

$A\;\bigcap\;1=1\;\bigcap\;A=A\;,\qquad A\;\bigcup\;1=1\;,\qquad
1\;\bigcup\;1=1\;.$

Also, there exists an impossible event $0\in\mathbb{A}$, which is identically
false, so that

$A\;\bigcap\;0=0\;\bigcap\;A=0\;,\qquad A\;\bigcup\;0=A\;,\qquad
0\;\bigcup\;1=1\;.$

The events for which $A\cap B=B\cap A=0$ are called disjoint or orthogonal.
Note that one often simplifies the above notation by denoting the addition as
$A\cup B\equiv A+B$ and the conjunction as $A\cap B\equiv AB$.

For each event $A\in\mathbb{A}$, there exists a complementary, or negating,
event $\overline{A}\in\mathbb{A}$, for which

$A\;\bigcup\;\overline{A}=1\;,\qquad
A\;\bigcap\;\overline{A}=\overline{A}\;\bigcap\;A=0\;,\qquad\overline{0}=1\;,\qquad\overline{1}=0\;.$

The absence of distributivity can be demonstrated by the simple example [62].
Consider two events $B_{1}$ and $B_{2}$ whose union forms unity, $B_{1}\cup
B_{2}=1$. And assume that both $B_{1}$ and $B_{2}$ are orthogonal to a non-
trivial event $A\neq 0$, so that $A\cap B_{1}=A\cap B_{2}=0$. By this
definition, $A\cap(B_{1}\cup B_{2})=A\cap 1=A$. If the property of
distributivity were true, then it would be $(A\cap B_{1})\cup(A\cap B_{2})=0$.
Since, by assumption, $A\neq 0$, the property of distributivity does not hold.

The concept of non-distributivity in quantum physics can be illustrated by the
example of spin measurements [63]. Let the spin projection of a particle with
spin 1/2 be measured. Suppose $B_{1}$ is the event of measuring the spin in
the up state with respect to the $z-$axis, whereas $B_{2}$ is the event of
measuring the spin in the down state along this axis. The spin can be either
up or down, hence $B_{1}\cup B_{2}=1$. Assume that $A$ is the event of
measuring the spin along an axis in the plane orthogonal to the $z-$axis, say
along the $x-$axis. Since the spin cannot be measured simultaneously along two
orthogonal axes, it is found either measured along one axis or along another
axis, but cannot have components on both axes simultaneously. Hence, $A\cap
B_{1}=A\cap B_{2}=0$. At the same time, $A\cap(B_{1}\cup B_{2})=A\neq 0$.
Therefore, there is no distributivity of events in the spin measurement.

###  2.2 Operationally testable events

An event is termed operationally testable, when it can be quantified by means
of measurements. In physics, one measures observable quantities. For example,
in quantum physics, one can measure the eigenvalues of a Hermitian operator
corresponding to an observable. In decision making, one makes decisions by
choosing preferable alternatives from the given set. Quantum measurements can
be treated as a kind of decision making [50, 51, 52, 58, 64, 65].

Let us consider a set of alternatives (2.1) representing, e.g., a set of
eigenvalues of an operator in quantum physics or a set of alternatives in
decision theory. In quantum theory, each $A_{n}$ can be put into
correspondence to a vector (state of an alternative) $|A_{n}\rangle$ in a
Hilbert space $\mathcal{H}_{A}$. For simplicity, we keep in mind nondegenerate
spectra of Hermitian operators. The vectors $|A_{n}\rangle$ can be
orthonormalized,

$\langle\;A_{m}\;|\;A_{n}\;\rangle=\delta_{mn}\;.$ (2.2)

Here and in what follows, the Dirac bracket notation [66] is employed. The
Hilbert space $\mathcal{H}_{A}$ can be defined as the closed linear envelope

${\cal H}_{A}={\rm span}\;\\{|\;A_{n}\;\rangle\\}\;.$ (2.3)

In quantum decision theory, this is called the space of alternatives.

Each alternative $A_{n}$ is represented by a projection operator

$\hat{P}(A_{n})=|\;A_{n}\;\rangle\langle\;A_{n}\;|$ (2.4)

enjoying the property

$\hat{P}(A_{m})\;\hat{P}(A_{n})=\delta_{mn}\;\hat{P}(A_{n})\;.$

The latter means that the projection operators are idempotent and the
alternatives of the ring $\mathbb{A}$ are mutually incompatible. The
projection operators satisfy the resolution of unity

$\sum_{n}\hat{P}(A_{n})=\hat{1}_{A}\;,$ (2.5)

where $\hat{1}_{A}$ is the identity operator in ${\cal H}_{A}$. The complete
family of the projection operators forms the projection-valued operator
measure on ${\cal H}_{A}$ with respect to set (2.1).

In quantum physics there can happen degenerate spectra, when for an $A_{n}$
there correspond several vectors $|A_{n_{i}}\rangle$, with $i=1,2,\ldots$.
Then the projection operator associated with $A_{n}$ is

$\hat{P}(A_{n})=\sum_{i}\hat{P}(A_{n_{i}})\;,\qquad\hat{P}(A_{n_{i}})\equiv|\;A_{n_{i}}\;\rangle\langle\;A_{n_{i}}\;|\;.$

If one wishes to avoid the problem of degeneracy, one slightly modifies the
considered system by introducing infinitesimal terms lifting the degeneracy
connected with some kind of symmetry, as has been mentioned by von Neumann
[50] and elaborated by Bogolubov [67, 68, 69]. Similarly, in decision theory,
the problem of degeneracy can be easily avoided by reclassifying the
alternatives under consideration [70, 71]. Thus for decision theory, it is
sufficient to consider the situation with no degeneracy.

###  2.3 Modes of intrinsic noise

Any measurement is accompanied by some kind of noise that can be of two types,
extrinsic and intrinsic [53, 54, 55, 56, 72]. Here we are interested in
intrinsic noise that is generated by the measurement device in the process of
measurement. Because of the intrinsic noise, what is measured is not a pure
result for an alternative, but a combination of the data related to the
testable event of interest and the influence of noise. The intrinsic noise
also can be called instrumental or self-induced. In decision theory, the
analogy of the intrinsic noise is the collection of emotions arising in the
process of decision making, of subconscious allusions, intuitive guesses, gut
feelings, and like that [57, 73].

Suppose the intrinsic noise is characterized by a set of elementary modes

$\mathbb{E}=\\{e_{\mu}:~{}\mu=1,2,\ldots\\}\;.$ (2.6)

In decision theory, this would be a family of different emotions. Each noise
mode $e_{\mu}$ is put into correspondence with a vector (noise state)
$|e_{\mu}\rangle$ of a Hilbert space of noise $\mathcal{H}_{E}$ that can be
represented as the closed linear envelope

${\cal H}_{E}={\rm span}\;\\{|\;e_{\mu}\;\rangle\\}\;.$ (2.7)

The vectors of elementary nodes are assumed to be orthonormalized,

$\langle\;e_{\mu}\;|\;e_{\nu}\;\rangle=\delta_{\mu\nu}\;.$ (2.8)

In quantum decision theory, space (2.7) is called the emotion space. Emotion
modes represent different types of emotions, such as joy, sadness, anger,
fear, disgust, trust, etc.

The projection operator for a mode $e_{\mu}$ is

$\hat{P}(e_{\mu})=|\;e_{\mu}\;\rangle\langle\;e_{\mu}\;|\;,$ (2.9)

which is idempotent and orthogonal to the projectors of other modes,

$\hat{P}(e_{\mu})\hat{P}(e_{\nu})=\delta_{\mu\nu}\;\hat{P}(e_{\mu})\;.$ (2.10)

The family of these projectors is complete satisfying the resolution of unity

$\sum_{\mu}\hat{P}(e_{\mu})=\hat{1}_{E}\;,$ (2.11)

where $\hat{1}_{E}$ is the unity operator in ${\cal H}_{E}$. Thus the family
of projectors (2.9) forms the projection-valued operator measure with respect
to the set (2.6).

The measurement of an alternative $A_{n}$ generates the intrinsic noise
$z_{n}$ represented by the vector

$|\;z_{n}\;\rangle=\sum_{\mu}a_{n\mu}\;|\;e_{\mu}\;\rangle\;.$ (2.12)

In decision making, this corresponds to the collection of emotions arising
under the choice between alternatives. The noise vector (2.12) can be
normalized,

$\langle\;z_{n}\;|\;z_{n}\;\rangle=\sum_{\mu}|\;a_{n\mu}\;|^{2}=1\;,$ (2.13)

although the noise vectors generated by different measurements are not
necessarily mutually orthogonal, so that the product

$\langle\;z_{m}\;|\;z_{n}\;\rangle=\sum_{\mu}a^{*}_{m\mu}a_{n\mu}$ (2.14)

is not obligatory a Kronecker delta. Equivalently, the collections of emotions
generated under the choice of different alternatives do not need to be
mutually exclusive.

Strictly speaking, emotions are contextual and are subject to temporal
variations, which means that the coefficients $a_{n\mu}$, generally, can vary
with time, depending on the state of a decision maker and the corresponding
surrounding.

The noise projectors

$\hat{P}(z_{n})=|\;z_{n}\;\rangle\langle\;z_{n}\;|$ (2.15)

are idempotent,

$\hat{P}^{2}(z_{n})=\hat{P}(z_{n})\;,$ (2.16)

however, generally, are not mutually orthogonal,

$\hat{P}(z_{m})\hat{P}(z_{n})=(\;\langle\;z_{m}\;|\;z_{n}\;\rangle\;)\;|\;z_{m}\;\rangle\langle\;z_{n}\;|$
(2.17)

because of property (2.14).

Note the important difference between projectors (2.9) and (2.15). The family
of projectors (2.9) is complete with respect to the set (2.6) due to the
resolution of unity (2.11). However, the set of projectors (2.15) is not
complete with respect to the set

$\mathbb{Z}=\\{z_{n}:~{}n=1,2,\ldots\\}\;,$ (2.18)

since the sum

$\sum_{n}\hat{P}(z_{n})=\sum_{n}\;\sum_{\mu\nu}a_{n\mu}a^{*}_{n\nu}|\;e_{\mu}\;\rangle\langle\;e_{\nu}\;|$
(2.19)

is not a unity operator in $\mathcal{H}_{E}$. The latter is clear from the
equality

$\langle\;e_{\mu}\;|\;\sum_{n}\hat{P}(z_{n})\;|\;e_{\nu}\;\rangle=\sum_{n}a^{*}_{n\mu}a_{n\nu}\;,$

which is not a Kronecker delta.

###  2.4 Noise-decorated alternatives

When a measurement of an alternative $A_{n}$ is accompanied by inevitable
intrinsic noise $z_{n}$, what is actually observed is not a pure event $A_{n}$
but this event decorated with the noise, that is the combined event
$A_{n}z_{n}$, whose representation is given by the vector

$|\;A_{n}z_{n}\;\rangle=|\;A_{n}\;\rangle\;\bigotimes\;|\;z_{n}\;\rangle=\sum_{\mu}a_{n\mu}\;|\;A_{n}e_{\mu}\;\rangle$
(2.20)

in the Hilbert space

${\cal H}={\cal H}_{A}\;\bigotimes\;{\cal H}_{E}={\rm
span}\;\\{\;|\;A_{n}e_{\mu}\;\rangle\;\\}.$ (2.21)

The vectors defined in equation (2.20) are mutually orthogonal,

$\langle\;z_{m}A_{m}\;|\;A_{n}z_{n}\;\rangle=\delta_{mn}\;.$ (2.22)

The set of the noise-decorated events

$\mathbb{A}_{Z}=\\{A_{n}z_{n}:~{}n=1,2,\ldots\\}$ (2.23)

is characterized by the family of the projectors

$\hat{P}(A_{n}z_{n})=|\;A_{n}z_{n}\;\rangle\langle\;z_{n}A_{n}\;|=\hat{P}(A_{n})\;\bigotimes\;\hat{P}(z_{n})$
(2.24)

that also can be written as

$\hat{P}(A_{n}z_{n})=\sum_{\mu\nu}a_{n\mu}a^{*}_{n\nu}\;|\;A_{n}e_{\mu}\;\rangle\langle\;e_{\nu}A_{n}\;|\;.$
(2.25)

These projectors are idempotent and mutually orthogonal,

$\hat{P}(A_{m}z_{m})\hat{P}(A_{n}z_{n})=\delta_{mn}\;\hat{P}(A_{n}z_{n}).$
(2.26)

However, since the vectors (2.20) do not form a basis in space (2.21), the
projectors(2.24) do not sum to one,

$\sum_{n}\hat{P}(A_{n}z_{n})=\sum_{n}\;\sum_{\mu\nu}a_{n\mu}a^{*}_{n\nu}\;|\;A_{n}e_{\mu}\;\rangle\langle\;e_{\nu}A_{n}\;|\;,$
(2.27)

which is seen from the equality

$\langle\;e_{\mu}A_{m}\;|\;\sum_{k}\hat{P}(A_{k}z_{k})\;|\;A_{n}e_{\nu}\;\rangle=\delta_{mn}\;a^{*}_{n\mu}a_{n\nu}\;.$

Thus the family of projectors (2.24) is idempotent, orthogonal, but not
complete, hence does not compose a standard operator-valued measure and
requires some additional conditions for introducing the probability of
alternatives [73, 74].

###  2.5 Quantum probability space

Statistical properties of the considered system are characterized by a
statistical operator $\hat{\rho}$ that depends on the context and the
observer’s knowledge about the state of the system, because of which it is
also called the state-of-knowledge operator. The operator $\hat{\rho}$ is a
positive-semidefinite trace-one operator. It is also called the system state,
or often simply a state.

The general representation of $\hat{\rho}$ in the basis
$\\{|A_{n}e_{\mu}\rangle\\}$ of orthonormalized vectors, that are not
necessarily eigenvectors of $\hat{\rho}$, has the form

$\hat{\rho}=\sum_{mn}\;\sum_{\mu\nu}\rho_{mn}^{\mu\nu}\;|\;A_{m}e_{\mu}\;\rangle\langle\;e_{\nu}A_{n}\;|\;,$
(2.28)

with

$\rho_{mn}^{\mu\nu}\equiv\langle\;e_{\mu}A_{m}\;|\;\hat{\rho}\;|\;A_{n}e_{\nu}\;\rangle\;.$
(2.29)

The trace normalization condition can be written as

${\rm Tr}_{\cal H}\hat{\rho}=\sum_{n\mu}\rho_{nn}^{\mu\mu}=1\;.$ (2.30)

A positive-semidefinite operator on a complex Hilbert space is necessarily
self-adjoint [75], which imposes the constraint

$\left(\rho_{mn}^{\mu\nu}\right)^{*}=\rho_{nm}^{\nu\mu}\qquad(\hat{\rho}^{+}=\hat{\rho})\;.$
(2.31)

Also, let us require that the family of projectors (2.24) be complete on
average, so that

${\rm Tr}_{\cal
H}\hat{\rho}\;\left[\;\sum_{n}\hat{P}(A_{n}z_{n})\;\right]=1\;,$ (2.32)

which acquires the form

$\sum_{n}\;\sum_{\mu\nu}a_{n\mu}^{*}a_{n\nu}\rho^{\mu\nu}_{nn}=1\;,$ (2.33)

with the trace over the space (2.21).

To be self-consistent, the system of constraints should not be overdefined.
This means that the number of the involved parameters cannot be smaller than
number of the constraint equations. The vectors $|z_{n}\rangle$ include the
complex coefficients $a_{n\mu}$ containing $2d_{A}d_{E}$ real components,
where

$d_{A}\equiv{\rm dim}\;{\cal H}_{A}\;,\qquad d_{E}\equiv{\rm dim}\;{\cal
H}_{E}\;.$ (2.34)

The statistical operator $\hat{\rho}$ comprises the coefficients
$\rho_{mn}^{\mu\nu}$ containing $d_{A}d_{E}$ real diagonal elements and
$d_{A}^{2}d_{E}^{2}-d_{A}d_{E}$ complex off-diagonal elements, hence in total
$2d_{A}^{2}d_{E}^{2}-d_{A}d_{E}$ real components. Thus the total number of
real components in $\hat{\rho}$ is $2d_{A}^{2}d_{E}^{2}-d_{A}d_{E}$.
Conditions (2.31) impose $d_{A}^{2}d_{E}^{2}-d_{A}d_{E}$ restrictions. In
addition, there are two normalization conditions (2.30) and (2.33). In this
way, there are in total $2d_{A}^{2}d_{E}^{2}+d_{A}d_{E}$ real parameters and
$d_{A}^{2}d_{E}^{2}-d_{A}d_{E}+2$ equations. From here, the condition of self-
consistency becomes

$d_{A}^{2}d_{E}^{2}+2d_{A}d_{E}\geq 2\;,$ (2.35)

which holds for any $d_{A}d_{E}\geq 1$.

The pair $\\{{\cal H},\hat{\rho}\\}$ is called quantum statistical ensemble
[76]. Adding here the family

${\cal P}_{AZ}=\\{\hat{P}(A_{n}z_{n}):~{}n=1,2,\ldots\\}$ (2.36)

of projectors (2.24) gives the quantum probability space

$\\{{\cal H},\;\hat{\rho},\;{\cal P}_{AZ}\\}\;.$

The probability of observing an event $A_{n}z_{n}$ reads as

$p(A_{n}z_{n})={\rm Tr}_{\cal H}\;\hat{\rho}\;\hat{P}(A_{n}z_{n})\;.$ (2.37)

The normalization condition (2.32) guarantees the validity of the
normalization condition

$\sum_{n}p(A_{n}z_{n})=1\;,\qquad 0\leq p(A_{n}z_{n})\leq 1\;.$ (2.38)

Explicitly, equation (2.37) takes the form

$p(A_{n}z_{n})=\sum_{\mu\nu}a^{*}_{n\mu}a_{n\nu}\rho^{\mu\nu}_{nn}\;.$ (2.39)

The form (2.37) defines the quantum probability of observing an alternative
$A_{n}$ decorated by intrinsic noise.

###  2.6 Quantum-classical correspondence

Quantum theory reduces to classical under the effect of decoherence [77, 78].
Then quantum decision theory reduces to classical and the quantum probability
(2.37) reduces to classical probability [70, 79] .

In expression (2.39), it is possible to separate the diagonal part

$f(A_{n}z_{n})=\sum_{\mu}|\;a_{n\mu}\;|^{2}\;\rho^{\mu\mu}_{nn}$ (2.40)

and the remaining off-diagonal part

$q(A_{n}z_{n})=\sum_{\mu\neq\nu}a^{*}_{n\mu}a_{n\nu}\;\rho^{\mu\nu}_{nn}\;.$
(2.41)

Then probability (2.37) becomes the sum

$p(A_{n}z_{n})=f(A_{n}z_{n})+q(A_{n}z_{n})\;.$ (2.42)

The first term here is semi-positive (non-negative), while the second one is
not sign-defined. The term $q(A_{n}z_{n})$ is due to the interference of noise
modes and is zero if there is just a single mode or when $\rho_{nn}^{\mu\nu}$
is diagonal in the upper indices. This is why it can be called quantum
interference term or quantum coherence term. In the present case, it is caused
by the noise interference.

The disappearance of the quantum coherence term is named decoherence, when
there occurs the reduction of quantum probability to the classical form [77,
78, 79] associated with expression (2.40). Interpreting the latter as
classical probability implies the validity of the conditions

$\sum_{n}f(A_{n}z_{n})=1\;,\qquad 0\leq f(A_{n}z_{n})\leq 1\;.$ (2.43)

Therefore, in view of conditions (2.38) and (2.43), the interference term
satisfies the conditions

$\sum_{n}q(A_{n}z_{n})=0\;,\qquad-1\leq q(A_{n}z_{n})\leq 1\;.$ (2.44)

More precisely, it fulfills the inequality

$-f(A_{n}z_{n})\leq q(A_{n}z_{n})\leq 1-f(A_{n}z_{n})\;.$ (2.45)

In decision theory, the first equation in (2.44) is called the alternation law
[80, 81].

The quantum-classical correspondence can be formulated as the reduction of
quantum probability to classical under decoherence, when

$p(A_{n}z_{n})\mapsto f(A_{n}z_{n})\;,\qquad q(A_{n}z_{n})\mapsto 0\;.$ (2.46)

Thus the appearance of an additional term $q(A_{n}z_{n})$ is due to the
interference of noise modes. The phenomenon of mode interference is well known
in quantum physics [31, 82, 83, 84, 85, 86]. The absence of intrinsic noise
accompanying measurements corresponds to the absence of emotions in the choice
between alternatives.

###  2.7 Probability of superposition states

For the purpose of quantum information processing [87, 88, 89, 90, 91, 92,
93], one creates quantum states in the form of superpositions. Then it is
admissible to define the probability of observing these states. For
illustration, let us consider the binary combinations of states

$A_{mn}z_{mn}=A_{m}z_{m}\;\bigcup\;A_{n}z_{n}\qquad(m\neq n)\;.$ (2.47)

Following the general procedure, each member (2.47) can be put into
correspondence to the vector

$|\;A_{mn}z_{mn}\;\rangle=c_{m}\;|\;A_{m}z_{m}\;\rangle+c_{n}\;|\;A_{n}z_{n}\;\rangle$
(2.48)

and characterized by the projector

$\hat{P}(A_{mn}z_{mn})=|\;A_{mn}z_{mn}\;\rangle\langle\;z_{mn}A_{mn}\;|\;.$
(2.49)

Vector (2.48) is assumed to be normalized to one, which requires the condition

$|\;c_{m}\;|^{2}+|\;c_{n}\;|^{2}=1\;.$

It is worth emphasizing that this type of composite states could be introduced
by combining the members from two different sets, say $\\{A_{n}\\}$ and
$\\{B_{k}\\}$ such that their related vectors $|A_{n}\rangle$ and
$|B_{k}\rangle$ pertain to the same basis in the Hilbert space, which is
required by the necessity of defining the vector (2.48) as a superposition
with respect to a common basis. In the physics language, this means that, if
$A_{n}$ and $B_{k}$ are the eigenvalues of some operators $\hat{A}$ and
$\hat{B}$, then these operators have to commute with each other, since only
then they enjoy the common family of orthonormalized eigenvectors.
Noncommuting operators cannot form such linear combinations. In that sense,
the corresponding events and observables are called incompatible as far as
they cannot be measured simultaneously [50, 86].

The projector (2.49) reads as

$\hat{P}(A_{mn}z_{mn})=|\;c_{m}\;|^{2}\hat{P}(A_{m}z_{m})+|\;c_{n}\;|^{2}\hat{P}(A_{n}z_{n})\;+$
$+\;c_{m}c^{*}_{n}\;|\;A_{m}z_{m}\;\rangle\langle\;z_{n}A_{n}\;|+c_{n}c^{*}_{m}\;|\;A_{n}z_{n}\;\rangle\langle\;z_{m}A_{m}\;|\;.$
(2.50)

Then the probability of observing the composite state, corresponding to
$A_{mn}z_{mn}$, becomes

$p(A_{mn}z_{mn})=|\;c_{m}\;|^{2}p(A_{m}z_{m})+|\;c_{n}\;|^{2}p(A_{n}z_{n})\;+$
$+\;2{\rm
Re}\;\left(c^{*}_{m}c_{n}\sum_{\mu\nu}a^{*}_{m\mu}a_{n\nu}\;\rho^{\mu\nu}_{mn}\right)\qquad(m\neq
n)\;.$ (2.51)

This expression includes the terms due to the interference of noise modes as
well as to the interference of the alternatives $A_{m}$ and $A_{n}$. Even when
there is just a single noise mode, hence there is no noise interference, there
remains the interference of alternatives. Thus in the case of a single noise
mode $e_{0}$, when

$a_{m\mu}=\delta_{\mu 0}\qquad(z_{n}\mapsto e_{0})\;,$ (2.52)

so that the noise interference disappears, probability (2.51) reduces to the
form

$p(A_{mn}z_{mn})\mapsto|\;c_{m}\;|^{2}p(A_{m}e_{0})+|\;c_{n}\;|^{2}p(A_{n}e_{0})+2{\rm
Re}\;\left(c^{*}_{m}c_{n}\;\rho_{mn}\right)\qquad(m\neq n)\;,$ (2.53)

where

$\rho_{mn}\;\equiv\;\langle\;e_{0}A_{m}\;|\;\hat{\rho}\;|\;A_{n}e_{0}\;\rangle\;.$

The last term in probability (2.53) describes the interference of
alternatives.

It is natural to ask whether the linear combinations of alternatives, hence
the alternative interference, exist in human decision making, similarly to
quantum physics. In the latter, the superpositions of wave functions
representing quantum states do exist. However this quantum notion does not
exist in human decision making. For instance, we can consider a set of fruits
and vegetables deciding what to buy, an apple or a banana. However the seller
will be quite astonished if we ask him/her to give us a quantum superposition
of a banana and an apple. It looks that quantum superpositions of alternatives
do not exist in the real life outside quantum experiments.

Note that in many works of quantum cognition, one considers exactly the
interference of alternatives, but not the interference of emotions (intrinsic
noise). This is the principal difference between our approach and the works of
other authors. In our approach, the operationally testable events, that is the
observed alternatives, do not interfere. These are emotions that can
interfere.

###  2.8 Alternative-noise entanglement

The notion of entanglement plays an important role in quantum information
processing and quantum computing [87, 88, 89, 90, 91, 92, 93]. Entanglement
happens when the considered Hilbert space is represented as a tensor product
of several Hilbert spaces. In our case, the Hilbert space (2.21) is the
product of the spaces characterizing the alternatives and intrinsic noise.
Therefore there may exist entanglement between alternatives and noise.
Generally, the statistical operator (2.28) is entangled. The state that is not
entangled is termed separable. In the present case, the state would be
separable if it would have the form

$\hat{\rho}_{sep}=\sum_{i}\lambda_{i}\;\hat{\rho}^{i}_{A}\bigotimes\hat{\rho}^{i}_{E}\;,$
(2.54)

where the first factor in (2.54) is a state acting on the space ${\cal
H}_{A}$, while the second is a state acting on the space $\mathcal{H}_{E}$,
and

$\sum_{i}\lambda_{i}=1\;,\qquad 0\leq\lambda_{i}\leq 1\;.$

The factor states can be represented as

$\hat{\rho}^{i}_{A}=\sum_{mn}\rho^{i}_{mn}|\;A_{m}\;\rangle\langle\;A_{n}\;|\;,\qquad\hat{\rho}^{i}_{E}=\sum_{\mu\nu}\rho_{i}^{\mu\nu}|\;e_{\mu}\;\rangle\langle\;e_{\nu}\;|\;,$
(2.55)

with the normalization

$\sum_{n}\rho_{nn}^{i}=1\;,\qquad\sum_{\mu}\rho^{\mu\mu}_{i}=1\;.$

Then the state (2.54) reads as

$\hat{\rho}_{sep}=\sum_{i}\lambda_{i}\;\sum_{mn}\;\sum_{\mu\nu}\rho^{i}_{mn}\;\rho_{i}^{\mu\nu}\;|\;A_{m}e_{\mu}\;\rangle\langle\;e_{\nu}A_{n}\;|\;.$
(2.56)

This means that in the representation (2.28), the coefficient is

$\rho_{mn}^{\mu\nu}=\sum_{i}\lambda_{i}\;\rho^{i}_{mn}\;\rho_{i}^{\mu\nu}\;.$
(2.57)

For the separable state (2.54), the classical limit (2.40) becomes

$f(A_{n}z_{n})=\sum_{i}\lambda_{i}\;\sum_{\mu}|\;a_{n\mu}\;|^{2}\;\rho^{i}_{nn}\;\rho_{i}^{\mu\mu}$
(2.58)

and the quantum noise interference term is

$q(A_{n}z_{n})=\sum_{i}\lambda_{i}\;\sum_{\mu\neq\nu}a^{*}_{n\mu}\;a_{n\nu}\;\rho^{i}_{nn}\;\rho_{i}^{\mu\nu}\;.$
(2.59)

Thus, generally, the alternatives and noise are entangled with each other. The
noise interference is not related to whether the state is entangled or not.
The state can be separable, while the noise interference present. In quantum
decision theory, the alternative-noise entanglement is equivalent to the
entanglement of alternatives and emotions [94, 95].

###  2.9 Entanglement production by measurements

It is necessary to keep in mind that, for operators, there exist two different
notions, state entanglement and operator entanglement production. An entangled
state is a state that is not separable, as is explained above. In that sense,
entanglement is the property of the state structure. While entanglement
production by an operator is the ability of generating entangled functions
from disentangled ones.

A vector of a tensor-product Hilbert space is named disentangled if it can be
represented as a tensor product of vectors pertaining to the factor Hilbert
spaces. Just as an example, the basis vector
$|A_{n}e_{\mu}\rangle=|A_{n}\rangle\otimes|e_{\mu}\rangle$ is disentangled.
Disentangled vectors are often called separable vectors.

An operator is called entangling, when there exists at least one disentangled
vector such that it becomes entangled under the action of this operator.
Conversely, one says that an operator is not entangling if its action on any
disentangled vector yields again a disentangled one. It has been proved [96,
97, 98] that the only operators preserving vector separability are the
operators having the form of tensor products of local operators and a swap
operator permuting Hilbert spaces in the tensor product describing the total
Hilbert space of a composite system. The action of the swap operator is
trivial, in the sense that it merely permutes the indices labeling the spaces.
The vector separability preservation by product operators has been proved for
binary [96, 99, 100] as well as for multipartite vectors [97, 98, 101]. The
operators preserving vector separability are called nonentangling [102, 103].
While an operator transforming at least one disentangled vector into an
entangled vector is termed entangling [104, 105]. The strongest type of an
entangling operator is a universal entangler that makes all disentangled
vectors entangled [106].

Entanglement of vectors can be generated in the process of measurements by the
action of statistical operators. The measure of entanglement production for
arbitrary operators has been introduced in Refs. [107, 108]. This measure is
applicable to any system, whether bipartite or multipartite, and to any trace-
class operator. It has been applied for characterizing different physical
systems [109, 110, 111, 112, 113], as reviewed in Ref. [114]. Entanglement
production generated in the process of decision making is studied in Ref.
[115].

The measure of entanglement production by the statistical operator (2.28)
acting on the Hilbert space (2.21) is calculated as follows. We define the
partially traced operators

$\hat{\rho}_{A}\equiv{\rm
Tr}_{E}\;\hat{\rho}=\sum_{mn}\rho_{mn}|\;A_{m}\;\rangle\langle\;A_{n}\;|\;,\qquad\hat{\rho}_{E}\equiv{\rm
Tr}_{A}\;\hat{\rho}=\sum_{\mu\nu}\rho^{\mu\nu}|\;e_{\mu}\;\rangle\langle\;e_{\nu}\;|\;,$
(2.60)

in which the traces are over $\mathcal{H}_{E}$ or $\mathcal{H}_{A}$,
respectively, and

$\rho_{mn}\equiv\sum_{\mu}\rho^{\mu\mu}_{mn}\;,\qquad\rho^{\mu\nu}\equiv\sum_{n}\rho^{\mu\nu}_{nn}\;.$

The nonentangling state is given by the product

$\hat{\rho}^{\otimes}\equiv\hat{\rho}_{A}\;\bigotimes\;\hat{\rho}_{E}\;.$
(2.61)

Comparing the action of the statistical operator $\hat{\rho}$ with that of its
nonentangling counterpart (2.61) we have the measure of entanglement
production by the statistical operator

$\varepsilon(\hat{\rho})\equiv\log\;\frac{||\;\hat{\rho}\;||}{||\;\hat{\rho}^{\otimes}\;||}\;.$
(2.62)

Here

$||\;\hat{\rho}^{\otimes}\;||=||\;\hat{\rho}_{A}\;||\cdot||\;\hat{\rho}_{E}\;||.$

Keeping in mind the spectral norm yields

$||\;\hat{\rho}\;||=\sup_{n\mu}\rho^{\mu\mu}_{nn}$ (2.63)

and

$||\;\hat{\rho}_{A}\;||=\sup_{n}\;\sum_{\mu}\rho^{\mu\mu}_{nn}\;,\qquad||\;\hat{\rho}_{E}\;||=\sup_{\mu}\;\sum_{n}\rho^{\mu\mu}_{nn}\;.$
(2.64)

Therefore the measure of entanglement production (2.62) turns into

$\varepsilon(\hat{\rho})=\log\;\frac{\sup_{n\mu}\rho^{\mu\mu}_{nn}}{(\sup_{n}\sum_{\mu}\rho^{\mu\mu}_{nn})(\sup_{\mu}\sum_{n}\rho^{\mu\mu}_{nn})}\;.$
(2.65)

In this way, the statistical operator produces the alternative-noise
entanglement by acting on the vectors of the Hilbert space. As an
illustration, it is easy to show that even a separable state can produce
entanglement. Thus, acting by the separable state (2.54) on the disentangled
basis vector $|A_{n}e_{\mu}\rangle$ gives the vector

$\hat{\rho}_{sep}\;|\;A_{n}e_{\mu}\;\rangle=\sum_{m\nu}\;\rho^{\mu\nu}_{mn}\;|\;A_{m}e_{\nu}\;\rangle\;,$
(2.66)

where $\rho_{mn}^{\mu\nu}$ is given by (2.57). This vector is entangled if at
least two $\lambda_{i}$ are not zero. Similarly, in the process of making
decisions the alternatives and emotions become entangled.

###  2.10 Time dependence of probability

In the previous sections, the measurements as well as decision making have
been treated as occurring during so short time that it could be neglected.
However, these processes do require some finite time. In addition, one can
accomplish measurements or decisions at different moments of time. Therefore,
for the correct description of measurements, as well as decision making, it is
necessary to take account of time dependence of quantum probabilities.

Time enters the probability through the time dependence of statistical
operator $\hat{\rho}(t)$, whose time evolution is given by means of a unitary
evolution operator $\hat{U}(t)$ according to the rule

$\hat{\rho}(t)=\hat{U}(t,0)\;\hat{\rho}(0)\;\hat{U}^{+}(t,0)\;.$ (2.67)

Alternatives decorated by intrinsic noise are represented by the family (2.36)
of projectors (2.24) acting on the Hilbert space $\mathcal{H}$ defined in
(2.21). The quantum probability space reads as

$\\{{\cal H},\;\hat{\rho}(t),\;{\cal P}_{AZ}\\}\;,$ (2.68)

with the probability becoming dependent on time,

$p(A_{n}z_{n},t)={\rm Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n}z_{n})\;.$ (2.69)

Here and in what follows, the trace operation, without a notation of the
related Hilbert space, is assumed to be over the whole space (2.21).

As early, the probability is represented as the sum

$p(A_{n}z_{n},t)=f(A_{n}z_{n},t)+q(A_{n}z_{n},t)$ (2.70)

of the classical limit

$f(A_{n}z_{n},t)=\sum_{\mu}|\;a_{n\mu}\;|^{2}\;\rho^{\mu\mu}_{nn}(t)$ (2.71)

and the quantum term caused by the noise interference

$q(A_{n}z_{n},t)=\sum_{\mu\neq\nu}a^{*}_{n\mu}a_{n\nu}\;\rho^{\mu\nu}_{nn}(t)\;.$
(2.72)

The dependence on time comes from the time dependence of the matrix elements
(2.29). The noise vector $|z_{n}\rangle$ can also depend on time through the
coefficients $a_{n\mu}$, which we do not stress explicitly for the sake of
notation compactness. The dependence of noise on time, as well as the time-
dependence of emotion properties is rather natural, since they can vary with
time.

Because of the unitarity of the evolution operator, the normalization
condition (2.38) remains true:

$\sum_{n}p(A_{n}z_{n},t)=1\;,\qquad 0\leq p(A_{n}z_{n},t)\leq 1\;.$ (2.73)

Similarly, the normalization conditions (2.43), (2.44), and (2.45) also remain
valid.

###  2.11 Quantum state reduction

Suppose, the system, at time $t=0$, is prepared in a state $\hat{\rho}(0)$ and
develops in time following the evolution equation (2.67). Then at time $t_{0}$
it is subject to a measurement procedure for an observable allowing for the
set of alternatives (2.1). In the same way, one could be talking about taking
a decision at time $t_{0}$ by choosing an alternative from the set of possible
alternatives.

At time $t_{0}-0$, just before the measurement, the a priori probabilities of
the alternatives are given by the equation

$p(A_{n}z_{n},t_{0}-0)={\rm Tr}\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n}z_{n})\;.$
(2.74)

Let us assume that at the moment of time $t_{0}$ an alternative $A_{n}$ is
certainly observed. In decision making, this would imply that an alternative
$A_{n}$ is certainly chosen. In any case, this means that, as a result of the
interaction between the studied system and a measuring device, the a priori
state has been reduced to an a posteriori state,

$\hat{\rho}(t_{0}-0)\mapsto\hat{\rho}(A_{n},t_{0}+0)\;,$ (2.75)

so that the a posteriori probability

$p(A_{n}z_{n},t_{0}+0)={\rm
Tr}\;\hat{\rho}(A_{n},t_{0}+0)\;\hat{P}(A_{n}z_{n})$ (2.76)

becomes unity, thus describing a certain event,

$p(A_{n}z_{n},t_{0}+0)=1\;.$ (2.77)

The above condition in the explicit form reads as

${\rm Tr}\;\hat{\rho}(A_{n},t_{0}+0)\;\hat{P}(A_{n}z_{n})=1\;.$ (2.78)

It is easy to verify that the solution to this equation can be written in the
form

$\hat{\rho}(A_{n},t_{0}+0)=\frac{\hat{P}(A_{n}z_{n})\hat{\rho}(t_{0}-0)\hat{P}(A_{n}z_{n})}{{\rm
Tr}\hat{\rho}(t_{0}-0)\hat{P}(A_{n}z_{n})}\;.$ (2.79)

The transformation (2.75) is called quantum state reduction [50, 116, 117] and
the form (2.79) is named the von Neumann-Lüders state. This state becomes the
initial state for the following state dynamics at times $t>t_{0}$,

$\hat{\rho}(A_{n},t)=\hat{U}(t,t_{0})\;\hat{\rho}(A_{n},t_{0}+0)\;\hat{U}^{+}(t,t_{0})\;.$
(2.80)

The a priori probability of measuring an alternative $A_{m}$ for $t>t_{0}$ is

$p(A_{m}z_{m},t)={\rm
Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(A_{m}z_{m})\qquad(t>t_{0})\;.$ (2.81)

Thus the state reduction (2.75), caused by the measurement process, implies
the change of the initial condition for the state, hence the change of the
state evolution at later times, which in turn presumes the alteration of the
quantum probability space,

$\\{{\cal H},\;\hat{\rho}(t),\;{\cal P}_{AZ}\\}\mapsto\\{{\cal
H},\;\hat{\rho}(A_{n},t),\;{\cal P}_{AZ}\\}$ (2.82)

and, respectively, the reduction of the probability,

$p(A_{n}z_{n},t_{0}-0)\mapsto p(A_{n}z_{n},t_{0}+0)\;.$ (2.83)

It is important that the existence of intrinsic noise does not disturb the
standard scheme of quantum state reduction.

The quantum state reduction is nothing but the change of an a priori
probability to an a posteriori probability due to the received information.

###  2.12 Consecutive measurements of alternatives

Assume that at time $t_{0}$ an alternative $A_{n}$ has been certainly
observed, as is described in the previous section. Then state (2.79) plays the
role of an initial condition for the following state dynamics. For times
$t>t_{0}$, we have the state (2.80).

Suppose that after the moment of time $t_{0}$, we are interested in measuring
another observable corresponding to the new set of alternatives

$\mathbb{B}=\\{B_{k}:~{}k=1,2,\ldots\\}\;.$ (2.84)

The related operators $\hat{A}$ and $\hat{B}$ are not necessarily commuting,
since their measurements are accomplished at different times. These
alternatives are again assumed to be decorated by intrinsic noise. The
projectors

$\hat{P}(B_{k}z_{k})=|\;B_{k}z_{k}\;\rangle\langle\;z_{k}B_{k}\;|$ (2.85)

compose the family

${\cal P}_{BZ}=\\{\hat{P}(B_{k}z_{k}):~{}k=1,2,\ldots\\}\;.$ (2.86)

After the time $t_{0}$, the quantum probability space is

$\\{{\cal H},\;\hat{\rho}(A_{n},t),\;{\cal P}_{BZ}\\}\qquad(t>t_{0})\;.$
(2.87)

The probabilities of alternatives from the set (2.84) read as

$p(B_{k}z_{k},t)={\rm
Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(B_{k}z_{k})\qquad(t>t_{0})$ (2.88)

and, as any probability, they are normalized:

$\sum_{k}p(B_{k}z_{k},t)=1\;.$ (2.89)

From the other side, probability (2.88) can be interpreted as a conditional
probability of measuring an alternative $B_{k}$ at time $t$, after the
alternative $A_{n}$ at time $t_{0}$ has been certainly observed. Thus the
conditional probability is defined as the straightforward renotation

$p(B_{k}z_{k},t)\equiv p(B_{k}z_{k},t|A_{n}z_{n},t_{0})\qquad(t>t_{0})\;,$
(2.90)

with the related renotation of normalization (2.89),

$\sum_{k}p(B_{k}z_{k},t|A_{n}z_{n},t_{0})=1\;.$ (2.91)

Substituting the state (2.80) into the expression

$p(B_{k}z_{k},t|A_{n}z_{n},t_{0})={\rm
Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(B_{k}z_{k})$ (2.92)

and using the notation

$p(B_{k}z_{k},t,A_{n}z_{n},t_{0})\equiv{\rm
Tr}\;\hat{U}(t,t_{0})\;\hat{P}(A_{n}z_{n})\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n}z_{n})\;\hat{U}^{+}(t,t_{0})\;\hat{P}(B_{k}z_{k})$
(2.93)

results in the probability that can be called conditional,

$p(B_{k}z_{k},t|A_{n}z_{n},t_{0})=\frac{p(B_{k}z_{k},t,A_{n}z_{n},t_{0})}{p(A_{n}z_{n},t_{0}-0)}\;.$
(2.94)

Employing normalization (2.91), we get the relation

$\sum_{k}p(B_{k}z_{k},t,A_{n}z_{n},t_{0})=p(A_{n}z_{n},t_{0}-0)\;.$ (2.95)

From here, the normalization condition follows:

$\sum_{nk}p(B_{k}z_{k},t,A_{n}z_{n},t_{0})=1\;.$

These formulas suggest that probability (2.93) can be named as joint
probability. Similar equations often are considered for a fixed moment of time
$t=t_{0}$, which brings problems dealing with incompatible events
corresponding to the simultaneous measurement of noncommuting operators [118].
These problems do not arise when considering a realistic situation of
measurements at different moments of time. Taking into account the presence of
intrinsic noise also does not complicate much the consideration [57, 73].

Note that, for $t>t_{0}$, neither the joint probability (2.93) nor the
conditional probability (2.92) or (2.94) are symmetric with respect to the
interchange of the events $A_{n}$ and $B_{k}$. This asymmetry can explain the
so-called order effects in decision theory, when the probability of choice
depends on the order of choosing alternatives [84].

###  2.13 Immediate consecutive measurements

One often considers two measurements occurring immediately one after another
[50, 116]. This is the limiting case of the consecutive measurements treated
in the previous subsection, when at the moment of time $t_{0}$ an alternative
$A_{n}$ has been certainly observed and the second measurement of another
observable corresponding to the set of alternatives (2.84) is measured at the
time $t_{0}+0$ immediately following $t_{0}$.

In the case of these immediate measurements, the evolution operator reduces to
unity operator,

$\hat{U}(t_{0}+0,t_{0})=\hat{1}\;.$ (2.96)

Then for the conditional probability (2.92), we have

$p(B_{k}z_{k},t_{0}+0|A_{n}z_{n},t_{0})={\rm
Tr}\;\hat{\rho}(A_{n},t_{0}+0)\;\hat{P}(B_{k}z_{k})$ (2.97)

and the joint probability (2.93) becomes

$p(B_{k}z_{k},t_{0}+0,A_{n}z_{n},t_{0})={\rm
Tr}\;\hat{P}(A_{n}z_{n})\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n}z_{n})\;\hat{P}(B_{k}z_{k})\;.$
(2.98)

The conditional probability (2.94) takes the form

$p(B_{k}z_{k},t_{0}+0|A_{n}z_{n},t_{0})=\frac{p(B_{k}z_{k},t_{0}+0,A_{n}z_{n},t_{0})}{p(A_{n}z_{n},t_{0}-0)}\;,$
(2.99)

which can be called von Neumann-Lüders probability. The explicit expression
for the joint probability (2.98) turns into

$p(B_{k}z_{k},t_{0}+0,A_{n}z_{n},t_{0})=|\;\langle\;z_{k}B_{k}\;|\;A_{n}z_{n}\;\rangle\;|^{2}\;p(A_{n}z_{n},t_{0}-0)\;.$
(2.100)

This transforms the conditional probability (2.99) into the symmetric form

$p(B_{k}z_{k},t_{0}+0|A_{n}z_{n},t_{0})=|\;\langle\;z_{k}B_{k}\;|\;A_{n}z_{n}\;\rangle\;|^{2}\;,$
(2.101)

where

$\langle\;z_{k}B_{k}\;|\;A_{n}z_{n}\;\rangle=\sum_{\mu}a^{*}_{k\mu}a_{n\mu}\;\langle\;B_{k}\;|\;A_{n}\;\rangle\;.$

If the repeated measurement is accomplished with respect to the same
observable, so that $B_{k}=A_{k}$, then the conditional probability (2.101)
reduces to

$p(A_{k}z_{k},t_{0}+0|A_{n}z_{n},t_{0})=\delta_{nk}\;.$ (2.102)

This is in agreement with the principle of reproducibility in quantum theory,
according to which, when the choice, among the same set of alternatives, is
made twice, immediately one after another, the second choice has to reproduce
the first one [50]. This also is in agreement with decision making: when a
decision maker accomplishes a choice from the same set of alternatives
immediately after another choice, so that there is no time for deliberation,
then this decision maker should repeat the previous choice [57, 73].

Generally, the joint probability (2.98) is not symmetric with respect to the
interchange of the events $A_{n}$ and $B_{k}$. It becomes symmetric only when
there is no noise and the corresponding operators commute with each other,
hence enjoy the common basis of eigenvectors. At the same time, the
conditional probability (2.101) is always symmetric, whether for commuting or
noncommuting observables, and whether in the presence or absence of noise.
Therefore the immediate consecutive probabilities, with the symmetry
properties

$p(B_{k}z_{k},t_{0}+0,A_{n}z_{n},t_{0})\neq
p(A_{n}z_{n},t_{0}+0,B_{k}z_{k},t_{0})\;,$
$p(B_{k}z_{k},t_{0}+0|A_{n}z_{n},t_{0})=p(A_{n}z_{n},t_{0}+0|B_{k}z_{k},t_{0})$
(2.103)

cannot be accepted as a generalization of classical Kolmogorov-type
probabilities, where the joint probability is symmetric, while the conditional
one is not. This fact should not be of surprise as far as the definitions of
quantum consecutive probability of two events occurring at different times and
classical probability for two events occurring synchronously are principally
different. Classical probability contains no mentioning of state evolution,
while quantum probability connects two measurements realized at different
times and involving the state evolution.

###  2.14 Synchronous noiseless measurements

Quantum theory, as it is usually formulated, is not directly analogous to
classical probability theory in the sense of Kolmogorov [119], but is much
closer to the theory of stochastic processes [56]. In nonrelativistic quantum
mechanics, states at different times are related by dynamics, generally
represented as a completely positive map. In that sense, consecutive
measurements correspond to dynamic probability with the underlying causal
structure. This type of theory is closely analogous to a classical stochastic
process, in which a state is a probability distribution over a set of random
variables representing the properties of a system at a given time and the
states at different times are related by dynamics.

In contrast, classical probability spaces make no assumptions about the causal
structure of the events on which probabilities are defined. Two disjoint
events might refer to properties of two different subsystems at a given time,
or they might refer to properties of the same subsystem at two different
times. In full generality, classical events need have no interpretation in
terms of causal structure at all.

A variant of quantum probability enjoying the same symmetry properties as
classical probability should be noiseless and allowing for the accomplishment
of simultaneous measurements. This type of probability is defined as follows
[57].

Let us consider two sets of alternatives

$\mathbb{A}=\\{A_{n}:~{}n=1,2,\ldots\\}\;,\qquad\mathbb{B}=\\{B_{k}:~{}k=1,2,\ldots\\}\;.$
(2.104)

A simultaneous measurement of two observables can be realized provided they
pertain to two different Hilbert spaces, for instance they are located at two
different spatial regions, or act in the spaces of different variables, e.g.
momenta and spins. Speaking about simultaneous measurements at different
spatial locations, we keep in mind a nonrelativistic situation when the notion
of synchronously occurring events or measurements is well defined. In the
relativistic case, we could use the notion of spacelike separated measurements
or events. The corresponding Hilbert space is the tensor product

${\cal H}={\cal H}_{A}\;\bigotimes\;{\cal H}_{B}\;.$ (2.105)

For the moment, we do not include intrinsic noise.

Alternatives $A_{n}$ and $B_{k}$ are represented by the projectors
$\hat{P}(A_{n})$ and $\hat{P}(B_{k})$ in the related spaces. The Hilbert space
$\mathcal{H}$, the statistical operator $\hat{\rho}(t)$, and the family of the
projectors

${\cal
P}_{AB}=\\{\hat{P}(A_{n})\otimes\hat{P}(B_{k}):~{}n=1,2,\ldots;~{}k=1,2,\ldots\\}$
(2.106)

compose the quantum probability space

$\\{{\cal H}_{A}\otimes{\cal H}_{B},\;\hat{\rho}(t),\;{\cal P}_{AB}\\}\;.$
(2.107)

The probability of measuring the alternatives $A_{n}$ and $B_{k}$ in different
spaces is

$p(A_{n}B_{k},t)={\rm
Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n})\bigotimes\hat{P}(B_{k})\;.$ (2.108)

The defined probability possesses the same properties as classical
probability. Thus the marginal probabilities are given by the partial
summation

$p(A_{n},t)=\sum_{k}p(A_{n}B_{k},t)\;,\qquad
p(B_{k},t)=\sum_{n}p(A_{n}B_{k},t)\;,$ (2.109)

and they are normalized,

$\sum_{n}p(A_{n},t)=1\;,\qquad\sum_{k}p(B_{k},t)=1\;.$ (2.110)

If the measurements are not correlated, such that

$\hat{\rho}(t)=\hat{\rho}_{A}(t)\;\bigotimes\;\hat{\rho}_{B}(t)\;,$

the joint probability becomes the product

$p(A_{n}B_{k},t)=p(A_{n},t)p(B_{k},t)\;.$

It is possible to introduce the conditional probability

$p(B_{k}|A_{n},t)\equiv\frac{p(B_{k}A_{n},t)}{p(A_{n},t)}\;.$ (2.111)

The defined joint and conditional quantum probabilities of synchronous events,
happening in different Hilbert spaces, possess the same symmetry properties as
the classical probability: the joint probability is symmetric with respect to
the event interchange, while the conditional probability is not symmetric,

$p(A_{n}B_{k},t)=p(B_{k}A_{n},t)\;,\qquad p(A_{n}|B_{k},t)\neq
p(B_{k}|A_{n},t)\;.$ (2.112)

Strictly speaking, the measurements of two events, simultaneously occurring at
two different spatial locations, is possible only if the measuring device is
sufficiently large, such that it includes several parts allowing for the
synchronous measurement of two different events. In decision making, in order
to accept that a subject is able to decide on two alternatives simultaneously,
it is necessary to assume that either there are different parts of the brain
thinking synchronously or what seems to be synchronous is actually a fast
temporal reswitching from one object to another [120, 121, 122].

###  2.15 Synchronous measurements under noise

Synchronous measurements in different Hilbert spaces, e.g. at different
spatial locations, can be straightforwardly generalized by including intrinsic
noise. Then the system is defined in the Hilbert space

${\cal H}={\cal H}_{A}\;\bigotimes\;{\cal H}_{E}\;\bigotimes\;{\cal
H}_{B}\;\bigotimes\;{\cal H}^{\prime}_{E}\;,$ (2.113)

where ${\cal H}^{\prime}_{E}$ is a copy of ${\cal H}_{E}$. The events are
characterized by the family of the projectors

${\cal
P}_{ABZ}=\\{\hat{P}(A_{n}z_{n})\otimes\hat{P}(B_{k}z_{k}):~{}n=1,2,\ldots;~{}k=1,2,\ldots\\}\;.$
(2.114)

The quantum probability space becomes

$\\{{\cal H},\;\hat{\rho}(t),\;{\cal P}_{ABZ}\\}\;.$ (2.115)

The probability of two synchronously occurring events in the presence of
intrinsic noise is

$p(A_{n}z_{n}B_{k}z_{k},t)={\rm
Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n}z_{n})\bigotimes\hat{P}(B_{k}z_{k})$ (2.116)

that is required to be normalized,

$\sum_{nk}p(A_{n}z_{n}B_{k}z_{k},t)=1\;.$ (2.117)

Notice that this probability is symmetric with respect to the order swap of
the alternatives,

$p(A_{n}z_{n}B_{k}z_{k},t)=p(B_{k}z_{k}A_{n}z_{n},t)\;.$ (2.118)

This probability, similarly to the probability of a single event, can be
represented as a sum

$p(A_{n}z_{n}B_{k}z_{k},t)=f(A_{n}z_{n}B_{k}z_{k},t)+q(A_{n}z_{n}B_{k}z_{k},t)\;,$
(2.119)

of a diagonal part

$f(A_{n}z_{n}B_{k}z_{k},t)=\sum_{\mu\lambda}|\;a_{n\mu}\;|^{2}\;|\;a_{k\lambda}\;|^{2}\;\langle\;e_{\lambda}e_{\mu}B_{k}A_{n}\;|\;\hat{\rho}(t)\;|\;A_{n}B_{k}e_{\mu}e_{\lambda}\;\rangle$
(2.120)

and an off-diagonal part comprising all interference terms,

$q(A_{n}z_{n}B_{k}z_{k},t)=\sum_{\mu\nu\gamma\lambda}a_{n\mu}a^{*}_{n\nu}a_{k\gamma}a^{*}_{k\lambda}\langle\;e_{\lambda}e_{\nu}B_{k}A_{n}\;|\;\hat{\rho}(t)\;|\;A_{n}B_{k}e_{\mu}e_{\gamma}\;\rangle\;,$
(2.121)

where

$\sum_{\mu\nu\gamma\lambda}~{}\mapsto~{}\sum_{\mu\neq\nu}\;\sum_{\gamma\lambda}\delta_{\gamma\lambda}~{}+~{}\sum_{\mu\nu}\;\sum_{\gamma\neq\lambda}\delta_{\mu\nu}~{}+~{}\sum_{\mu\neq\nu}\;\sum_{\gamma\neq\lambda}\;.$

Note that if the parts of the synchronous measurement are not correlated, so
that

$\hat{\rho}(t)=\hat{\rho}_{AZ}(t)\;\bigotimes\;\hat{\rho}_{BZ}(t)\;,$

then probability (2.116) separates into two factors

$p(A_{n}z_{n}B_{k}z_{k},t)=p(A_{n}z_{n})p(B_{k}z_{k})\;.$

The conditional probability can be defined as

$p(A_{n}z_{n}|B_{k}z_{k},t)\equiv\frac{p(A_{n}z_{n}B_{k}z_{k},t)}{p(B_{k}z_{k},t)}\;.$
(2.122)

This probability is not swap order symmetric,

$p(A_{n}z_{n}|B_{k}z_{k},t)\neq p(B_{k}z_{k}|A_{n}z_{n},t)\;.$

The synchronous joint probability (2.116) is a natural generalization of
classical probability to the case of quantum measurements under intrinsic
noise. In decision theory, it plays the role of a behavioural probability of
taking a decision on two events simultaneously occurring in two different
spatial locations.

###  2.16 Swap order relations

The symmetry properties of probabilities with respect to the swap of the order
of events makes it straightforward to derive some relations that can be
checked experimentally. However one has to be very cautious distinguishing
necessary and sufficient conditions for such relations.

Let us consider the alternatives $A_{n}$ and $B_{k}$, with $n,k=1,2$, and the
joint probability of immediate consecutive measurements (2.98). Define the
swap function

$S[\;p(A_{n}z_{n},t_{0}+0,B_{k}z_{k},t_{0})\;]\equiv
p(A_{1}z_{1},t_{0}+0,B_{2}z_{2},t_{0})-p(B_{2}z_{2},t_{0}+0,A_{1}z_{1},t_{0})\;+$
$+\;p(A_{2}z_{2},t_{0}+0,B_{1}z_{1},t_{0})-p(B_{1}z_{1},t_{0}+0,A_{2}z_{2},t_{0})\;.$
(2.123)

Using the normalization conditions (2.73) and (2.91), and the symmetry
property (2.103) of the conditional probability, it is easy to get the
relation

$S[\;p(A_{n}z_{n},t_{0}+0,B_{k}z_{k},t_{0})\;]=0\;.$ (2.124)

Clearly, the same swap order relation is valid for the case when there is no
intrinsic noise,

$S[\;p(A_{n},t_{0}+0,B_{k},t_{0})\;]=0\;.$ (2.125)

Relation (2.125) has been discussed by many authors, e.g. [35]. In a number of
experimental studies in psychology, it has been found that the probability of
answers to two consecutive questions depends on the question order, and
relation (2.125) holds true for empirical probabilities, because of which the
following conclusion has been advocated: Since the validity of relation
(2.125) for a joint empirical probability has been confirmed in a vast number
of experimental studies in psychology, and this relation has been derived for
a quantum probability, this means that the empirical data prove that
consciousness obeys quantum rules, that is, consciousness is quantum.

This claim, though, is not correct, since here one confuses necessary and
sufficient conditions. If some $p$ enjoys the properties of the quantum
probability of immediate consecutive measurements, then this is sufficient for
the relation $S[p]=0$ to be valid. However this is not a necessary condition,
as far as, if the relation $S[p]=0$ holds, it follows from nowhere that $p$
must be a particular quantum probability.

Really, not only the probability (2.98) of consecutive measurements satisfies
the same relation, but also the probability (2.108) of synchronous noiseless
measurements,

$S[\;p(A_{n}B_{k},t)\;]=0,$ (2.126)

as well as the probability of synchronous noisy measurements (2.118),

$S[\;p(A_{n}z_{n}B_{k}z_{k},t)\;]=0$ (2.127)

do satisfy the same relation. The validity of the latter relations is the
direct result of the swap order symmetry of these probabilities, as in
(2.112).

Moreover, if we consider a classical probability $f(A_{n}B_{k})$ that, by
definition is swap order symmetric, $f(A_{n}B_{k})=f(B_{k}A_{n})$, and study
the swap function

$S[\;f(A_{n}B_{k})\;]=f(A_{1}B_{2})-f(B_{2}A_{1})+f(A_{2}B_{1})-f(B_{1}A_{2})$
(2.128)

then, because of the swap order symmetry, the same relation immediately
follows:

$S[\;f(A_{n}B_{k})\;]=0\;.$ (2.129)

The swap order symmetry of quantum conditional probability of consecutive
events is a sufficient condition for the validity of the swap order relation
for the joint quantum probability of consecutive events. The swap order
symmetry of the classical joint probability is also a sufficient condition for
the validity of the swap order relation. However none of these symmetry
properties separately is a necessary condition for the validity of the swap
order relation. That is, different quantum as well as classical probabilities
can satisfy the same swap order relation. However the validity of this
relation tells us nothing on the nature of probability, whether it is quantum
or classical.

###  2.17 Quantum versus classical probabilities

In the literature advocating the use of quantum techniques for describing
consciousness, it is customary to counterpose quantum to classical approaches,
arguing in favour of quantum theory that, supposedly, is more versatile in
characterizing, e.g., such phenomena as non-commutativity of consecutive
events. In doing this, one usually compares the classical Kolmogorov
probability with the von Neumann-Lüders probability of consecutive
measurements. However comparing these probabilities is not correct, since the
classical Kolmogorov probability contains no dynamics, while the von Neumann-
Lüders approach considers the dynamic evolution from one measurement to
another. For the correct comparison of quantum and classical probabilities, it
is necessary to remember that there are several types of the latter, so that
the comparison has sense only for the probabilities from the same class. There
are the following classes of probabilities.

(i) Probability of single events. For the quantum case, under events we mean
quantum measurements, and in the classical case, some occurring events or the
acts of taking decisions. The quantum probability $p(A_{n},t)$ of a single
event $A_{n}$ differs from the classical Kolmogorov probability $f(A_{n})$ by
including temporal evolution and by taking account of intrinsic noise. Here
and below we assume that quantum probability includes intrinsic noise, but for
the sake of compactness, we do not show this explicitly.

(ii) Probability of synchronous events. Two or more synchronous events can be
observed in the quantum case, provided they are happening in different Hilbert
spaces, for instance in different spatial locations. In the classical
Kolmogorov theory, the events are always synchronous. Both these
probabilities, quantum as well as classical, enjoy the same symmetry
properties.

(iii) Probability of consecutive events. Events are happening one after
another at different times. The times have to be treated as different even
when one event occurs immediately after another. For quantum probability, the
temporal evolution is incorporated in the evolution operators. Consecutive
events in the quantum case are treated by the von Neumann-Lüders theory. In
the classical case, the evolution can be imposed by the equations called
Kolmogorov equations or master equations.

One often claims that the classical Kolmogorov probability is inferior to
quantum von Neumann-Lüders probability because the classical joint probability
does not depend on the order of events, being swap-order symmetric, while the
quantum von Neumann-Lüders theory gives for the joint probability of two
consecutive events at different times a non-symmetric order dependent
probability. However this comparison is not appropriate, since it considers
the probability from different classes. Quantum consecutive probabilities have
to be collated with classical consecutive probabilities that, in general, are
also not swap-order symmetric.

To illustrate the asymmetry of the classical consecutive probabilities, let us
consider the probability of two events, one $A_{n}$ occurring at time $t_{0}$,
after which the other event $B_{k}$ can happen at time $t$. The classical
joint probability
$f(B_{k},t,A_{n},t_{0})=f(B_{k},t|A_{n},t_{0})f(A_{n},t_{0}-0)$ is expressed
through the related conditional probability satisfying the master equation (or
Kolmogorov forward equation)

$\frac{d}{dt}\;f(B_{k},t|A_{n},t_{0})=\sum_{l=1}^{N_{B}}\gamma_{kl}f(B_{l},t|A_{n},t_{0})\;,$
(2.130)

in which $\gamma_{kl}$ is a transition rate matrix, or generator matrix,
characterizing the transition rate from the event $B_{l}$ to $B_{k}$. The
transition rate matrix has the properties

$\gamma_{kl}\geq 0\qquad(k\neq l)$ (2.131)

and

$\sum_{k=1}^{N_{B}}\gamma_{kl}=0\;.$ (2.132)

The latter property can be rewritten as

$\gamma_{ll}+\sum_{k(\neq l)}^{N_{B}}\gamma_{kl}=0\;,$ (2.133)

which allows us to represent equation (2.130) in the equivalent form

$\frac{d}{dt}\;f(B_{k},t|A_{n},t_{0})=\sum_{l(\neq
k)}^{N_{B}}[\;\gamma_{kl}f(B_{l},t|A_{n},t_{0})-\gamma_{lk}f(B_{k},t|A_{n},t_{0})\;]\;.$
(2.134)

It is instructive to observe an explicit solution, for example, considering
two events $\\{A_{1},A_{2}\\}$ and two events $\\{B_{1},B_{2}\\}$ under the
initial condition

$f(B_{k},t_{0}|A_{n},t_{0})=f_{kn}\;.$ (2.135)

Then the solution reads as

$f(B_{1},t|A_{1},t_{0})=\left(f_{11}\;-\;\frac{\gamma_{1}}{\gamma_{1}+\gamma_{2}}\right)e^{-(\gamma_{1}+\gamma_{2})(t-t_{0})}+\frac{\gamma_{1}}{\gamma_{1}+\gamma_{2}}\;,$
$f(B_{1},t|A_{2},t_{0})=\left(f_{12}\;-\;\frac{\gamma_{1}}{\gamma_{1}+\gamma_{2}}\right)e^{-(\gamma_{1}+\gamma_{2})(t-t_{0})}+\frac{\gamma_{1}}{\gamma_{1}+\gamma_{2}}\;,$
$f(B_{2},t|A_{1},t_{0})=\left(f_{21}\;-\;\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right)e^{-(\gamma_{1}+\gamma_{2})(t-t_{0})}+\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}\;,$
$f(B_{2},t|A_{2},t_{0})=\left(f_{22}\;-\;\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right)e^{-(\gamma_{1}+\gamma_{2})(t-t_{0})}+\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}\;,$
(2.136)

where

$\gamma_{1}\equiv\gamma_{12}=-\gamma_{22}\;,\qquad\gamma_{2}\equiv\gamma_{21}=-\gamma_{11}\;.$

Inverting the order of the events leads to the probability satisfying the
equation

$\frac{d}{dt}\;f(A_{n},t|B_{k},t_{0})=\sum_{m=1}^{N_{A}}\alpha_{nm}f(A_{m},t|B_{k},t_{0})\;.$
(2.137)

The transition rate matrix $\alpha_{mn}$ describes the transition from an
event $A_{n}$ to $A_{m}$ and possesses the properties

$\alpha_{mn}\geq 0\quad(m\neq n)$ (2.138)

and

$\sum_{m=1}^{N_{A}}\alpha_{mn}=0\;.$ (2.139)

In the case of binary sets of alternatives, the solution of equation (2.137),
under an initial condition

$f(A_{n},t_{0}|B_{k},t_{0})=g_{nk}\;,$ (2.140)

is similar by form to solution (2.136).

From these equations, it is clearly seen that inverting the order of the
events results in principally different expressions for the probability. This
is because the initial conditions are different and the transition rate
matrices are different. Generally, these matrices are different even in their
size, since the size of the rate $\gamma_{kl}$ is $N_{B}\times N_{B}$, while
the size of the rate $\alpha_{nm}$ is $N_{A}\times N_{A}$. The matrices of
initial values, in general, are also different in their form, since $f_{kn}$
is a matrix of the size $N_{B}\times N_{A}$, while $g_{nk}$ is a matrix of the
size $N_{A}\times N_{B}$.

Thus the classical consecutive probabilities, whether conditional or joint,
are not swap-order symmetric

$f(B_{k},t|A_{n},t_{0})\neq f(A_{n},t|B_{k},t_{0})\;,\qquad
f(B_{k},t,A_{n},t_{0})\neq f(A_{n},t,B_{k},t_{0})$ (2.141)

for any $t\geq t_{0}$. In this way, classical consecutive probabilities can
perfectly explain the so-called order effects observed in human behaviour.

###  2.18 Quantum decision theory

The analysis of quantum probabilities described above and their interpretation
in the language of decision theory is, strictly speaking, what composes the
basis of the so-called Quantum Decision Theory [57, 58, 59, 73, 80, 83, 94].
Summarizing this part of the review, it is necessary to make several remarks.

Everywhere above the system state has been represented by a statistical
operator $\hat{\rho}$. In particular cases, this operator could have the form
of a pure state

$\hat{\rho}(t)=|\;\psi(t)\;\rangle\langle\;\psi(t)\;|\;,$

where the wave function can be expanded over the given basis as

$|\;\psi(t)\;\rangle=\sum_{n\mu}b_{n\mu}(t)\;|\;A_{n}e_{\mu}\;\rangle\;.$

The description by wave functions is appropriate for isolated systems.
However, strictly speaking, quantum systems cannot be absolutely isolated, but
can only be quasi-isolated [123, 124, 125]. This means that, even if a system
is prepared in a pure state described by a wave function, there always exist
uncontrollable external perturbations from the surrounding that result in the
system decoherence beyond a decoherence time, which makes the system state
mixed. Moreover, to confirm that the considered system is to some extent
isolated, it is necessary to check this by additional control measurements
which again disturb the system’s isolation. In that way, the system can be
only quasi-isolated.

In addition, decision-makers are the members of a society, hence, they
correspond to non-isolated open systems that have to be described by
statistical operators. One could think that in laboratory tests, it would be
admissible to treat decision-makers as closed systems characterized by wave
functions. However, in laboratory tests, even when being separated from each
other, decision-makers do communicate with the investigators performing the
test. Moreover, even when being for some time locked in a separate room, any
decision-maker possesses the memory of interactions with many other people
before. From the physiological point of view, memory is nothing but delayed
interactions. Therefore, no decision maker can be treated as an absolutely
isolated system, which excludes the use of wave functions. It looks that the
most general and correct description of any decision-maker requires to
consider him/her as an open system, hence, characterized by a statistical
operator.

When representing the considered alternatives as the vectors of a Hilbert
space, we have assumed a nondegenerate representation with a one-to-one
correspondence between each alternative $A_{n}$ and the representing it vector
$|A_{n}\rangle$. Generally, in quantum theory there can occur the effect of
degeneracy, when an operator eigenvalue can correspond to several
eigenfunctions. In the present case, this would imply that an alternative
$A_{n}$ would correspond to several vectors $|A_{ni}\rangle$, where
$i=1,2,\ldots$. The existence of degeneracy in decision theory is sometimes
supposed for removing the contradiction between the reciprocal symmetry of von
Neumann-Lüders probability (2.101), that is the symmetry with respect to the
interchange of the events $A_{n}$ and $B_{k}$, and the experimentally observed
absence of this reciprocal symmetry [45, 46]. Really, if at least one of the
considered alternatives say $A_{n}$, is degenerate, such that $A_{n}$ is
represented by a set of vectors $|A_{ni}\rangle$, with $i=1,2,\ldots$, then
the related projector becomes the sum

$\hat{P}(A_{n})=\sum_{i}|\;A_{ni}\;\rangle\langle\;A_{ni}\;|\;.$

Considering, for simplicity, the noiseless case, for the von Neumann-Lüders
probability (2.99) we get

$p(B_{k},t_{0}+0|A_{n},t_{0})=\frac{\sum_{ij}\langle
A_{ni}|\hat{\rho}(t_{0}-0)|A_{nj}\rangle\langle A_{nj}|B_{k}\rangle\langle
B_{k}|A_{ni}\rangle}{\sum_{i}\langle
A_{ni}|\hat{\rho}(t_{0}-0)|A_{ni}\rangle}\;.$

Reversing the order of events yields a different expression

$p(A_{n},t_{0}+0|B_{k},t_{0})=\sum_{i}|\;\langle\;A_{ni}\;|\;B_{k}\;\rangle\;|^{2}\;.$

However, although the occurrence of degenerate operator spectra is natural for
quantum systems, in decision theory the appearance of degenerate alternatives
has no sense. If there occur several vectors $|A_{ni}\rangle$, it is always
admissible to reclassify the given alternatives so that each vector
$|A_{ni}\rangle$ would correspond to a single alternative $A_{ni}$ [70, 71].
This is equivalent to the breaking of symmetry in physics [50, 67, 68, 69].

The measurement procedure has been described by using projection-valued
measurements of alternatives decorated by intrinsic noise. In general, it
could be possible to invoke positive operator-valued measurements (POVM) [87,
88, 89, 90, 91, 92, 93]. This, probably, could be useful for some quantum
systems. However, as has been stressed above, we do not assume that the brain
is a quantum system, but we are analyzing the possibility of employing quantum
techniques for describing the operation of the decision-making process. For
this purpose, there is no reason to complicate the consideration by invoking
POVM bringing additional problems, such as the nonuniqueness of the post-
measurement states and the absence of reproducibility of immediately repeated
measurements.

Moreover, already on the level of projection-valued measurements, we meet a
number of difficulties in the attempts of applying quantum theory for
describing conscious processes, although certainly there are many similarities
on the general qualitative level, as has been mentioned many times above.
Summarizing, it is necessary to separate grains from tares by clearly
formulating what are the useful recipes following from the quantum theory of
measurements and what are the limitations in the attempts of their use for
characterizing the operation of artificial intelligence. The main conclusions
that can be derived from the analogies between quantum measurements and
behavioural decision making, are as follows.

(i) First of all, the process of decision making has to be treated in a
probabilistic way. The probabilistic description better corresponds to real
life, where in any sufficiently large group of subjects, deciding on a choice
among the same set of alternatives, not all prefer the same choice, but always
there are those who choose other alternatives. Any such group separates into
subgroups preferring different alternatives. The fractions of people
preferring different alternatives are nothing but frequentist probabilities.
Even a single person at different moments of time can choose different
alternatives. In that case, the frequentist probability shows the ratio of
particular choices to the total number of accomplished choices.

(ii) Generally, decision making is not a purely rational choice, but it is
accompanied by intrinsic noise representing irrational subconscious sides of
decision process, including emotions, gut feelings, and intuitive allusions.
In that sense, decision making is characterized by cognition-emotion duality,
or conscious-subconscious duality, or rational-irrational duality. Emotions
and the related characteristics can be modeled by intrinsic noise in quantum
measurements.

(iii) Quantum probability for measurements in the presence of intrinsic noise
consists of two terms, one that can be called classical limit and the other
caused by the interference of noise. The former can be associated with the
rational choice and the other is induced by the existence of emotions. In
decision theory, the occurrence of the additional interference term is
especially noticeable under uncertainty [84, 126].

(iv) Alternatives and noise, or alternatives and emotions, generally are
entangled being connected with each other. Measurement procedure as well as
decision making produce additional entanglement between alternatives and
noise.

(v) It is necessary to distinguish between two types of quantum probabilities
for two events. One type is the quantum probability of consecutive events
happening at different times and the other type is the quantum probability of
synchronous events occurring at different spatial locations or at different
spaces of variables.

Despite a number of hints on the general structure and main properties of
probability, which could be used in developing decision theory, quantum theory
provides no explicit rules allowing for the calculation of the probability for
the purpose of decision making, thus possessing no predictive power. It is of
course possible to fit the interference term for interpreting some particular
events, however fitting is not explanation. In order to supply quantum
decision theory with the ability of making quantitative estimates, it is
necessary to invoke a number of assumptions not related to quantum theory [79,
127].

It is also necessary to understand whether the usage of quantum theory is
compulsory for the development of adequate decision theory taking account of
behavioural effects or this usage is just a much more complicated trendy way
of describing what could be much easier described employing classical
terminology.

A great goal would be to develop a mathematical formulation of an approach,
taking into account the general peculiarities of quantum measurements and the
properties of quantum probabilities, but explicitly involving no quantum
techniques, at the same time providing the ability of quantitative
predictions. Such a formalized approach would be indispensable for the
creation of affective artificial intelligence.

##  3 Affective decision making

Before formulating the general approach to decision making, combining the
rational choice with the influence of irrational emotions, it is useful to
remind the origin of emotions and to explain what are the problems in the
earlier attempts to formalize the process of behavioural decision making.

###  3.1 Evolutionary origin of emotions

Emotions are common for humans as well as for animals. They evolved in the
process of evolution and were adapted over time like other traits found in
animals. Darwin [128] was, probably, the first to seriously study the
appearance and adaptation of emotions in the process of natural selection. He
discussed not only facial expressions in animals and humans, but attempted to
point out parallels between behaviours in humans and other animals [129].
According to evolutionary theory, different emotions evolved at different
times. Primal emotions, such as fear, are associated with ancient parts of the
brain and presumably evolved among our premammal ancestors. Filial emotions,
such as a human mother’s love for her offspring, seem to have evolved among
early animals. Social emotions, such as guilt and pride, evolved among social
primates.

Since emotions evolved and adapted during the years of evolution, they
appeared for some reason, and as other features, they should be useful for
animals and humans. For example, they facilitate communication by sending
signals to other members of the social group. Such an emotion as fear has
helped humans to survive, warning about a danger and forcing to take actions
before the cognitive logical part of the brain gives more detailed
information. Having emotions may mean the difference between life and death.
Certain emotions are universal to all humans, regardless of culture: anger,
fear, surprise, disgust, happiness and sadness. Emotions can be defined as a
specialized mechanism, shaped by natural selection, that increases fitness in
specific situations. The physiological, psychological, and behavioural
characteristics of emotions can be understood as possible design features that
increase the ability to cope with the threats and opportunities present in the
corresponding situations. Every emotion has been developed individually in the
course of biological evolution, and they all have been evolved to maintain the
survival needs.

Emotions play an important role in decision making. It would be not an
exaggeration to say that emotions shape decisions. As in the mentioned above
example of fear that saves lives, the fear can also save from bankruptcy.
Thus, compelling scientific evidence comes from emotionally impaired patients
who have sustained injuries to the ventromedial prefrontal cortex, a key area
of the brain for integrating emotion and cognition. Studies find that such
neurological impairments reduce both the patients’ ability to feel emotion
and, as a result, reduce the optimality of their decisions. Participants with
these injuries repeatedly select a riskier financial option over a safer one,
even to the point of bankruptcy, despite their cognitive understanding of the
suboptimality of their choices. These participants behave this way because
they do not experience the emotional signals that lead normal decision makers
to have a reasonable fear of high risks [11].

Living in a world where events cannot be predicted with certainty, agents must
select actions based on limited information, i.e., they often must make risky
decisions. Emotional information has a special weight in decision-making, as
it automatically triggers adaptive behavioral modules selected during the
course of evolution, driving agents to move toward attractive goals while
avoiding threats. Emotional information is critical, because on the one hand
it could prevent potential physical harm or unpleasant social interactions, on
the other hand it could promote physical pleasure or pleasant social
interactions [130].

Emotions are generally classified onto positive and negative [19],
respectively making alternatives more attractive or more repulsive. In the
choice between alternatives, there is no one-to-one correspondence between
alternatives and emotions, but there appears a multitude of different
connected emotions. Researchers exploring the subjective experience of
emotions have noted that emotions are highly intercorrelated both within and
between the subjects reporting them. Subjects rarely describe feeling a
specific positive or negative emotion without also claiming to feel other
positive or negative emotions [131].

In the process of decision making, emotions have great influence on multiple
cognitive phenomena, such as attention, perception, memory encoding, storage,
and retrieval of information, and associative learning [132]. Emotions
activate the motivational system of action tendencies. Recall that the word
emotion comes from Latin “emovere”, which means to move. The origin of the
word emotion already emphasizes its relevance to behavioural drive.

Although emotions seem to be similar to noise, they in fact help to optimize
decisions in two ways. First, they are faster than logical rational
deliberations, thus being of crucial importance in the case of urgent
decisions. Second, emotions reflect subconscious feelings based on your past
experiences and beliefs. This might serve to protect you from danger or
prevent your repeating past mistakes.

Notice that in physical measurements noise also is not always an obstacle, but
sometimes the detection of signals can be boosted by noise so that their
detection can even be facilitated [53, 54]. Measurements of thermal noise had
been used to measure the Boltzmann constant and measurements of shot noise had
been used to measure the charge on the electron [133]. Noise plays beneficial
role in the functioning of neural systems in the framework of stochastic
facilitation of signal processing [134].

###  3.2 Problems in decision making

The predominant theory describing individual behaviour under uncertainty is
nowadays the expected utility theory of preferences over uncertain prospects.
This theory was axiomatized by von Neumann and Morgenstern [135] and
integrated with the theory of subjective probability by Savage [136]. The
theory was shown to possess great analytical power by Arrow [137] and Pratt
[138] in their work on risk aversion and by Rothschild and Stiglitz [139, 140]
in their work on comparative risk. Friedman and Savage [141] and Markowitz
[142] demonstrated its tremendous flexibility in representing decision makers
attitudes toward risk. It is fair to state that the expected utility theory
has provided solid foundations to the theory of games, the theory of
investment and capital markets, the theory of search, and other branches of
economics, finance, and management.

However, a number of economists and psychologists have uncovered a growing
body of evidence that individuals do not always conform to the prescriptions
of the expected utility theory and indeed very often depart from the theory in
a predictable and systematic way [29]. Many researchers, starting with the
works by Allais [143], Edwards [144, 145], and Ellsberg [146] and continuing
through the present, experimentally confirmed pronounced and systematic
deviations from the predictions of the expected utility theory leading to the
appearance of many paradoxes. These paradoxes are often called behavioural,
since the behaviour of subjects contradicts the prescriptions of the utility
theory. Large literature on this topic can be found in the review articles
[147, 148, 149, 150].

There were many attempts to change the expected utility approach, which were
classified as non-expected utility theories. There are a number of such non-
expected utility theories, among which we may mention a few of the most known
ones: prospect theory [144, 151, 152], weighted-utility theory [153, 154,
155], regret theory [156], optimism-pessimism theory [157], ordinal-
independence theory [158], quadratic-probability theory [159], opportunity-
threat theory [160], and state-dependent utility theory [161]. The general
discussion of these theories can be found in the review by Machina [150].

However, non-expected utility theories are descriptive requiring fitting of
several parameters from empirical data. Moreover, as was shown by Safra and
Segal [162], none of the non-expected utility theories can explain all
paradoxes permanently arising in behavioural decision making. The best that
could be achieved is a kind of fitting for interpreting just one or, at best,
a few paradoxes, with other paradoxes remaining unexplained. In addition,
spoiling the structure of expected utility theory results in the appearance of
several complications and inconsistences [163]. As was concluded in the
detailed analysis of Al-Najjar and Weinstein [164, 165], any variation of the
classical expected utility theory ”ends up creating more paradoxes and
inconsistences than it resolves”.

An attempt of taking into account unconscious feelings has been undertaken by
the approach called the dual-process theory [166, 167, 168, 169, 170].
According to this theory, decisions arise in the human brain as a result of
two different processes that can be distinguished by one of the following
characteristic pairs [170]: slow/fast, rational/irrational,
conscious/unconscious, logical/intuitive, reasoning/reasonless,
deliberate/emotional, intentional/unintentional, voluntary/involuntary,
explicit/implicit, analyzing/sensuous, controlled/uncontrolled,
operated/automatic, regulated/impulsive, effortful/effortless,
comprehensive/perceptional, precise/impressionistic, objective/subjective,
verbal/nonverbal. Of course, not all these characteristics have to be present
in one or another process. Some of them can be shared to some extent by both
ways of thinking. Detailed discussions of the description of the two processes
can be found in the literature [166, 167, 168, 169, 170, 171, 172, 173] and
are well exposed in the review articles [174, 175, 176]. The existence of two
ways of thinking finds support in neuropsychological studies [177, 178, 179,
180, 181], although such a separation is not very strict [16, 17, 18].

Thus, in the dual-process theory one accepts the existence of two ways of
thinking, which to some extent finds support in some psychological and
neurological studies. These ways, for brevity, can be termed one as cognitive,
rational, logical and the second as emotional, irrational, intuitive. This
separation has to be understood in the conditional operational sense applied
to the process of decision making. The rational way of thinking is normative,
being based on clearly prescribed logical rules, while the irrational way is
poorly controlled, representing emotions induced by intuition and a kind of
gut feelings. This distinction does not assume that one of the ways, say
cognitive, is more correct.

In psychophysical and neurophysiological studies it has been found that the
influence of emotions leads to random choice, with the randomness caused by
generic variability and local instability of neural networks [182, 183, 184,
185, 186, 187, 188, 189]. The choice varies not only for different
individuals, but also for the same subject at different moments of time.
Moreover, even a given subject when making a single decision, experiences
intrinsic noise in the brain neural network, because of which the subject
decision becomes probabilistic. Cognitive imprecision is due to the inevitable
internal noise in the nervous system. Stochasticity is the unavoidable feature
of the human brain functioning. As a result, the choice in decision making is
not deterministic, being based on the comparison of utilities, but it is
rather stochastic and based on the comparison of probabilities. The recent
review [190] summarizes the modern point of view that considers randomness as
an internal feature of functioning of the human brain, where decisions are
formed on the basis of noisy internal representation.

The standard method employed in the attempts of taking into account irrational
effects in human decision making is a modification of the utility functional
[189, 191, 192]. In that sense, the dual-process models are reduced to
variants of non-expected utility theories, sharing with the latter the same
deficiencies.

Summarizing, in order to develop a decision theory comprising both ways of
decision process, conditionally labeled as cognitive (rational) and emotional
(irrational), one should respect the following premisses:

  1. 1.

Since the presence of emotions results in random decisions, behavioural
decision making has to be treated as a generically probabilistic process.
Therefore the main quantity to be determined is the behavioural probability of
events. It is never happens that among a given group of people all without
exception would make the identical choice prescribed by the standard
deterministic utility theory. There always exist fractions of subjects
preferring different alternatives. That is, there always exists a distribution
of decisions over the set of the given alternatives.

  2. 2.

The behavioural probability of an event should reflect the superposition of
two operational aspects in decision making, rational (cognitive), defined by
the prescribed rules evaluating the utility of the event, and irrational
(emotional), taking account of irrational effects.

  3. 3.

As far as emotions randomly vary for different decision makers, as well as for
the same decision maker at different moments of time, their quantitative
influence cannot be predicted exactly for each subject and for each choice.
However the approach should provide quantitative predictions at the aggregate
level, when the behavioural probabilities could be compared with the empirical
average fractions of decision makers choosing the related alternatives.

  4. 4.

The efficiency of the approach should be proved by demonstrating the absence
of paradoxes that should find quantitative resolution in the framework of the
general methodology without fitting parameters.

  5. 5.

The last, but not the least, the approach should not be overloaded by
unnecessary complicated theorizing. Thus, borrowing from the theory of quantum
measurements and quantum decision theory some general ideas, it is desirable
to avoid the explicit use of quantum techniques. In that sense, an artificial
intelligence could accomplish quantum operation without the necessity of
involving quantum formulas [193].

###  3.3 Behavioural probabilities of alternatives

As is emphasized above, behavioural decision making is a principally
probabilistic process. Hence the pivotal role is played by the notion of
probability of alternatives. In decision theory, probabilistic approach is
usually based on the random utility model, where the expected utility of
alternatives is complimented by an additive random error characterized by a
postulated distribution [194, 195, 196]. This approach, being based on
expected utility theory, contains the same deficiencies as the underlying
utility theory: it does not take into account emotions and does not explain
behavioural paradoxes. In addition, it contains several fitting parameters
making the approach descriptive but not predictive.

The classical approach, axiomatically formulated by Kolmogorov [119], defines
the probabilities satisfying three axioms, non-negativity, normalization, and
additivity. For behavioural probabilities, in general, it is sufficient to
satisfy only two axioms, non-negativity and normalization, with the additivity
property as a compulsory condition, being dropped out [197, 198, 199, 58, 70,
80, 200]. Throughout the paper, the standard classical probabilities,
sometimes called just probabilities, are distinguished from behavioural
probabilities. To be precise, the basic points of the approach are formulated
in axiomatic way that, although keeping in mind the properties of quantum
behavioural probabilities, at the same time does not involve any quantum
notions explicitly.

Let us consider a ring $\\{A_{n}\\}$ of alternatives $A_{n}$. Assume that
decision making consists in the choice between the alternatives of a set

$\mathbb{A}=\\{A_{n}:~{}n=1,2,\ldots,N_{A}\\}\;.$ (3.1)

Axiom 1. Each alternative $A_{n}$ is equipped with its behavioural probability
$p(A_{n})$, whose family forms a probability measure over the set
$\\{A_{n}\\}$, with the properties of non-negativity and normalization

$\sum_{n=1}^{N_{A}}p(A_{n})=1\;,\qquad 0\leq p(A_{n})\leq 1\;.$ (3.2)

It is assumed that each alternative is decorated by emotions, which for the
sake of notation compactness is not explicitly marked.

Axiom 2. The alternatives are connected by relations defined through the
relations between their behavioural probabilities. The set of alternatives
(3.1) enjoys the following properties.

  1. 1.

Ordering: For any two alternatives $A_{1}$ and $A_{2}$, one of the relations
necessarily holds: either $A_{1}\prec A_{2}$, in the sense that
$p(A_{1})<p(A_{2})$, or $A_{1}\preceq A_{2}$, when $p(A_{1})\leq p(A_{2})$, or
$A_{1}\succ A_{2}$, if $p(A_{1})>p(A_{2})$, or $A_{1}\succeq A_{2}$, when
$p(A_{1})\geq p(A_{2})$, or $A_{1}\sim A_{2}$, if $p(A_{1})=p(A_{2})$.

  2. 2.

Linearity: The relation $A_{1}\preceq A_{2}$ implies $A_{2}\succeq A_{1}$.
These and the relations below are to be understood as relations between the
corresponding probabilities $p(A_{n})$.

  3. 3.

Transitivity: For any three alternatives, such that $A_{1}\preceq A_{2}$, with
$p(A_{1})\leq p(A_{2})$, and $A_{2}\preceq A_{3}$, when $p(A_{2})\leq
p(A_{3})$, it follows that $A_{1}\preceq A_{3}$, in the sense that
$p(A_{1})\leq p(A_{3})$.

  4. 4.

Completeness: In the set of alternatives (3.1), there exist a minimal
$A_{min}$ and a maximal $A_{max}$ elements, for which
$p(A_{min})=\min_{n}p(A_{n})$ and, respectively,
$p(A_{max})=\max_{n}p(A_{n})$.

The ordered set of alternatives (3.1), enjoying these properties, is called a
complete lattice.

Definition 1. An alternative $A_{1}$ is called stochastically preferable to
$A_{2}$ if and only if

$p(A_{1})>p(A_{2})\qquad(A_{1}\succ A_{2})\;.$ (3.3)

Definition 2. Two alternatives are stochastically indifferent if and only if

$p(A_{1})=p(A_{2})\qquad(A_{1}\sim A_{2})\;.$ (3.4)

Definition 3. The alternative $A_{opt}$ is called stochastically optimal if it
corresponds to the maximal behavioural probability,

$p(A_{opt})=\max_{n}p(A_{n})\;.$ (3.5)

Behavioural decision making includes both rational reasoning, following
prescribed logical rules, as well as irrational inclinations not rationalized
by explicit logical argumentation, such as emotions, subconscious guesses, gut
feelings, intuition, etc, all of which we shall, for brevity, call emotions.

Definition 4. Rational reasoning for an alternative $A_{n}$ is described by a
rational fraction named utility factor $f(A_{n})$ that is a classical
probability of choosing an alternative $A_{n}$, being based on rational rules.
A collection of rational fractions for a given set of alternatives forms a
classical probability measure over the set $\\{A_{n}\\}$ with the properties

$\sum_{n=1}^{N_{A}}f(A_{n})=1\;,\qquad 0\leq f(A_{n})\leq 1\;.$ (3.6)

Emotion categories are fuzzy and are labeled with words, expressions, and
metaphors [23, 24, 7]. When comparing the emotions induced by different
alternatives, one cannot quantify them by exact numbers but one can only
characterize them in descriptive terms, e.g., as attractive or repulsive,
pleasant or unpleasant, and like that. Emotion processes do not enjoy clear
categorical boundaries [23, 24, 7].

Definition 5. Emotional impulses in choosing an alternative $A_{n}$ are
characterized by an attraction factor $q(A_{n})$ lying in the interval

$-1\leq q(A_{n})\leq 1\;.$ (3.7)

Axiom 3. Behavioural probability of choosing an alternative $A_{n}$, taking
account of rational reasoning as well as emotion influence, is a functional of
the utility factor $f(A_{n})$ and attraction factor $q(A_{n})$ satisfying the
limiting condition

$p(A_{n})\;\mapsto\;f(A_{n})\;,\qquad q(A_{n})\;\mapsto\;0\;.$ (3.8)

This is an analog of decoherence in quantum theory, when the quantum term,
causing interference, vanishes and the quantum quantity tends to its classical
form.

Axiom 4. Behavioural probability of choosing an alternative $A_{n}$, taking
account of emotions and satisfying the limiting condition (3.8), is the sum

$p(A_{n})=f(A_{n})+q(A_{n})\;.$ (3.9)

From the inequality $0\leq p(A_{n})\leq 1$, it follows that

$-f(A_{n})\leq q(A_{n})\leq 1-f(A_{n})$ (3.10)

in agreement with inequality (3.7).

The value of the rational fraction $f(A_{n})$ shows how useful the alternative
$A_{n}$ is, because of which it is called the utility factor. The magnitude
and sign of the attraction factor $q(A_{n})$ characterize how attractive the
alternative $A_{n}$ is, hence $q(A_{n})$ is termed the attraction factor.

###  3.4 Quantification of utility factor

The utility factor, describing the rational utility of each alternative, has
to be defined by prescribed rules. Here we show how these fractions can be
determined through expected utilities or other value functionals. For
concreteness, we shall be talking about the expected utility of an alternative
$U(A_{n})$, although in the place of the expected utility one can take any
value functional.

Let the alternatives be represented by the lotteries

$A_{n}=\\{x_{i},\;p_{n}(x_{i}):~{}i=1,2,\ldots,N_{n}\\}\;,$ (3.11)

being the probability distributions over the payoff set $\\{x_{i}\\}$ with the
properties

$\sum_{i=1}^{N_{n}}p_{n}(x_{i})=1\;,\qquad 0\leq p_{n}(x_{i})\leq 1\;.$

The probabilities $p_{n}(x)$ can be either objective [135] or subjective
[136]. Following the classical utility theory [135], one can introduce a
dimensionless utility function $u(x)$ and the expected utility

$U(A_{n})=\sum_{i}u(x_{i})p_{n}(x_{i})\;.$ (3.12)

The utility factor reflects the utility of a choice, hence it has to be a
functional of the expected utility. As a functional of the expected utility,
the utility factor has to satisfy the evident conditions

$f(A_{n})\rightarrow 1\;,\qquad U(A_{n})\rightarrow\infty$ (3.13)

and

$f(A_{n})\rightarrow 0\;,\qquad U(A_{n})\rightarrow-\infty\;.$ (3.14)

The utility factor plays the role of classical probability expressed through
the expected utility $U(A_{n})$ or another value functional. The general
approach of defining the probability distribution describing utility factor is
the minimization of an information functional including imposed constraints
[201, 202]. The first such a natural constraint is the normalization condition
(3.6). Another constraint is the existence of a global mean

$\sum_{n}f(A_{n})U(A_{n})=U\qquad(|\;U\;|<\infty)\;.$ (3.15)

The explicit expression for an information functional can be taken in the
Kullback–Leibler form [203, 204, 205]. The Shore-Jonson theorem [205] states
that, given a prior (or trial) probability density $f_{0}(A_{n})$ and
additional constraints, there is only one posterior density $f(A_{n})$
satisfying these constraints and the conditions of uniqueness, coordinate
invariance, and system independence, such that this unique posterior can be
obtained by minimizing the Kullback-Leibler information functional. The
posterior probability $f(A_{n})$ is the minimizer of the Kullback–Leibler
functional provided the imposed constraints do not contain singularities,
which requires a not divergent value of the global mean $U$.

It is important to stress that the existence of the global mean $U$ does not
impose any constraints on the expected utilities $U(A_{n})$ that can be
divergent, as for instance in the St. Petersburg paradox to be considered
below. The existence of the global $U$ is required for the uniqueness of the
probability $f(A_{n})$ in the Shore-Jonson theorem [205].

In the present case, the information functional for the posterior probability
distribution $f(A_{n})$, under a prior distribution $f_{0}(A_{n})$ and the
given constraints, is

$I[\;f(A_{n})\;]=\sum_{n}f(A_{n})\ln\;\frac{f(A_{n})}{f_{0}(A_{n})}+\alpha\left[\;1-\sum_{n}f(A_{n})\;\right]\;+$
$+\;\beta\left[\;U-\sum_{n}f(A_{n})U(A_{n})\;\right]\;,$ (3.16)

where $\alpha$ and $\beta$ are Lagrange multipliers. The minimization of the
information functional (3.16) yields

$f(A_{n})=\frac{f_{0}(A_{n})e^{\beta U(A_{n})}}{\sum_{n}f_{0}(A_{n})e^{\beta
U(A_{n})}}\;.$ (3.17)

The trial distribution $f_{0}(A_{n})$ can be defined employing the Luce rule
[206, 207, 208]. Let the attribute of an expected utility $U(A_{n})$ be
characterized by an attribute value $a_{n}$ assumed to be non-negative. Then,
according to the Luce rule [206, 207, 208], the trial utility factor can be
defined as

$f_{0}(A_{n})=\frac{a_{n}}{\sum_{n=1}^{N_{A}}a_{n}}\qquad(a_{n}\geq 0)\;.$
(3.18)

The attribute value depends on whether the corresponding utility is positive
(semi-positive) or negative. For a semi-positive utility the attribute value
can be defined [209] as

$a_{n}=U(A_{n})\;,\qquad U(A_{n})\geq 0\;,$ (3.19)

while for a negative expected utility it can be given by

$a_{n}=\frac{1}{|\;U(A_{n})\;|}\;,\qquad U(A_{n})<0\;.$ (3.20)

For example, in the case of two lotteries, we have

$f_{0}(A_{n})=\frac{U(A_{n})}{U(A_{1})+U(A_{2})}\;,\qquad U(A_{n})\geq 0\;$
(3.21)

for semi-positive utilities, and

$f_{0}(A_{n})=1-\frac{|\;U(A_{n})\;|}{|\;U(A_{1})\;|+|\;U(A_{2})\;|}\;,\qquad
U(A_{n})<0\;$ (3.22)

for negative utilities.

Definition 6. A lottery $A_{1}$ is more useful than $A_{2}$ if and only if

$f(A_{1})>f(A_{2})\;.$ (3.23)

Definition 7. Two lotteries $A_{1}$ and $A_{2}$ are equally useful if and only
if

$f(A_{1})=f(A_{2})\;.$ (3.24)

As is evident, a lottery can be more useful but not preferable, since the
behavioural probability consists of two terms, including a utility factor and
an attraction factor. Generally, the rational fraction could be taken in a
different form. However the considered Luce form, probably, is the simplest
containing no fitting parameters and sufficient for providing quite reliable
estimates, as will be shown below.

The utility factor (3.17), with the trial distribution (3.18) and attributes
(3.19) and (3.20), for non-negative utilities, reads as

$f(A_{n})=\frac{U(A_{n})e^{\beta U(A_{n})}}{\sum_{n}U(A_{n})e^{\beta
U(A_{n})}}\;;\qquad U(A_{n})\geq 0$ (3.25)

and for negative utilities, as

$f(A_{n})=\frac{|U(A_{n})|^{-1}e^{-\beta|U_{n}|}}{\sum_{n}|U(A_{n})|^{-1}e^{-\beta|U_{n}|}}\;;\qquad
U(A_{n})<0\;.$ (3.26)

The parameter $\beta$, called belief parameter, characterizes the level of
certainty of a decision maker in the fairness of the decision task and in the
subject confidence with respect to his/her understanding of the overall rules
and conditions of the decision problem. The absolute certainty of a decision
maker is characterized by $\beta\rightarrow\infty$, when

$\displaystyle
f(A_{n})=\left\\{\begin{array}[]{ll}1,{}&U(A_{n})=\max_{n}U(A_{n})\\\ \\\
0,{}&U(A_{n})\neq\max_{n}U(A_{n})\end{array}\right.\qquad(\beta\rightarrow\infty)\;,$
(3.30)

so that we return to the standard deterministic decision theory prescribing to