Processing Information in Quantum Decision Theory

This section serves as an introduction to the problem, describing the historical retrospective, related studies, and basic ideas of the quantum approach to decision making. First of all, in order that the reader would not be lost in details, we need to stress the main goal of the approach.

Principal Goal: The principal goal of the quantum approach to decision making is to develop a unified theory that, from one side, could formalize the process of taking decisions by human decision makers in terms of quantum language and, from another side, would suggest a scheme of thinking quantum systems that could be employed for creating artificial intelligence.

Generally, decision theory is concerned with identifying what are the optimal decisions and how to reach them. Traditionally, it is a part of discrete mathematics. Most of decision theory is normative and prescriptive, and assumes that people are fully-informed and rational. These assumptions have been questioned early on with the evidence provided by the Allais paradox [1] and many other behavioral paradoxes [2], showing that humans often seem to deviate from the prescription of rational decision theory due to cognitive and emotion biases. The theories of bounded rationality [3], of behavioral economics and of behavioral finance have attempted to account for these deviations. As reviewed by Machina [4], alternative models of preferences over objectively or subjectively uncertain prospects have attempted to accommodate these systematic departures from the expected utility model, while retaining as much of its analytical power as possible. In particular, non-additive nonlinear probability models have been developed to account for the deviations from objective to subjective probabilities observed in human agents [5,6,7,8,9,10]. However, many paradoxes remain unexplained or are sometimes rationalized on an ad hoc basis, which does not provide much predictive power.

Another approach to decision theory can be proposed, being part of the mathematical theory of Hilbert spaces [11] and employing the mathematical techniques that are used in quantum theory. (see, e.g., the special issue [12] and references therein). However, no self-consistent quantum theory of decision making has been developed, which would have predictive power.

Recently, we introduced a general framework, called the Quantum Decision Theory (QDT), in which decisions involve composite intended actions which, as we explain below, provides a unifying explanation of many paradoxes of classical decision theory in a quantitative predictive manner [13]. Such an approach can be thought of as the mathematically simplest and most natural extension of objective probabilities into nonlinear subjective probabilities. The proposed formalism allows one to explain quantitatively different anomalous phenomena, e.g., the disjunction and conjunction effects. The disjunction effect is the failure of humans to obey the sure-thing principle of classical probability theory. The conjunction effect is a logical fallacy that occurs when people assume that specific conditions are more probable than a single general one. The QDT unearths a deep relationship between the conjunction fallacy and the disjunction effect, the former being sufficient for the latter to exist.

QDT uses the same underlying mathematical structure as the one developed to establish a rigorous formulation of quantum mechanics [14]. Based on the mathematical theory of separable Hilbert spaces on the continuous field of complex numbers, quantum mechanics showed how to reconcile and combine the continuous wave description with the fact that waves are organized in discrete energy packets, called quanta, which behave in a manner similar to particles. Analogously, in the QDT framework, the qualifier quantum emphasizes the fact that a decision is a discrete selection from a large set of entangled options. The key idea of QDT is to provide the simplest generalization of the classical probability theory underlying decision theory, so as to account for the complex dynamics of the many nonlocal hidden variables that may be involved in the cognitive and decision making processes of the brain. The mathematical theory of complex separable Hilbert spaces provides the simplest direct way to avoid dealing with the unknown hidden variables, and at the same time reflecting the complexity of nature [15]. In decision making, unknown states of nature, emotions, and subconscious processes play the role of hidden variables.

Before presenting the QDT approach, it is useful to briefly summarize previous studies of decision making and of the associated cognitive processes of the brain which, superficially, could be considered as related to the QDT approach. This exposition will allow us to underline the originality and uniqueness of the approach. We do not touch here purely physiological aspects of the problem, which are studied in medicine and physiological cognitive sciences. Concerning the functional aspects of the brain, we focus our efforts towards its formal mathematical modeling.

One class of approaches is based on the theory of neural networks and of dynamical systems (see, e.g., [16,17,18,19]). These bottom-up approaches suffer from the obvious difficulties of modeling the emergence of upper mental faculties from a microscopic constructive neuron-based description.

Two main classes of theories invoke the qualifier “quantum”. In the first class, one finds investigations which attempt to represent the brain as a quantum or quantum-like object [20,21,22], for which several mechanisms have been suggested [23,24,25,26,27,28,29]. The existence of genuine quantum effects and the operation of any of these mechanisms in the brain remain however controversial and have been criticized by Tegmark as being unrealistic [30]. Another approach in this first class appeals to the mind-matter duality, treating mind and matter as complementary aspects and considering consciousness as a separate fundamental entity [31,32,33,34]. This allows one, without insisting on the quantum nature of the brain processes, if any, to ascribe quantum properties solely to the consciousness itself, as has been advocated by Stapp [35,36]. Actually, the basic idea that mental processes are similar to quantum-mechanical phenomena goes back to the founder of the old quantum mechanics, Niels Bohr. One of the first publications on this analogy is his paper [37]. Later on, he returned many times to the similarity between quantum mechanics and the function of the brain, for instance in [38,39,40]. This analogy proposes that mental processes could be modeled by a quantum-mechanical wave function, whose evolution would be characterized by a dynamical equation, like the Schrödinger equation.

The second class of theories does not necessarily assume quantum properties of the brain or that consciousness is a separate entity with quantum characteristics. Rather, these approaches use quantum techniques, as a convenient language to generalize classical probability theory. An example is provided by the quantum games [41,42,43,44,45,46,47,48,49,50]. With the development of quantum game theory, it has been shown that many quantum games can be reformulated as classical games by allowing for a more complex game structure [51,52,53,54]. But, in the majority of cases, it is more efficient to play quantum game versions, as less information needs to be exchanged. Another example is the Shor algorithm [55], which is purely quantum-mechanical but is solving the classical factoring problem. This shows that there is no contradiction in using quantum techniques to describe classical problems. Here “classical” is contrasted with “quantum”, in the sense consecrated by decades of discussions on the interpretation of quantum mechanics. In fact, some people go as far as stating that quantum mechanics is nothing but an effective theory describing very complicated classical systems [56,57,58]. Interpretations of this type have been made, e.g., by de Broglie and Bohm. An extensive literature in this direction can be found in de Broglie [59] and Bohm [60]. In any case, whether we deal really with a genuinely quantum system or with an extremely complex classical system, the language of quantum theory can be a convenient effective tool for describing such systems [15]. In the case of decision making performed by real people, the subconscious activity and the underlying emotions, which are difficult to quantify, play the role of the hidden variables appearing in quantum theory.

The QDT belongs to this second class of theories, i.e., we use the construction of complex separable Hilbert spaces as a mathematical language that is convenient for characterizing the complicated processes in the mind, which are associated with decision making. This approach encompasses in a natural way several delicate features of decision making, such as its probabilistic nature, the existence of entangled decisions, the possible non-commutativity of decisions, and the interference between several different decisions. These terms and associated concepts are made operationally clear in the sequel.

As a bonus, the QDT provides natural algorithms which could be used in the future for quantum information processing, the operation of quantum computers, and in creating artificial intelligence.

The classical approaches to decision making are based on utility theory [61,62]. Decision making in the presence of uncertainty about the states of nature is formalized in the statistical decision theory [63,64,65,66,67,68,69,70,71,72,73]. Some problems, occurring in the interpretation of the classical utility theory and its application to real human decision processes have been discussed in numerous literature (e.g., [4,68,74,75]).

Quantum approach to decision making, suggested in Reference [13], is principally different from the classical utility theory. In this approach, the action probability is defined as is done in quantum mechanics, using the mathematical theory of complex separable Hilbert spaces. This proposition can be justified by invoking the following analogy. The probabilistic features of quantum theory can be interpreted as being due to the existence of the so-called nonlocal hidden variables. The dynamical laws of these nonlocal hidden variables could be not merely extremely cumbersome, but even not known at all, similarly to the unspecified states of nature in decision theory. The formalism of quantum theory is then formulated in such a way as to avoid dealing with unknown hidden variables, but at the same time, to reflect the complexity of nature [15]. In decision making, the role of hidden variables is played by unknown states of nature, by emotions, and by subconscious processes, for which quantitative measures are not readily available.

In the following sections, we develop the detailed description of the suggested program, explicitly constructing the action probability in quantum-mechanical terms. The probability of an action is intrinsically subjective, as it must characterize intended actions by human beings. For brevity, an intended action can be called an intention or just an action. And, in compliance with the terminology used in the theories of decision-making, a composite set of intended actions, consisting of several sub-actions, is called a prospect. An important feature of the quantum approach is that, in general, it deals not with separate intended actions, but with composite prospects, including many incorporated intentions. Only then it becomes possible, within the frame of one general theory, to describe a variety of interesting unusual phenomena that have been reported to characterize the decision making properties of real human beings.

The pivotal point of the approach, formalized in QDT, is that mathematically it is based on the von Neumann theory of quantum measurements [14]. The formal relation of the von Neumann measurement theory to quantum information processing has been considered by [76]. QDT generalizes the quantum measurement theory to be applicable not merely to simple actions, but also to composite prospects, which is of paramount importance for the appearance of decision interference. The principal difference of QDT from the measurement theory is the existence of a specific strategic state characterizing each particular decision maker.

A brief account of the axiomatics of QDT has been published in the recent letter [13]. The principal scheme of functioning of a thinking quantum system, imitating the process of decision making, has been advanced [77]. The applicability of the suggested quantum approach for analyzing the phenomena of dynamic inconsistency has been illustrated [78]. The aim of the present survey is to provide a detailed explanation of the theory and to demonstrate that it can be successfully applied to real-life problems of decision making. We also show that the method of conditional entropy maximization, which is equivalent to the minimization of an information functional, yields an explicit relation between the quantum decision theory and the classical decision theory based on the standard notion of expected utility.

In order to formulate in precise mathematical terms the scheme of information processing and decision making in quantum decision theory, it is necessary to introduce several definitions. To better understand these definitions, we shall give some very simple examples, although much more complicated cases can be invented. The entity concerned with the decision making task can be a single human, a group of humans, a society, a computer, or any other system that is able or enables to make decisions. Throughout the paper, for the operations with intended actions, we shall use the notations that are accepted in the literature on decision theory [62,63,64,65,66,67,68,69,70,71,72,73] and for the physical states, we shall employ the Dirac notations widely used in quantum theory [79].

The process of taking decisions implies that one is deliberating between several admissible actions with different outcomes, in order to decide which of the intended actions to choose. Therefore, the first element arising in decision theory is an intended action A.

An intended action which, for brevity, can be called an intention or an action, is a particular thought about doing something. Examples of intentions could be as follows: “I would like to marry” or “I would like to be rich” or “I would like to establish a firm”. There can be a variety of intentions, which we assume to be enumerated by an index $i=1,2,3,\dots ,N$, where the total number N of actions can be finite or infinite.

The whole family of all these actions forms the action set The elements of this set are assumed to be endowed with two binary operations, addition and multiplication, so that, if A and B pertain to $\mathcal{A}$, then $AB$ and $A+B$ also pertain to $\mathcal{A}$. The addition is associative, such that $A+(B+C)=(A+B)+C$, and reversible, in the sense that $A+B=C$ implies $A=C-B$. The multiplication is distributive, $A(B+C)=AB+BC$. The multiplication is not necessarily commutative, so that, generally, $AB$ is not the same as $BA$.

Among the elements of the action set (1), there is an identity action 1, for which $A1=1A=A$. The identity action 1 is not to “do nothing”, since inaction is actually an action. This is well recognized for instance in the field of risk management. Consider for instance the famous quotes: “The man who achieves makes many mistakes, but he never makes the biggest mistake of all—doing nothing” (Benjamin Franklin), or “Life is inherently risky. There is only one big risk you should avoid at all costs, and that is the risk of doing nothing” (Denis Waitley). This also resonates with the standard recommendations in risk management: “If you do not actively attack risks, they will attack you” or “Risk prevention is cheaper than reconstruction.” Thus, “not acting” is not the identity action. We interpret the identity action 1 as the action of keeping running the present action an individual is involved in. For instance, if action A is “to marry someone”, the action $1A$ is to marry someone and to confirm this action. The action $A1$ can be interpreted as first “being open to decide an action” and then to “decide to marry someone”.

And there exists an impossible action 0, for which $A0=0A=0$. Two actions are called disjoint, when their joint action is impossible, giving $AB=BA=0$. The action set (1), with the described structure, is termed the action ring.

We recall that, in mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element (closure property). This closure property here embodies the fact that choosing between alternative actions or combining several actions still correspond to actions.

In the algebra of the elements of the action ring, the meaning of the operations of addition and multiplication is the same as is routinely used in the literature [62,63,64,65,66,67,68,69,70,71,72,73]. The sum $A+B$ means that either the action A or action B is intended to be realized. And the product of actions $AB$ implies that both these actions are to be accomplished together. Instead of writing the sum $A_1+A_2+\cdots$, it is often convenient to use the shorter summation symbol ${\bigcup }_i{A}_i\equiv A_1+A_2+\cdots$, which is also the standard abbreviation. Similarly, for a long product $A_1{A}_2\cdots$, it is convenient to use the shorter notation ${\bigcap }_i{A}_i\equiv A_1{A}_2\cdots$. The use of these standard notations can lead to no confusion.

An action is simple, when it cannot be decomposed into the sum of other actions. An action is composite, when it can be represented as a sum of several other actions. If an action is represented as a sum whose terms are mutually incompatible, then these terms are named the action modes. Here, $M_i$ denotes the number of modes in action $A_i$. The modes correspond to different possible ways of realizing an action. According to the meaning emphasized above, the summation symbol in Equation (2) implies that one of the actions is intended to be realized.

Action representations, or action modes, are concrete implementations of an intended action. For instance, the intention “to marry” can have as representations the following variants: “to marry A” or “to marry B", and so on. The intention “to be rich” can have as representations “to be rich by working hard” or “to be rich by becoming a bandit”. The intention “to establish a firm" can have as representations “to establish a firm producing cars” or “to establish a firm publishing books” and so on. We number all representations of an i-intended action by the index $\mu =1,2,3,\dots$. Note that intention representations may include not only positive intention variants “to do something” but also negative variants such as “not to do something”. For example, Hamlet’s hesitation “to be or not to be” is the intended action consisting of two representations, one positive and the other negative.

Generally, decision taking is not necessarily associated with a choice of just one action among several simple given actions, but it involves a choice between several complex actions. The simplest such complex action is defined as follows. Let the multi-index $n=\{{\nu }_1,{\nu }_2,...,{\nu }_N\}$ be a set of indices enumerating several chosen modes, under the condition that each action is represented only by one of its modes. The elementary prospect is the conjunction of the chosen modes, one for each of the actions from the action ring (1). The total set of all elementary prospects will be denoted as $\left\{e_n\right\}$.

A prospect is composite, when it cannot be represented as an elementary prospect (3). Generally, a composite prospect is a conjunction of several composite actions of form (2), where each of the factors $A_{j_n}$ pertains to the action ring (1). While expression (4) has the similar form as (3), the difference is that the actions $A_{j_n}$ in (4) are composite while the actions $A_{i{\nu }_i}$ in (3) are elementary action modes.

A prospect is a set of several intended actions or several intention representations. In reality, a decision maker is always motivated by a variety of intentions, which are mutually interconnected. Even the realization of a single intention always involves taking into account many other related intended actions.

All possible prospects, among which one needs to make a choice, form a set The set is assumed to be equipped with the binary relations $>,<,=,\ge ,\le$, so that each two prospects ${\pi }_i$ and ${\pi }_j$ in $\mathcal{L}$ are related as either ${\pi }_i>{\pi }_j$, or ${\pi }_i={\pi }_j$, or ${\pi }_i\ge {\pi }_j$, or ${\pi }_i<{\pi }_j$, or ${\pi }_i\le {\pi }_j$. For a while, it is sufficient to assume that such an ordering exists. Then, the ordered set (5) is called a lattice. The explicit ordering procedure associated with decision making will be given below.

To each action mode $A_{i\mu }$, there corresponds the mode state $|A_{i\mu }\rangle$, which is a complex function $\mathcal{A}\to \mathcal{C}$, and its Hermitian conjugate $\langle A_{i\mu }|$. Here we employ the Dirac notation [79]. We assume that a scalar product is defined, such that the mode states, pertaining to the same action, are orthonormalized: The mode space is the closed linear envelope spanning all mode states. By this definition, the mode space, corresponding to an i-action $A_i$, is a Hilbert space of dimensionality $M_i$. The elements of the mode space will be called the intention states.

To each elementary prospect $e_n$, there corresponds the basic state $|e_n\rangle$, which is a complex function ${\mathcal{A}}^N\to \mathcal{C}$, and its Hermitian conjugate $\langle e_n|$. The structure of a basic state is The scalar product is assumed to be defined, such that the basic states are orthonormalized: The mind space is the closed linear envelope spanning all basic states (8). Hence, the mind space is a Hilbert space of dimensionality The vectors of the mind space represent all possible actions and prospects considered by a decision maker.

The family of the basic states forms the mind basis $\left\{\right|e_n>\}$ in the mind space. Different states belonging to the mind basis are assumed to be disjoint, in the sense of being orthogonal. Since the modulus of each state has no special meaning, these states are also normalized to one. This is formalized as the orthonormality of the basis.

To each prospect ${\pi }_j$, there corresponds a state $|{\pi }_j\rangle \in \mathcal{M}$ that is a member of the mind space (10). Hence, the prospect state can be represented as an expansion over the basic states The expansion coefficients in Equation (11) are assumed to be defined by the decision maker, so that $|a_{jn}{|}^2$ gives the weight of the state $|e_n>$ into the general prospect.

The prospects are enumerated with the index $j=1,2,\dots$. The total set $\left\{\right|{\pi }_j>\}$ of all prospect states $|{\pi }_j>$, corresponding to all admissible prospects, forms a subset of the space of mind. The set $\left\{\right|{\pi }_j>\}\subset \mathcal{M}$ can be called the prospect-state set.

The prospect states are not required to be mutually orthogonal and normalized to one, so that the scalar product is not necessarily a Kronecker delta. The normalization condition will be formulated for the prospect probabilities to be defined below.

The fact that different prospect states are not necessarily orthogonal assumes that the related prospects are not necessarily incompatible. The incompatibility is supposed only for the elementary prospects (3), whose states form the basis in the mind space (10) and are orthogonal to each other, according to Equation (9). But an arbitrarily defined composite prospect, generally, is not required to be orthogonal to all other considered prospects. In particular, this can be so, but, in general, we do not need this property.

The prospect states are not normalized to one, since, imposing such a condition would over define these states. The normalization condition will be imposed below on the prospect probabilities. Generally, imposing two normalization conditions could make them inconsistent with each other. So, we need just one normalization condition for the prospect probabilities, which is necessary for the correct definition of the related probability measure.

Being, generally, not orthonormalized, the prospect states do not form a basis in the mind space.

Among the states of the mind space, there exists a special fixed state $|s\rangle \in \mathcal{M}$, playing the role of a reference state, which is termed the strategic state. The strategic state of mind is a fixed vector characterizing a particular decision maker, with his/her beliefs, habits, principals, etc., that is, describing each decision maker as a unique subject. Hence, each space of mind possesses a unique strategic state. Different decision makers possess different strategic states.

Being in the mind space (10), this state can be represented as the decomposition Being a unique state, characterizing each decision maker like its fingerprints, it can be normalized to one:

From Equations (12) and (13), it follows that The existence of the strategic state, uniquely defining each particular decision maker, is the principal point distinguishing an active thinking quantum system from a passive quantum system subject to measurements from an external observer. For a passive quantum system, predictions of the outcome of measurements are performed by summing (averaging) over all possible statistically equivalent states, which can be referred to as a kind of “annealed” situation. In contrast, decisions and observations associated with a thinking quantum system occur in the presence of this unique strategic state, which can be thought of as a kind of fixed “quenched” state. As a consequence, the outcomes of the applications of the quantum-mechanical formalism will thus be different for thinking versus passive quantum systems.

Each prospect state $|{\pi }_j\rangle$, together with its Hermitian conjugate $\langle {\pi }_j|$, defines the prospect operator By this definition, the prospect operator is self-adjoint. The family of all prospect operators forms the involutive bijective algebra that is analogous to the algebra of local observables in quantum theory. Since the prospect states, in general, are neither mutually orthogonal nor normalized, the squared operator contains the scalar product which does not equal to one. This tells us that the prospect operators, generally, are not idempotent, thus, they are not projection operators. It is only when the prospect is elementary that the related prospect operator becomes idempotent and is a projection operator. But, in general, this is not so.

The properties of the prospect operators follow immediately from those of the prospect states and definition (14). Recall that the prospect operators are analogous to the operators of local observables in quantum theory. The latter operators are not required to be idempotent. So, the prospect operators are also not required to be such. The intuition of why the prospect operators are not idempotent could be justified by understanding that, in general, a prospect realized twice results in the consequences that are not necessarily the same as a sole prospect realization. For instance, to marry twice is not the same as to marry once.

In quantum theory, the averages over the system state, for the operators from the algebra of local observables, define the observable quantities. In the same way, the averages, over the strategic state, for the prospect operators define the observable quantities, the prospect probabilities These are assumed to be normalized to one: where the summation is over all prospects from the prospect lattice (5). By their definition, the quantities (15) are non-negative, since Equation (15) reduces to the modulus of the transition amplitude squared The normalization in Equation (16) is necessary for the set $\left\{p\left({\pi }_j\right)\right\}$ be the scalar probability measure. In plane words, the fact that all prospects probabilities are summed to one implies that one of them is to be certainly realized.

The diagonal form plays the role of the expected utility in classical decision making, justifying its name as the utility factor. In order to be generally defined and to be independent of the chosen units of measurement, the utility factor (17) can be normalized as The fact that the utility factor (17) is really equivalent to the classical expected utility follows from noticing that hence Equation (17) acquires the form where $<e_n\left|\widehat{P}\left({\pi }_j\right)\right|e_n>$ plays the role of a utility function, weighted with the probability $|c_n{|}^2$.

The nondiagonal term corresponds to the quantum interference effect. Its appearance is typical of quantum mechanics. Such nondiagonal terms do not occur in classical decision theory. This term can be called the interference factor. Interpreting its meaning in decision making, we can associate its appearance as resulting from the system deliberation between several alternatives, when deciding which of the latter is more attractive. Thence, the name “attraction factor”. Using expansion (12) in Equation (19) yields which shows that the interference occurs between different elementary prospects in the process of considering a composite prospect ${\pi }_j$. It is worth stressing that the interference factor is nonzero only when the prospect ${\pi }_j$ is composite. If it were elementary, say ${\pi }_j=e_k$ then, since we would have and no interference would arise.

Between two prospects, the one which enjoys the larger attraction factor is more attractive.

In defining the prospect lattice (5), we have assumed that the prospects could be ordered. Now, after introducing the scalar probability measure, we are in a position to give an explicit prescription for the prospect ordering. We say that the prospect ${\pi }_1$ is preferable to ${\pi }_2$ if and only if Two prospects are called indifferent if and only if And the prospect ${\pi }_1$ is preferable or indifferent to ${\pi }_2$ if and only if These binary relations provide us with an explicit prospect ordering making the prospect set (5) a lattice.

Since all prospects in the lattice are ordered, it is straightforward to find among them that one enjoying the largest probability. This defines the optimal prospect ${\pi }_{*}$ for which Finding the optimal prospect is the final goal of the decision-making process. Since the prospect probabilities are non-negative, it is possible to find the minimal prospect in the lattice (5) with the smallest probability. And the largest probability defines the optimal prospect ${\pi }_{*}$. Therefore the prospect set (5) is a complete lattice.

Remark. Generally speaking, all states of the mind space can depend on time t. We do not write explicitly the time dependence, when this makes no difference for the considerations developed below. When this is important, we shall denote the time dependence explicitly.

Prospect states can be of two qualitatively different types.

• A disentangled prospect state is a prospect state which is represented as the tensor product of the intention states:

  $$|f>={\otimes }_i|{\psi }_i>.$$

  (24)
• An entangled prospect state is any prospect state that cannot be reduced to the tensor product form of the disentangled prospect states (24).

We define the disentangled set as the collection of all admissible disentangled prospect states of form (24):

In quantum theory, it is possible to construct various entangled and disentangled states (see, e.g., [80,81]). In order to explain how entangled states appear in the quantum theory of decision making, let us illustrate the above definitions by an example of a prospect consisting of two intended actions with two mode representations each. Let us consider the prospect of the following two intentions: “to get married” and “to become rich”. And let us assume that the intention “to get married" consists of two representations, “to marry A”, with the mode state $|A>$, and “to marry B”, with the mode state $|B>$. And let the intention “to become rich” be formed by two representations, “to become rich by working hard”, with the mode state $|W>$, and “to become rich by being a gangster”, with the mode state $|G>$. Thus, there are two intention states of the following type:

The general prospect state has the form where the coefficients $c_{ij}$ belong to the field of complex numbers.

Depending on the values of the coefficients $c_{ij}$, the prospect state (27) can be either disentangled or entangled. If it is disentangled, it must be of the tensor product type (24), which for the present case reads Both states (27) and (28) include four basic states:

• “to marry A and to work hard”, $|AW>$,
• “to marry A and become a gangster”, $|AG>$,
• “to marry B and to work hard”, $|BW>$,
• “to marry B and become a gangster”, $|BG>$.

However, the structure of states (27) and (28) is different. The prospect state (27) is more general and can be reduced to state (28) for special values of the coefficients $c_{ij}$, but the opposite may not be possible. For instance, the prospect state which is a particular example of state (27) cannot be reduced to any of the states (28), provided that both coefficients $c_{12}$ and $c_{21}$ are non-zero. In quantum mechanics, this state would be called the Einstein-Podolsky-Rosen state, one of the most famous examples of an entangled state [82]. Another example is the prospect state whose quantum-mechanical analog would be called the Bell state [83]. In the case where both $c_{11}$ and $c_{22}$ are non-zero, the Bell state cannot be reduced to any of the states (28) and is thus entangled.

In contrast with the above two examples, the prospect states are disentangled, since all of them can be reduced to the form (28).

Other examples of entangled prospect states are where all coefficients are assumed to be non-zero.

Since the coefficients $c_{ij}=c_{ij}\left(t\right)$ are, in general, functions of time, it may happen that a prospect state at a particular time is entangled, but becomes disentangled at another time or, vice versa, a disentangled prospect state can be transformed into an entangled state with changing time [84].

The state of a human being is governed by its physiological characteristics and the available information [85,86]. These properties are continuously changing in time. Hence the strategic state (12), specific of a person at a given time, may also display temporal evolution, according to different homeostatic processes adjusting the individual to the changing environment [87].

The process of quantum decision making possesses several features that make it rather different from the classical decision making. These main features are emphasized below.

Interference is the effect that is typical of all those phenomena which are described by wave equations. Following the Bohr’s idea [37,38,39,40] of describing mental processes in terms of quantum mechanics, one is immediately confronted with the interference effect, since the physical states in quantum mechanics

are characterized by wave functions. The possible occurrence of interference in the problems of decision making has been discussed before on different grounds (see, e.g., [93]). However, no general theory has been suggested, which would explain why and when such a kind of effect would appear, how to predict it, and how to give a quantitative analysis of it that can be compared with empirical observations. In our approach, interference in decision making arises only when one takes a decision involving composite intentions. The corresponding mathematical treatment of these interferences within QDT is presented in the following subsections.

The disjunction effect was first specified by Savage [62] as a violation of the “sure-thing principle”, which can be formulated as follows: If the alternative A is preferred to the alternative B, when an event $X_1$ occurs, and it is also preferred to B, when an event $X_2$ occurs, then A should be preferred to B, when it is not known which of the events, either $X_1$ or $X_2$, has occurred.

The conjunction fallacy constitutes another example revealing that intuitive estimates of probability by human beings do not conform to the standard probability calculus. This effect was first studied by Tversky and Kahneman [109,110] and then discussed in many other works (see, e.g., [101,111,112,113,114]). Despite an extensive debate and numerous attempts to interpret this effect, there seems to be no consensus on the origin of the conjunction fallacy [114].

Here, we show that this effect finds a natural explanation in QDT. It is worth emphasizing that we do not invent a special scheme for this particular effect, but we show that it is a natural consequence of the general theory we have developed. In order to claim to explain the conjunction fallacy in terms of an interference effect in a quantum description of probabilities, it is necessary to derive the quantitative values of the interference terms, amplitudes and signs, as we have done above for the examples illustrating the disjunction effect. This has never been done before. Our QDT provides the necessary ingredients, in terms of the uncertainty-aversion principle, the theorem on interference alternations, and the interference-quarter law. Only the establishment of these general laws can provide an explanation of the conjunction fallacy, that can be taken as a positive step towards validating QDT, according to the general methodology of validating theories [115]. Finally, in our comparison with available experimental data, we analyze a series of experiments and demonstrate that all their data substantiate the validity of the general laws of the theory.