## Abstract

A rigorous general definition of quantum probability is given, which is valid not only for elementary events but also for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting observables in addition to commutative observables. Our proposed definition of quantum probability makes it possible to describe quantum measurements and quantum decision-making on the same common mathematical footing. Conditions are formulated for the case when quantum decision theory reduces to its classical counterpart and for the situation where the use of quantum decision theory is necessary.

## 1. Introduction

A general and mathematically correct definition of quantum probability is necessary for several important applications: theory of quantum measurements, theory of quantum information processing and quantum computing, quantum decision theory (QDT), and creation of artificial quantum intelligence. Although the definition of quantum probability for operationally testable events is well known and used from the beginning of quantum theory [1], such a definition for composite events, corresponding to non-commuting observables, has been a long-standing problem. This problem becomes especially important in the application of the quantum approach to psychological and cognitive sciences, where there exist not only operationally testable events, but also decisions under uncertainty, corresponding to operationally uncertain events. Moreover, for decision-making in real life, decisions under uncertainty are not exceptions, but rather are common typical situations.

Classical decision theory, based on the notion of utility [2], is known to yield numerous paradoxes in realistic decision-making [3]. This is why a variety of quantum models have been suggested for applications in psychological and cognitive sciences, as can be inferred from books [4–7] and reviews [8–10].

Applying quantum theory to psychological and cognitive sciences, researchers have often constructed special models designed specifically to treat particular cases of decision-making. However, to our firm understanding, the theory of quantum decision-making has to be formulated as a general theory valid for arbitrary cases. Moreover, such a theory should have the same mathematical grounds as the theory of quantum measurements. Really, the latter can be interpreted as decision theory [1]. Between measurements and decisions, there is a direct correspondence requiring just a slight language change: measurements correspond to events; operationally testable measurements are analogous to certain events; undefined measurements can be matched to uncertain events; composite measurements are equivalent to composite decisions.

The aim of this paper is to present a general theory, with a unique well-defined mathematical basis, which would be valid for both quantum measurements as well as for quantum decision-making. The main point of such an approach lies in a correct definition of quantum probability that would be applicable for any type of measurements and events, operationally testable or inconclusive, elementary or composite, corresponding to commuting or non-commuting observables. The theory has to be valid for closed as well as for open systems, for individual as well as for social decision-makers. Also, it has to be more general than classical theory, including the latter as a particular case and clearly distinguishing the conditions necessarily requiring the use of quantum techniques and those when the classical approach is sufficient. Finally, it should not be just a descriptive way of modelling, but it must allow for quantitative predictions.

## 2. Main preliminary notions

### (a) Quantum–classical correspondence principle

In order to constrain and anchor the general quantum theory, we require the validity of the quantum–classical correspondence principle. This principle was put forward by Bohr [11,12] for a particular case related to atomic spectra. Later, its applicability was extended to other problems of quantum mechanics, with the Ehrenfest equations being one of the illustrations [13]. Nowadays, this principle is understood in the generalized sense as the requirement that classical theory be a particular case of quantum theory [14]. In the present context, it implies that the theory of quantum measurements should include the theory of classical measurements, that QDT should include classical decision theory, and that classical probability should be a particular case of quantum probability.

In what follows, we use the term event, implying that this can be an event in decision theory or probability theory, or the result of a measurement in the quantum theory of measurements.

### (b) Quantum logic of events

The algebra of events is prescribed by quantum logic [15]. Events form an event ring possessing two binary operations, addition and conjunction. Addition is such that for any , there exists with the properties: Conjunction means that for any , there exists satisfying the properties: But, generally, conjunction is not commutative and not distributive:

The fact that distributivity is absent in quantum logic was emphasized by Birkhoff & von Neumann [15], who illustrated this by the following example. Suppose there are two events *B*_{1} and *B*_{2} that, when combined, form unity, *B*_{1}∪*B*_{2}=1. Moreover, *B*_{1} and *B*_{2} are such that each of them is orthogonal to a non-trivial event *A*≠0, hence *A*∩*B*_{1}=*A*∩*B*_{2}=0. According to this definition, *A*∩(*B*_{1}∪*B*_{2})=*A*∩1=*A*. But if the property of distributivity was true, then one would get (*A*∩*B*_{1})∪(*A*∩*B*_{2})=0. This implies that *A*=0, which contradicts the assumption that *A*≠0.

It is easy to illustrate the concept of non-distributivity in quantum physics by numerous examples. The simplest of these is as follows [16]. Let us measure the spin projection of a particle with spin 1/2. Let *B*_{1} be the event of measuring the spin in the up state with respect to the *x*-axis, whereas *B*_{2} is the event of measuring the spin in the down state along this axis. Because the spin can be either up or down, *B*_{1}∪*B*_{2}=1. Let *A* be the event of measuring the spin along an axis in the plane orthogonal to the *z*-axis. According to the rules of quantum mechanics, 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 at the same time. Hence, *A*∩*B*_{1}=*A*∩*B*_{2}=0, whereas *A*∩(*B*_{1}∪*B*_{2})≠0. Therefore, there is no distributivity of events in the spin measurement.

Thus, the non-distributivity of events is an important concept that should not be forgotten in applying quantum theory to cognitive sciences.

### (c) Decision-maker state

In quantum theory, systems can be closed or open. Respectively, their states can be described by wave functions or as statistical operators. How should one interpret the state of a decision-maker, as a wave function or as a statistical operator? Such a state, characterizing the given decision-maker, can be called a strategic decision-maker state [17–19].

Recall the notion of an isolated system in quantum theory. Strictly speaking, quantum systems cannot be absolutely isolated, but can only be quasi-isolated [20,21], which means the following. At initial time *t*=0, one can prepare a system in a pure state described by a wave function. However, there always exist uncontrollable external perturbations or noise from the surrounding, resulting in system decoherence beyond a time *t*_{dec}, which makes the system state mixed. In addition, to confirm that the considered system is to some extent isolated, it is necessary to check this by additional control measurements starting at time *t*_{con}, which again disturbs the system's isolation. In this way, one can assume that the system is quasi-isolated during the interval of time .

Decision-makers, generally, 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 and to characterize them by wave functions. This, however, is not correct. First of all, 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 interacting with many people before as well as his/her expectations of future interactions, which influences his/her decisions. From the physiological point of view, memory is nothing but delayed interactions. Therefore, no decision-maker can be treated as an isolated system, which excludes the validity of using a wave function description. The correct treatment of any decision-maker requires to consider him/her as an open system, hence, characterized by a statistical operator.

### (d) Operationally testable events

In the theory of quantum measurements or QDT, the simplest case occurs when one deals with a simple event corresponding to a single measurement, or a single action. Observable quantities in quantum theory are represented by self-adjoint operators, say , from the algebra of local observables. Measuring an eigenvalue *A*_{n} of the operator can be interpreted as the occurrence of an event *A*_{n}. The corresponding eigenvector |*n*〉 is termed a microstate in physics, or event mode in decision theory. Here and in what follows, the family of eigenvectors is assumed to be orthonormalized. Respectively, the operator is a measurement projector in physics, or an event operator in decision theory. The collection is a projector-valued measure.

The space of microstates, or the space of decision modes, is given by the Hilbert space
2.1
The considered quantum system state, or decision-maker strategic state, is characterized by a statistical operator . The pair is a statistical ensemble, or decision ensemble. The probability of measuring an eigenvalue *A*_{n}, or the probability of an event *A*_{n}, is given by the formula
2.2
where the trace operation is over space (2.1). This probability is uniquely defined for any Hilbert space (2.1) of dimensionality larger than two [22].

### (e) Problem of degenerate spectrum

The spectrum of the considered operator can happen to be degenerate, which implies that a single eigenvalue *A*_{n} corresponds to several eigenvectors |*n*_{j}〉, with *j*=1,2,…. Does this create any problem?

This is not a problem in quantum measurements. In the case of degeneracy, one introduces a projector
2.3
so that the probability of measuring *A*_{n} becomes
2.4

Degeneracy may seem to be an annoyance in decision theory. Really, if *A*_{n} is a degenerate event related to a degenerate spectrum, then what would be the meaning of the different modes associated with the same event? It is necessary to ascribe some meaning to these different modes; otherwise, the situation will be ambiguous.

Fortunately, the problem of degeneracy is easily avoidable, both in physics as well as in decision theory. In physics, degeneracy can be lifted by switching on arbitrarily weak external fields. In decision theory, this would correspond to reclassifying the events by adding small differences between the events. Mathematically, the procedure of lifting degeneracy is done by adding to the considered operator of an observable an infinitesimally small term breaking the symmetry that caused the degeneracy, which means the replacement 2.5

The related eigenvalues *A*_{nj}+*νΓ*_{nj} become non-degenerate. Then, the probability of each subevent can be defined as
2.6
Such a procedure of degeneracy lifting was mentioned by von Neumann [1] for quantum systems and developed as the method of quasi-averages by Bogolubov [23,24] for statistical systems.

In any case, neither in physics nor in decision theory is the problem of spectrum degeneracy actually a principal problem. One just needs to either ascribe a meaning to different modes of an event, or one can avoid the problem completely by lifting the degeneracy, which corresponds to a reclassification of events, as already mentioned. The latter way is preferable in decision theory, because it avoids the ambiguity in dealing with unspecified degeneracy.

### (f) Consecutive quantum measurements

In quantum theory, one considers the possibility of measuring two observables immediately one after the other. The standard treatment of this process is as follows. Suppose, first, that one accomplishes a measurement for an observable represented by an operator , with eigenvalues *B*_{α} and eigenvectors |*α*〉. The event *B*_{α} is represented by the projector . One assumes that, immediately after measuring *B*_{α}, the system state reduces from to the state
2.7

Immediately after the first measurement, one accomplishes a measurement for an observable represented by an operator , with eigenvalues *A*_{n} and eigenvectors |*n*〉. The event *A*_{n} is represented by the projector .

The probability of these consecutive measurements is the Lüders [25] probability 2.8 also called von Neumann–Lüders probability.

By introducing the Wigner [26] probability
2.9
one comes to the relation
2.10
This formula is reminiscent of the relation between the joint probability of two events and the conditional probability for these events. Because of this similarity, one interprets the Wigner probability *p*_{W} as a joint probability and the Lüders probability *p*_{L} as a conditional probability.

However, by direct calculations, assuming non-degenerate events, we have 2.11 This form is symmetric with respect to the interchange of events. Therefore, the Lüders probability cannot be treated as the generalization of the classical conditional probability that is not necessarily symmetric. Respectively, the Wigner probability cannot be considered as a joint probability of two events [27].

One could think that, by invoking degenerate events, it would be possible to avoid the problem. Suppose the events *A*_{n} and *B*_{α} are degenerate, so that their projectors are
2.12

Then, we have 2.13 Interchanging the events yields 2.14 This shows that the Lüders probability, generally, is not symmetric for degenerate events.

But let us remember the quantum–classical correspondence principle, according to which classical theory has to be a particular case of quantum theory. In classical theory, the field of events is commutative. In quantum theory, commuting observables share the same family of eigenvectors. This can be formulated as the property 〈*α*_{i}|*n*_{j}〉=*δ*_{ij}*δ*_{αβ}. Then, passing to commutative events, for the Lüders probability (2.8) we obtain
2.15
This is not merely symmetric, but even trivial. Contrary to this, classical conditional probabilities are neither symmetric nor trivial.

Thus, the quantum–classical correspondence principle does not hold, which means that the Lüders probability in no way should be accepted as a generalization of classical conditional probability. The Lüders probability is just a transition probability. If one wishes, one can use it as a transition probability in the frame of a narrow class of physical measurements. However, it is not a conditional probability in the general sense, and its use as such for cognitive sciences is not correct [27–29].

It is worth mentioning that the Kirkwood [30] form also cannot be accepted as a probability, because it is complex-valued.

Concluding this section, we stress that the standard von Neumann–Lüders transition probability cannot be treated as a generalization of classical conditional probability to the quantum region, because it does not satisfy the quantum–classical correspondence principle. The consideration of degenerate events does not save the situation.

### (g) Realistic measurement procedure

The problem with the von Neumann–Lüders probability lies in its oversimplified nature, giving only a cartoon of the much more complicated procedure of realistic measurements. This cartoon ignores the existence and influence of a measuring device, it ignores the finite time of any measurement, and it ignores that during measurements and between them, the system evolves. The correct description of a realistic measurement procedure is as follows [27].

Let us assume that we are interested in measuring two observables corresponding to the operators and , with eigenvalues *A*_{n} and *B*_{α} and eigenvectors |*n*〉 and |*α*〉, respectively. The related event representations are
2.16
According to equation (2.1), the corresponding mode spaces are
2.17

To measure anything, one needs a measuring device, whose internal states are the vectors of a Hilbert space . In decision theory, this state corresponds to internal states of a decision-maker. The total space, containing all possible microstates, is the tensor-product space 2.18

The measurement procedure consists of several channels. The first step of any measurement is the preparation of the device for measurement, which can be represented by the entangling channel
2.19
describing the formation from initial partial states, during the preparation time *t*_{1}, of an entangled total state of the system plus the measuring device.

Before the measurement starts, the total state evolves until time *t*_{2}, according to the channel
2.20
where
with being the evolution operator.

In the interval of time [*t*_{2},*t*_{3}], one measures the observable corresponding to the operator , which is described by the partially disentangling channel
2.21
where
Disentangling or separating from the total state is necessary for measuring the values related to the operator of the observable . According to the standard definition, separating a subsystem implies tracing out all other degrees of freedom, except those of the considered subsystem.

Then, until time *t*_{4}, the system again is getting entangled by the evolution channel
2.22
where

Finally, in the interval of time [*t*_{4},*t*_{5}], one accomplishes a measurement of the observable associated with the operator , which is characterized by the partially disentangling channel
2.23
where

Summarizing, the process of measurement of two observables is a procedure represented by the channel convolution 2.24 and consisting of five steps: 2.25 The evolution channels are unitary but entangling, whereas the measurement channels are disentangling but non-unitary. The measurement channels are non-unitary because they involve the trace operation that cannot be represented by a unitary operator. The realistic measurement procedure is more complicated than the von Neumann–Lüders scheme and, generally, cannot be reduced to the latter even if the involved intervals of time are rather short.

## 3. Joint quantum probability

### (a) Channel–state duality

As is explained in §2f, the von Neumann–Lüders scheme does not provide a general definition of conditional quantum probabilities and therefore does not lead to correct joint quantum probabilities. This is due to the fact that a realistic measurement procedure requires the five-step convolution channels described in the previous section. This multichannel measurement procedure looks quite complicated. Fortunately, there exists the Choi–Jamiolkowski [31,32] isomorphism establishing the channel–state duality 3.1 with a state defined on the Hilbert space 3.2 Thus, instead of dealing with the channel convolution, we can equivalently consider the composite state characterized by the space of microstates (3.2).

### (b) Prospects as composite events

Using the channel–state duality, we can interpret the measurement of two observables, or the occurrence of two events, as a composite event. For instance, let us consider events *A* and *B*. The corresponding composite event, called prospect, is , which is represented by the tensor product of two event operators as
3.3
with the event operators .

The joint probability of the prospect composed of two events is 3.4 This definition is employed from the beginning of the development of our approach named QDT [8,17–19,33,34]. We use the term prospect for a composite event, because when applying the QDT to decision-making, we calculate the classical part of the quantum probability by invoking the notion of utility [33–35].

### (c) Conditional quantum probabilities

Having defined the joint probability of events, it is straightforward to introduce the conditional probabilities 3.5 with the marginal probabilities 3.6 Here, and are unity operators in the corresponding spaces. Clearly, the conditional probabilities, in general, are not symmetric.

Note that this definition of conditional probabilities is self-consistent and does not meet the problem of connecting conditional and joint probabilities, as in the case when conditional probabilities are defined through the Lüders form [36].

### (d) Separable and entangled prospects

The property of entanglement is important for both quantum measurements as well as for quantum decision-making [37,38]. There are two types of prospects that qualitatively differ from each other, separable and entangled, whose rigorous definition is given below.

Let be an algebra of local observables defined on a Hilbert space ℋ_{A}. For any two operators and from , it is possible to introduce the scalar product by the rule
3.7
This scalar product generates the Hilbert–Schmidt norm . The triple of the algebra of observables , acting on the Hilbert space , and the above scalar product *σ*_{A} compose a Hilbert–Schmidt space
3.8

Let us introduce a composite Hilbert–Schmidt space by the tensor-product space 3.9 An operator in space (3.9) is called separable if and only if 3.10 while it is entangled if and only if it cannot be represented in the separable form: 3.11

Prospects, being composite events, are represented, in view of equation (3.3), by composite event operators. The structure of the prospect operators depends on how a composite Hilbert–Schmidt space is defined. Generally, the prospect operators can be separable or entangled. Then, the related prospects can also be termed separable or entangled.

It is easy to give an example of a separable prospect. Let the algebras and be composed of the corresponding projectors and . The prospect is represented by the relation 3.12 Here, the prospect operator is clearly separable. Hence, the prospect is called separable. Its probability is 3.13

In contrast, entangled prospects appear when measurements or decision-making are accomplished under uncertainty.

### (e) Measurements and decisions under uncertainty

An inconclusive event is a set *B*={*B*_{α}:*α*=1,2,…} that is represented by a vector |*B*〉 of a Hilbert space, such that
3.14
with the event operator
3.15
In quantum measurements, an inconclusive event implies that after a measurement there is no single measured value, but the result is a set of possible data *B*_{α} weighted with |*b*_{α}|^{2}. In that sense, it is not a certain operationally testable event. In decision-making, an inconclusive decision means that an exact decision is not yet actually taken, but it rather describes the process of deliberation between several possibilities, in that sense being an incomplete decision.

Let us emphasize that an inconclusive event is not a union. This is because an inconclusive event is represented as 3.16 while a union is represented by the relation 3.17 Therefore, the corresponding event operators are very different.

One may say that an inconclusive event, being not uniquely operationally testable, cannot be the final stage of a measurement or decision-making. But inconclusive events can occur, and often do exist, at intermediate stages of measurements and decisions. Actually, this is a typical situation for decisions under uncertainty. There are many such illustrations in the processes of physical measurement [27,38,39] as well as in decision-making [8,19,33,34].

A typical prospect, describing a measurement or decision under uncertainty, has the form
3.18
where the final event *A*_{n} is operationally testable, and *B*={*B*_{α}} is an intermediate inconclusive event. This prospect is represented by the prospect state according to the relation
3.19
and induces the related prospect operator,
3.20
The explicit form of the latter is
3.21

The prospect states |*π*_{n}〉 are not necessarily orthonormalized. Therefore, a prospect operator, generally, is not idempotent, because
3.22
Hence, it is not a projector. But the resolution of unity is required:
3.23
where is a unity operator in space (3.2). The family of the prospect operators forms a positive operator-valued measure [27,40].

The projectors and represent operationally testable events. Because of this, the algebras of observables are defined as the collections of these projectors. Thus, we have two algebras of observables 3.24 acting on the Hilbert spaces and , respectively. With these algebras of observables in mind, we construct the Hilbert–Schmidt space (3.9). Then, analysing the prospect operator (3.21), which can be written as 3.25 we see that this operator is entangled, because, although the first term is separable, the second term here is entangled. That is, prospect (3.18) is also called entangled.

## 4. Probability of uncertain prospects

Suppose we consider several prospects forming a lattice
4.1
The probability of a prospect is given by the quantum form
4.2
By construction, the probability is non-negative and normalized,
4.3
so that the family {*p*(*π*_{n})} is a probability measure.

With the prospect operator (3.25), it is straightforward to see that the prospect probability can be written as a sum of two terms:
4.4
The first term *f*(*π*_{n}) contains the diagonal part of equation (3.25). It describes the objective utility of the prospect, because of which it is called the utility factor. The second term *q*(*π*_{n}) is composed of the non-diagonal part of equation (3.25) caused by the quantum nature of the probability. From the quantum theory point of view, this term can be specified as an interference or coherence term. In decision theory, it characterizes subjective and subconscious feelings of the decision-maker, and can be named the attraction factor [8,19,33,34].

It is worth stressing that form (4.4) is not an assumption, but is the direct consequence of the definition for the prospect probability (4.2), with the prospect operator (3.25).

By the quantum–classical correspondence principle, when the quantum term becomes zero, the quantum probability reduces to the classical probability, so that 4.5 with the normalization 4.6 In quantum theory, this is called decoherence.

The attraction factor, by its construction, enjoys the following properties [8,17,19,33,34,41]. It lies in the range 4.7 and satisfies the alternation law 4.8 This law follows immediately from the form of probability (4.4), under the normalization equations (4.3) and (4.6).

For a large class of distributions, there exists the quarter law
4.9
The latter allows us to use as a non-informative prior the value |*q*(*π*_{n})|≈0.25, which makes it possible to give quantitative predictions.

Employing the definition of the conditional probability
4.10
for a prospect with an uncertain event *B*, we have
4.11

The use of quantum probabilities is required when the quantum term *q*(*π*_{n}) is not zero. As is clear from the above consideration, the necessary condition for this is the occurrence of decisions under uncertainty. More precisely, the following theorem is proved [27].

### Theorem 4.1

*For the quantum term q*(*π*_{n}) *to be non-zero, it is necessary that the corresponding prospect π*_{n} *be entangled and also the decision-maker state* *be entangled.*

In the case of decisions under uncertainty, the prospect probability (4.4) consists of two terms, utility factor and attraction factor. It is therefore possible to classify the prospects from the given lattice (4.1) in three ways. A prospect *π*_{1} is more useful than *π*_{2}, if and only if *f*(*π*_{1})>*f*(*π*_{2}). A prospect *π*_{1} is more attractive than *π*_{2}, if and only if *q*(*π*_{1})>*q*(*π*_{2}). A prospect *π*_{1} is preferable to *π*_{2}, if and only if *p*(*π*_{1})>*p*(*π*_{2}). In that way, a prospect can be more useful, but less attractive, as a result being less preferable, which explains all paradoxes in classical decision-making [8,19,33,34].

Let us stress that the principal difference of our approach in decision theory, from all other models involving quantum techniques, is the possibility to not merely qualitatively interpret empirical results, but, moreover, to give their quantitative description. As an example, let us briefly mention the Prisoner Dilemma game, where there are two prisoners who can either cooperate or defect (see details in [42–44]). Let *C*_{n} denote cooperation, whereas *D*_{n} defection. In our terminology, there are four separable prospects: , , and . The aim is to study the entangled uncertain prospects
corresponding to the choice between cooperation and defection for one of the prisoners, without knowing the decision of the other one. Empirical results of experiments, accomplished by Tversky & Shafir [45], yield *p*(*π*_{1})=0.37 and *p*(*π*_{2})=0.63. In our approach, using the prior attraction factor ±0.25, we get *p*(*π*_{1})=0.35 and *p*(*π*_{2})=0.65, which, with the given experimental accuracy, coincides with the empirical data. A detailed description of this example can be found in [27,46].

The prospect probabilities depend on the amount of available information. This happens because the decision-maker strategic state depends on this information. Let the information measure be denoted as *μ*. The decision-maker states with this information and without it are respectively and . By the Kadison [47] theorem, statistical operators, parametrized by a single parameter, are connected by means of a unitary operator as
4.12
The prospect probability, with information *μ*, is
4.13
Following the above consideration, we find that this probability is generally the sum of two terms:
4.14
The first term, that is, the utility factor characterizes the prospect utility, and is not influenced by additional information, provided the utility is objectively defined. But the attraction factor, which is subjective, does depend on the available information. Employing the techniques used for treating the evolution of quantum systems [48,49], it is possible to show [41] that the attraction factor decreases with the received additional information approximately as
4.15
where *μ*_{c} is the critical amount of information, after which the quantum term strongly decays.

The dependence of the attraction factor on the given information can explain the effect of preference reversal. This effect was noted by Tversky & Thaler [50], who illustrated it by the following example. Imagine that people are asked to decide, under given conditions, between two programmes, say *A* and *B*. It may happen that they choose *B* because it looks more useful. Then, additional information is provided characterizing the cost of these programmes. After getting this additional information, people choose *A* instead of *B*, thus demonstrating preference reversal. This effect is closely related to the planning paradox [19]. More detailed investigation of the preference reversal will be presented in a separate paper.

## 5. Conclusion

We have demonstrated the main mathematical points of a theory treating on the same grounds both quantum measurements as well as quantum decision-making. The quantum joint and conditional probabilities have been introduced, being valid for arbitrary events, elementary as well as composite, operationally testable, as well as inconclusive, for commutative observables, as well as for non-commuting observables. The necessity of treating decision makers as members of a society was emphasized. A pivotal point of the approach is the validity of the quantum–classical correspondence principle that provides a criterion for constructing a correct and self-consistent theory. The necessary conditions requiring the use of the quantum approach have been formulated. It was shown how additional information influences decision-making. The developed QDT does not meet paradoxes typical of classical decision-making and, moreover, makes it possible to give quantitative predictions.

## Authors' contributions

The authors equally contributed to the manuscript. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

Financial support from the Swiss National Foundation is appreciated.

## Acknowledgements

One of the authors (V.I.Y.) is grateful to E.P. Yukalova for discussions.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

- Accepted July 1, 2015.

- © 2015 The Author(s)