Tossing Quantum Coins and Dice

V.I. Yukalov1,2

1Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research,
Dubna 141980, Russia
E-mail<EMAIL_ADDRESS>
and
2Instituto de Fisica de São Carlos, Universidade de São Paulo,
CP 369, São Carlos 13560-970, São Paulo, Brazil

###### Abstract

The procedure of tossing quantum coins and dice is described. This case is an
important example of a quantum procedure because it presents a typical
framework employed in quantum information processing and quantum computing.
The emphasis is on the clarification of the difference between quantum and
classical conditional probabilities. These probabilities are designed for
characterizing different systems, either quantum or classical, and they,
generally, cannot be reduced to each other. Thus the Lüders probability cannot
be treated as a generalization of the classical conditional probability. The
analogies between quantum theory of measurements and quantum decision theory
are elucidated.

Keywords: quantum coins; quantum dice; quantum and classical probabilities;
quantum conditional probability; quantum decision theory

##  1 Introduction

Reformulation of classical processes into the quantum language is an
interesting and important trend in the recent endeavour of applying quantum
techniques to the problems of quantum information processing, quantum
computing, and quantum cognition [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18]. Quantum devices provide essentially more operational
power as compared to their classical counterparts. The widespread point of
view is that a quantum process is a straightforward generalization of a
classical process. However the situation is much more complicated. In some
cases, it is really a straightforward generalization, although in other cases
it is just a different process, but not a generalization in a strict sense.
Quite often, quantum and classical processes are so much different that the
former cannot be reduced to the latter in the sense that the latter is not a
particular case of the former.

The aim of this paper is to illustrate the said above by the example of a
quantum process translating the well known classical tossing of coins and dice
to the quantum language. Since this procedure allows for a clear cut
formulation in quantum terms, it enjoys three advantages. First, it serves as
an explicit example of how a classical process can be converted into the
corresponding quantum counterpart. Second, it demonstrates the difference
between quantum and classical procedures. At the same time, it explains the
principal difference between quantum and classical probabilities, especially
between conditional probabilities. The last, but not the least, this example
elucidates the relation between the theory of quantum measurements and quantum
decision theory.

The tossing of quantum coins or dice can be realized with different physical
objects, for instance with photons with left and right polarizations or with
particle spins having several projections. Coin flipping is often treated as a
cryptographic problem in which a pair of remote distrustful parties must
generate a random bit that they agree on [19, 20]. The present paper considers
other issues. One is the problem of repeated coin or dice tossing and the
relation between classical and quantum conditional probabilities. We show that
the von Neumann-Lüders probability cannot be accepted as a generalization of
the classical conditional probability. The other problem is the influence of
quantum noise on the results of measurements. Drawing a parallel between the
mathematical techniques of quantum measurement theory and quantum decision
theory makes it straightforward to develop a way of taking into account
irrational feelings imitating quantum random noise of measurements.

##  2 Quantum Coins and Dice

Quantum objects are usually defined in the frame of a Hilbert space
$\mathcal{H}$ that can be represented as a closed linear envelope over a
basis,

${\cal H}={\rm span}\\{|\;j\;\rangle\\}\;,$ (1)

where $j$ is a labeling index. Here and in what follows, the Dirac bracket
(bra-ket) notation is used [21, 22]. A quantum system is associated with a
statistical operator $\hat{\rho}$ which is a semi-positive trace-one operator
acting on ${\cal H}$. The statistical operator is also called the system
quantum state or simply the system state. The pair $\\{{\cal H},\hat{\rho}\\}$
forms a quantum statistical ensemble. Physical quantities are represented by
the algebra of local observables $\mathcal{A}$ that is a von Neumann algebra
on ${\cal H}$. Von Neumann algebras are self-adjoint, strong-operator closed
subalgebras (containing the identity operator) of the algebra of all bounded
operators on a Hilbert space. Observables compose a set of averages of the
operators from $\mathcal{A}$, termed the normal state that is given by the
trace

$\langle{\cal A}\rangle={\rm Tr}\;\hat{\rho}\;{\cal A}\;,$ (2)

where the trace operation is over $\mathcal{H}$. A quantum system is a quantum
statistical ensemble and the von Neumann algebra of observables $\\{{\cal
H},\hat{\rho},{\cal A}\\}$. An important subclass of $\mathcal{A}$ is the
projection ring $\mathcal{P}(\mathcal{A})$ of all projection operators
belonging to $\mathcal{A}$.

A subset of projection operators from $\mathcal{P}(\mathcal{A})$, forming an
orthomodular lattice, composes a projection-valued measure, provided a
probability measure is defined on this projection lattice. A quantum
statistical ensemble and a projection-valued measure form a quantum
probability space $\\{\mathcal{H},\hat{\rho},\mathcal{P}\\}$. Details of the
general description of the related constructions can be found in the papers
[23, 24, 25, 26, 27]. Here we aim at considering a concrete case associated
with the toss of coins and dice.

Let the observable of interest be represented by an operator
$\hat{A}\in\mathcal{A}$. The operators of observables are self-adjoint and
possess real-valued eigenvalues given by the eigenproblem

$\hat{A}\;|\;A_{n}\;\rangle=A_{n}\;|\;A_{n}\;\rangle\;.$ (3)

The eigenvectors $|A_{n}\rangle\in\mathcal{H}$ can always be orthonormalized
so that

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

We keep in mind a nondegenerate case because of two reasons. First, this
situation better suits the studied process of tossing, where each outcome is a
simple uniquely defined event. Second, the occurrence of degeneracy just
slightly complicates the consideration without changing conclusions [28].

The index $n$ enumerates the available alternatives, taking the values
$n=1,2,\ldots,N_{A}$. In the application to coins, an event $A_{n}$, with
$n=1,2$ signifies the occurrence of either heads or tails, while in the case
of a standard dice, $A_{n}$, with $n=1,2,3,4,5,6$, are the numbers coming up
on the sides of a die. There exist also multi-sided dice with $n$ varying from
$1$ to $120$. The projection-valued measure $\mathcal{P}(\hat{A})$ is composed
of the projection operators

$\hat{P}(A_{n})=|\;A_{n}\;\rangle\langle\;A_{n}\;|\;.$ (4)

The operator of an observable can be represented as

$\hat{A}=\sum_{n}A_{n}\hat{P}(A_{n})\;.$ (5)

The probability measure is given by the average

$p(A_{n})=\langle\;\hat{P}(A_{n})\;\rangle={\rm
Tr}\;\hat{\rho}\;\hat{P}(A_{n})\;$ (6)

and enjoys the normalization

$\sum_{n}\;p(A_{n})=1\;,\qquad 0\leq p(A_{n})\leq 1\;.$ (7)

With the projection operator (4), probability (6) takes the form

$p(A_{n})=\langle\;A_{n}\;|\;\hat{\rho}\;|\;A_{n}\;\rangle\;.$ (8)

Note that probability (6) is the a priori or predicted probability.

We need to compare result (8) with the outcomes of tossing classical coins or
dice, where we shall denote the classical probability of an event $A_{n}$ by
$f(A_{n})$ in order to distinguish it from the quantum probability $p(A_{n})$.
As we know, a fair coin or dice give $f(A_{n})=1/N_{A}$. So, generally, the
quantum and classical probabilities are different. However, it is possible to
point out a special case, when the quantum probability reduces to the
classical one. This is when the system state is pure, so that

$\hat{\rho}=|\;\psi\;\rangle\langle\;\psi\;|\qquad(|\;\psi\;\rangle\in{\cal
H})\;,$

and has the specially prepared wave function

$|\;\psi\;\rangle=\sum_{n}\;\frac{1}{\sqrt{N_{A}}}\;|\;A_{n}\;\rangle\;.$

Under this condition, $p(A_{n})=f(A_{n})=1/N_{A}$.

Since under special explicitly formulated conditions, the quantum probability
$p(A_{n})$ reduces to the classical probability $f(A_{n})$, one says that the
quantum probability is a generalization of the classical probability.

##  3 Probability of Repeated Tosses

The situation is more intricate in the case of repeated tosses of a coin or a
die, when one tries to define quantum conditional probabilities. The correct
consideration of repeated measurements requires a description of processes
developing in time [18]. Then the probability of an event $A_{n}$ occurring at
time $t$ writes as

$p(A_{n},t)={\rm Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n})\;.$ (9)

Dependence of the system state on time is regulated by the law

$\hat{\rho}(t)=\hat{U}(t,0)\;\hat{\rho}(0)\;\hat{U}^{+}(t,0)\;,$ (10)

where $\hat{U}(t,0)$ is the unitary evolution operator describing the
evolution of the initial state $\hat{\rho}(0)$ at time $t=0$ to a finite time
$t>0$. The a priori probability of an event $A_{n}$ in the interval $0\leq
t<t_{0}$ is given by expression (9).

Suppose at the moment of time $t_{0}$ there appears a new information on the
event $A_{n}$, because of which the system starts a new dynamics with the
changed initial condition,

$\hat{\rho}(t_{0}-0)\longmapsto\hat{\rho}(A_{n},t_{0})\;.$ (11)

This change is termed state reduction, although its meaning is just the
postulation of the new initial condition caused by the received information.

The new dynamics for time $t\geq t_{0}$ is regulated by the evolution law

$\hat{\rho}(A_{n},t)=\hat{U}(t,t_{0})\;\hat{\rho}(A_{n},t_{0})\;\hat{U}^{+}(t,t_{0})\;.$
(12)

Therefore the subsequent occurrence of an event $A_{m}$ at time $t\geq t_{0}$
happens with the probability

$p(A_{m},t\;|\;A_{n},t_{0})={\rm Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(A_{m})\;.$
(13)

The operator $\hat{U}(t,t_{0})$ is the evolution operator corresponding to the
considered quantum system. For instance, a quantum coin can be represented by
an electron with the spin either up or down. This electron has to be kept
inside a quantum system by means of trapping potentials and external fields.
During the time after a measurement, the state of the electron varies
depending on the concrete used trapping potentials and applied fields. These
concrete potentials and fields define the explicit form of the evolution
operator.

If the new information on the event $A_{n}$ consists in the knowledge that at
the time $t_{0}$ this event has certainly occurred, then one has the
requirement

$p(A_{n},t_{0}\;|\;A_{n},t_{0})=1\;,$ (14)

which implies

${\rm Tr}\;\hat{\rho}(A_{n},t_{0})\;\hat{P}(A_{n})=1\;.$ (15)

The solution to the latter equation, suggested by the von Neumann and Lüders,
[29, 30] is the state

$\hat{\rho}(A_{n},t_{0})=\frac{\hat{P}(A_{n})\hat{\rho}(t_{0}-0)\hat{P}(A_{n})}{{\rm
Tr}\hat{\rho}(t_{0}-0)\hat{P}(A_{n})}\;.$ (16)

By the direct substitution, it is straightforward to check that the von
Neumann-Lüders state (16) is really a solution to equation (15). This solution
is general in the sense of being valid for arbitrary $A_{n}$ and
$\hat{\rho}(t_{0}-0)$, under the initial condition change (11).

As an illustration of the state reduction, let us consider the case where
before the initial condition change (11) there was a pure state

$\hat{\rho}(t_{0}-0)=|\;\psi(t_{0})\;\rangle\langle\;\psi(t_{0})\;|\;.$

Then in view of the equalities

$\hat{P}(A_{n})\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n})=|\;\langle\;A_{n}\;|\;\psi(t_{0})\;\rangle\;|^{2}\;\hat{P}(A_{n})\;,\qquad{\rm
Tr}\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n})=|\;\langle\;A_{n}\;|\;\psi(t_{0})\;\rangle\;|^{2}\;,$

we get the pure state

$\hat{\rho}(A_{n},t_{0})=|\;A_{n}\;\rangle\langle\;A_{n}\;|\;.$

The probability (13) of a subsequent event $A_{m}$, occurring at a time
$t>t_{0}$, reads as

$p(A_{m},t|A_{n},t_{0})=\frac{{\rm
Tr}\hat{U}(t,t_{0})\hat{P}(A_{n})\hat{\rho}(t_{0}-0)\hat{P}(A_{n})\hat{U}^{+}(t,t_{0})\hat{P}(A_{m})}{{\rm
Tr}\hat{\rho}(t_{0}-0)\hat{P}(A_{m})}\;.$

The latter expression depends on the details of the concrete physical system,
with its trapping potentials and fields, chosen for representing the quantum
coin or dice, which define the explicit form of the evolution operator. Some
simplification of the expression can be achieved in the case of nondestructive
measurements [31, 32, 33, 34, 35, 36], when the matrix elements of the
evolution operator can be represented as [18]

$\langle\;m\;|\;\hat{U}(t,t_{0})\;|\;n\;\rangle=\delta_{mn}\exp\left\\{-i\int_{t_{0}}^{t}E_{n}(t^{\prime})\;dt^{\prime}\right\\}\;.$

In quantum theory, one considers successive measurements accomplished
immediately one after the other, when

$\lim_{t\rightarrow
t_{0}+0}\;\langle\;m\;|\;\hat{U}(t,t_{0})\;|\;n\;\rangle=\delta_{mn}\;.$

It is exactly this case that is usually treated [29, 30], since then the
consideration does not depend on the details of the evolution operator.

The probability of observing an event $A_{m}$ at time $t_{0}+0$, immediately
after the event $A_{n}$ has certainly happened at time $t_{0}$, becomes

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})={\rm
Tr}\;\hat{\rho}(A_{n},t_{0})\;\hat{P}(A_{m})\;,$ (17)

or explicitly

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})=\frac{{\rm
Tr}\;\hat{\rho}(t_{0}-0)\hat{P}(A_{n})\hat{P}(A_{m})\hat{P}(A_{n})}{{\rm
Tr}\;\hat{\rho}(t_{0}-0)\hat{P}(A_{n})}\;.$ (18)

This expression is termed the Lüders probability.

Since $\hat{P}(A_{m})$ and $\hat{P}(A_{n})$ commute with each other, we have

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})=\frac{{\rm
Tr}\;\hat{\rho}(t_{0}-0)\hat{P}(A_{m})\hat{P}(A_{n})}{{\rm
Tr}\;\hat{\rho}(t_{0}-0)\hat{P}(A_{n})}\;.$ (19)

This relation reminds us the expression for a classical conditional
probability. However this impression is misleading, since the left-hand and
right-hand sides of this relation are defined for different times, but not for
the same time as it should be for the correct definition of a conditional
probability. Really, in the left-hand side, we see an a posteriori quantity
determined at the time $t_{0}+0$, after the event $A_{n}$ has happened, while
in the right-hand side, we have the a priori probabilities ascribed to the
time $t_{0}-0$, before any event has occurred.

The explicit form of the Lüders probability can be easily found taking into
account that

${\rm
Tr}\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n})\;\hat{P}(A_{m})\;\hat{P}(A_{n})=|\;\langle\;A_{m}\;|\;A_{n}\;\rangle\;|^{2}\;{\rm
Tr}\;\hat{\rho}(t_{0}-0)\;\hat{P}(A_{n})\;.$

Therefore the Lüders probability becomes

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})=|\;\langle\;A_{m}\;|\;A_{n}\;\rangle\;|^{2}\;,$
(20)

which is a transition probability between the states $|A_{m}\rangle$ and
$|A_{n}\rangle$. Moreover, in view of the property $\langle
A_{m}|A_{n}\rangle=\delta_{mn}$, we get

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})=\delta_{mn}\;.$ (21)

Thus the Lüders probability for the case of tossing quantum coins or dice is
either one or zero,

$p(A_{n},t_{0}+0\;|\;A_{n},t_{0})=1\;,$
$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})=0\qquad(m\neq n)\;.$ (22)

This result is in the drastic difference with the case of tossing classical
coins or dice, when the conditional probability has the form

$f(A_{m}\;|\;A_{n})=\frac{f(A_{m}\cap A_{n})}{f(A_{n})}\;.$ (23)

Keeping in mind that different toss-ups are independent, hence

$f(A_{m}\cap A_{n})=f(A_{m})f(A_{n})\;,$ (24)

we come to the value

$f(A_{m}\;|\;A_{n})=f(A_{m})=\frac{1}{N_{A}}\;.$ (25)

As is evident, the quantum Lüders probability (21) in no way can be reduced to
the classical conditional probability (25). In that sense, the Lüders
probability cannot be named a generalization of a classical conditional
probability. The latter and the Lüders probability are different quantities
describing very different systems, a quantum system and a classical system.
The Lüders probability, contrary to the classical conditional probability,
possesses the following distinct properties.

(i) It is defined for different times, connecting an a posteriori expression
in the left-hand side with a priori probabilities in the right-hand side of
relation (19).

(ii) It does not satisfy the statistical independence rule, similar to the
classical property (25).

(iii) It cannot be reduced to the classical expression (25).

This result teaches us that, resorting to the Lüders procedure for the
interpretation of probabilistic events, one should not forget the true meaning
behind this procedure. This procedure is designed for describing a two-step
process of a repeated quantum measurement. The first measurement occurs for an
arbitrary given initial state of the system and involves no information on the
future second measurement. While the second measurement is accomplished over
the quantum state transformed by the first measurement. Any formal operations
neglecting this true meaning of the Lüders procedure have no sense.

The considered simple example of tossing quantum coins or dice demonstrates
that the classical conditional probabilities and the quantum probabilities
defined by the Lüders rule are principally different. Even for commutative
operators, when the Lüders probability formally reminds the classical
conditional probability, their meanings and values remain very distinct. The
Lüders probability is not a generalization of the classical conditional
probability simply because the former cannot be reduced to the latter. And
this is not surprising: quantum and classical systems have very different
properties, so that their characteristics are not obliged to identically
coincide. The basic difference between quantum and classical cases is that a
quantum measurement essentially changes the state of the considered quantum
system, while a measurement in the classical setup has no influence on the
system.

It is important to stress that Eqs. (21) and (22) are in agreement with the
principle of measurement reproducibility in quantum theory [29] according to
which: The second measurement, accomplished immediately after the first, has
to reproduce the result of the latter. No such principle exists for classical
systems.

##  4 Alternation of Coins and Dice

It is useful to consider a bit more general case, when the toss of a coin is
alternated with that of a die, or when two different types of dice are tossed
in turn. This means that, after tossing one object, say a coin, and getting
one of the events $A_{n}$, with $n=1,2,\ldots,N_{A}$, we toss another object,
say a die, with possible toss-up events $B_{k}$, where $k=1,2,\ldots,N_{B}$.

Similarly to the previous consideration, for the operator of an observable
$\hat{B}\in{\cal A}$ we have the eigenproblem

$\hat{B}\;|\;B_{k}\;\rangle=\hat{B}_{k}\;|\;B_{k}\;\rangle\;,$ (26)

with the orthonormalized eigenvectors,

$\langle\;B_{k}\;|\;B_{p}\;\rangle=\delta_{kp}\;.$

The related projection operators are

$\hat{P}(B_{k})=|\;B_{k}\;\rangle\langle\;B_{k}\;|\;.$ (27)

By assumption, in the time interval $0\leq t<t_{0}$, the system evolved as in
Eq. (10). At the moment of time $t_{0}$, the event $A_{n}$ was observed, so
that the system started a new dynamics with the initial condition (16) for the
system state. The quantum statistical ensemble, corresponding to the new
dynamics, is now given by the pair $\\{\mathcal{H},\hat{\rho}(A_{n},t)\\}$.
The projection-valued measure $\mathcal{P}(\hat{B})$ consists of the
projection operators (27). The quantum probability space, characterizing
measurements with respect to the observable $\hat{B}$, is formed by the new
statistical ensemble and the projection-valued measure $\mathcal{P}(\hat{B})$,
that is, this probability space is $\\{{\cal
H},\hat{\rho}(A_{n},t),\mathcal{P}(\hat{B})\\}$. In that way, the probability
of observing an event $B_{k}$, after the event $A_{n}$ has been certainly
observed, is

$p(B_{k},t\;|\;A_{n},t_{0})={\rm Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(B_{k})\;.$
(28)

Accomplishing the observation for $\hat{B}$, immediately after $A_{n}$ was
observed at the moment of time $t_{0}$, defines the probability

$p(B_{k},t_{0}+0\;|\;A_{n},t_{0})={\rm
Tr}\;\hat{\rho}(A_{n},t_{0})\;\hat{P}(B_{k})\;,$ (29)

which is the Lüders probability

$p(B_{k},t_{0}+0\;|\;A_{n},t_{0})=\frac{{\rm
Tr}\hat{\rho}(t_{0}-0)\hat{P}(A_{n})\hat{P}(B_{k})\hat{P}(A_{n})}{{\rm
Tr}\hat{\rho}(t_{0}-0)\hat{P}(A_{n})}\;.$ (30)

Substituting here the related projectors yields

$p(B_{k},t_{0}+0\;|\;A_{n},t_{0})=|\;\langle\;B_{k}\;|\;A_{n}\;\rangle\;|^{2}\;.$
(31)

The obtained Lüders probability has the meaning of the transition probability
from a quantum state $|A_{n}\rangle$ to the state $|B_{k}\rangle$. The
operators $\hat{A}$ and $\hat{B}$ do not necessarily commute with each other.
Their associated eigenvectors, in general, are not orthogonal. Hence,
depending on the final and initial quantum states, the Lüders probability (31)
can be any number between $0$ and $1$. The transition probability is symmetric
with respect to the interchange of the initial and final states, such that

$p(B_{k},t_{0}+0\;|\;A_{n},t_{0})=p(A_{n},t_{0}+0\;|\;B_{k},t_{0})\;.$ (32)

Commuting operators enjoy the same eigenvectors, hence for commuting operators
the probability (32) reduces to $\delta_{nk}$.

The corresponding classical conditional probability

$f(B_{k}\;|\;A_{n})=f(B_{k})=\frac{1}{N_{B}}$ (33)

does not depend on the initial quantum state. Also, the classical conditional
probability is not symmetric, since

$f(A_{n}\;|\;B_{k})=f(A_{n})=\frac{1}{N_{A}}\;,$ (34)

where $N_{A}$ does not equal $N_{B}$.

Thus again we come to the conclusion that the Lüders probability cannot be
reduced to the classical conditional probability, hence it cannot be named a
generalization for the latter, in agreement with the earlier results [28, 37].
These probabilities are both correct in the regions of their applicability.
But they are principally different, being designed for very different systems,
either quantum or classical. A single quantum measurement can be reduced to a
single classical measurement, but several consecutive quantum measurements, in
the frame of the standard quantum-mechanical approach, are not reducible to
consecutive classical measurements. The root of this basic difference lies in
the fact that a quantum measurement essentially changes the system state,
producing the so-called state reduction, while a classical measurement has no
influence on the studied system.

##  5 Degenerate Coins or Dice

As is shown in the previous section, the Lüders probability is symmetric with
respect to the interchange of events, provided the quantum coins and dice are
described by non-degenerate quantum models. Such a reciprocal symmetry,
generally, does not happen for the classical conditional probability, as well
as contradicts experimental observations [38, 39]. The reciprocal symmetry can
be broken when the considered objects are characterized by degenerate models.

The quantum degeneracy implies that some eigenvalues of the studied operator
are associated with several eigenvectors. Let us illustrate the degenerate
case for an operator $\hat{B}$ for which the eigenproblem is degenerate, when

$\hat{B}\;|\;B_{k\alpha}\;\rangle=\hat{B}_{k}\;|\;B_{k\alpha}\;\rangle\qquad(\alpha=1,2,\ldots)\;.$
(35)

The eigenvectors corresponding to the same eigenvalue can be orthonormalized
by means of the Gram-Schmidt orthogonalization, so that

$\langle\;\hat{B}_{k\alpha}\;|\;B_{k\beta}\;\rangle=\delta_{\alpha\beta}\;.$

The degenerate projection operator takes the form

$\hat{P}(B_{k})=\sum_{\alpha}\hat{P}(B_{k\alpha})\;,$ (36)

where

$\hat{P}(B_{k\alpha})=|\;B_{k\alpha}\;\rangle\langle\;\hat{B}_{k\alpha}\;|\;.$

The Lüders probability of a degenerate event $B_{k}$, occurring after a non-
degenerate event $A_{n}$ has happened, is

$p(B_{k},t_{0}+0\;|\;A_{n},t_{0})=\sum_{\alpha}|\;\langle\;B_{k\alpha}\;|\;A_{n}\;\rangle\;|^{2}\;.$
(37)

Reversing the order of the events yields

$p(A_{n},t_{0}+0\;|\;B_{k},t_{0})=\frac{\sum_{\alpha\beta}\langle\;B_{k\alpha}\;|\;\hat{\rho}(t_{0}-0)\;|\;B_{k\beta}\;\rangle\langle\;B_{k\beta}\;|\;A_{n}\;\rangle\langle\;A_{n}\;|\;B_{k\alpha}\;\rangle}{\sum_{\alpha}\langle\;B_{k\alpha}\;|\;\hat{\rho}(t_{0}-0)\;|\;B_{k\alpha}\;\rangle}\;.$
(38)

As is clear, generally, there is no reciprocal symmetry for these
probabilities.

However, breaking the reciprocal symmetry in no way makes it clear how the
Lüders probabilities could be reduced to the classical conditional
probabilities. The use of degenerate models brings the following problems.

First, the meaning of degeneracy for an event not always can be defined. For
instance, how a coin could be degenerate? There is too much of ambiguity in
possible interpretations of the degeneracy and in its quantification.

Second, probabilities (33) and (34) are defined solely by the properties of
the last events, while probabilities (37) and (38) depend on both conditional
and conditioned events.

Third, trying to reduce a quantum object to a classical one by introducing
even more quantum features looks absurd.

Finally, if there exists at least one example where the quantum Lüders
probability cannot in principle be reduced to a classical conditional
probability, then the former cannot be treated as the generalization of the
latter.

It is necessary to accept that the Lüders probability and the classical
conditional probability are basically different notions. No one of them is a
generalization of the other. Each of them allows for a correct mathematical
definition and is valid for different classes of problems, one for quantum,
the other for classical systems. The origin of the principal difference is the
state reduction in the case of quantum theory, but its absence in classical
theory. The quantum state reduction does not exist for classical systems.
Moreover, the principally quantum notion of state reduction cannot be
translated into the classical language.

##  6 State Reduction in Decision Theory

Theory of quantum measurements can be straightforwardly reinterpreted as
quantum decision theory [28, 37, 40]. However, it seems that the above
analysis puts doubt on the possibility of using quantum techniques in decision
theory because it looks quite unreasonable to imagine that the state of a
decision maker, after a decision has been made, completely collapses to the
state describing just that particular decision. Then, after each taken
decision, a human subject would be drastically changing, reminding not a self-
consistent personality but a kind of a weather vane spontaneously turning
after each decision.

The problem lies in the fact that in the standard picture, considered above,
only the space of alternatives has been taken into account. However any
observation consists not merely of a state of alternatives, but involves as
well measuring instruments, including observers [18]. In cognitive sciences,
the role of measuring instruments plus an observer is played by a decision
maker, with the related subject space of mind that can be represented by a
Hilbert space

${\cal H}_{S}={\rm span}\\{\;|\;\alpha\;\rangle\;\\}\;.$ (39)

This space characterizes the invariant features of a subject, actually making
him/her an individual with more or less invariant properties. When considering
the process of decision making with respect to the set of alternatives
$\\{A_{n}\\}$, the subject space of mind has to be extended by including the
state of alternatives

${\cal H}_{A}={\rm span}\\{\;|\;A_{n}\;\rangle\;\\}\;.$ (40)

Thus the total system decision space becomes

${\cal H}={\cal H}_{S}\bigotimes{\cal H}_{A}\;.$ (41)

The system state is given by a statistical operator $\hat{\rho}(t)$ acting on
the decision space (41).

To explain better the difference of the subject space of mind ${\cal H}_{S}$
and the space of alternatives ${\cal H}_{A}$, let us take the situation
typical of tossing coins or dice. The space of alternatives describes the
possible outcomes of tossing. An event $A_{n}$ of getting one of the outcomes
is represented by the vector $|A_{n}\rangle$ in the space of alternatives.
However, the space of outcomes of tossing does not constitute the whole player
who possesses a variety of other interests, feelings, and planned actions that
characterize the player as a human individual. All these additional attributes
describing the player as an individual can be represented by vectors in the
subject space of mind.

When an event $A_{n}$, corresponding to the observation of the related outcome
of tossing, occurs, the state in the space of alternatives reduces. For
instance, as is explained in Sec. 3, if the state before the observation were
pure, say $|\psi(t_{0})\rangle\langle\psi(t_{0})|$, after the observation it
reduces to the state $|A_{n}\rangle\langle A_{n}|$. However this does not mean
that the whole player as a separate personality has suffered from the
reduction to another personality. This only implies the reduction in the space
of the considered alternatives ${\cal H}_{A}$, but not in the subject space of
mind ${\cal H}_{S}$ that can stay intact. The player remains the same person
almost independently of the tossing outcome, although in some life situations
the result of tossing can influence the state of the player, for instance
changing his/her mood or influencing the planned actions. Vice versa, the
subjective feelings and emotions of a player can influence the results of
tossing. The interrelation between the subject space of mind and the space of
alternatives will be more in detail explained in Section 10.

The process of tossing coins or dice is characterized by the probability
$p(A_{n})$ of observing an outcome $A_{n}$. In the frequentist picture, this
is equivalent to saying that the fraction of people who get the outcome
$A_{n}$ is $p(A_{n})$. This probability is

$p(A_{n},t)={\rm Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n})={\rm
Tr}_{A}\;\hat{\rho}_{A}(t)\;\hat{P}(A_{n})\;,$ (42)

with the state in the space of alternatives

$\hat{\rho}_{A}(t)={\rm Tr}_{S}\;\hat{\rho}(t)\equiv{\rm Tr}_{{\cal
H}_{S}}\;\hat{\rho}(t)\;.$ (43)

The subject state of mind is

$\hat{\rho}_{S}(t)={\rm Tr}_{A}\;\hat{\rho}(t)\equiv{\rm Tr}_{{\cal
H}_{A}}\;\hat{\rho}(t)\;.$ (44)

The certain occurrence of an event $A_{n}$ at time $t_{0}$ implies the
condition

$p(A_{n},t_{0}+0\;|\;A_{n},t_{0})={\rm
Tr}_{A}\;\hat{\rho}_{A}(A_{n},t_{0})\;\hat{P}(A_{n})=1$ (45)

and the reduction

$\hat{\rho}_{A}(t_{0}-0)\longmapsto\hat{\rho}_{A}(A_{n},t_{0})$ (46)

of the state of alternatives to the state

$\hat{\rho}_{A}(A_{n},t_{0})=\frac{\hat{P}(A_{n})\;\hat{\rho}_{A}(t_{0}-0)\;\hat{P}(A_{n})}{{\rm
Tr}_{A}\;\hat{\rho}_{A}(t_{0}-0)\;\hat{P}(A_{n})}\;.$ (47)

In this way, the state reduction happens only in the space of alternatives
(40), while there is no reduction in the subject space (39). Therefore the
individual invariance of the decision maker is left intact. To better
illustrate this, let us consider the system state in the form

$\hat{\rho}(t)=\hat{\rho}_{S}(t)\bigotimes\hat{\rho}_{A}(t)\;,$ (48)

which implies the absence of correlations between the space of alternatives
and the subject space. Then the reduction of the state of alternatives (46)
results in the reduction of the total system state

$\hat{\rho}(t_{0}-0)\longmapsto\rho_{S}(t_{0})\bigotimes\hat{\rho}_{A}(A_{n},t_{0})\;.$
(49)

As we see, the state reduction touches only the state of alternatives, while
the subject state is left invariant.

When the state reduction occurs only in the space of alternatives, but the
decision maker, as an individual, remains invariant, this means that the total
system state changes not much, thus preserving a decision maker as a
personality. Hence this removes contradictions in applying quantum techniques
to cognitive sciences.

If a subject decides with respect to the same set of alternatives twice, so
that the second decision follows immediately after the first one, then it is
not surprising if the decision is the same. Really, we have

$p(A_{m},t_{0}+0\;|\;A_{n},t_{0})={\rm
Tr}_{A}\;\hat{\rho}_{A}(A_{n},t_{0})\hat{P}(A_{m})=\delta_{mn}\;.$ (50)

The quantum state reduction, leading to this result, does not contradict
realistic cognitive processes. This is because it is not the subject who
collapses but the reduction happens only in the space of alternatives, while
the subject, as an individual, stays invariant.

##  7 Correlated and Separated Measurements

Although expression (50) finds a reasonable interpretation in the frame of
decision making, nevertheless it is not clear whether there exist such
conditions when formula (50) could be turned into (25), or more generally,
when a quantum conditional probability of two events could be reduced to the
classical form (33) and (34). At this point, we again have to recollect that
measurements or decision processes develop in time [18, 28, 37]. Let one
measurement (decision) be accomplished at time $t_{0}$. Before another
measurement (decision) could be accomplished, some time necessarily elapses,
when the first measurement is being fixed, the system evolves, suffers from
the disturbance of measuring devices, from the influence of surrounding, and
from the actions of observers preparing another measurement. Let us denote
this total time of evolution and preparation as $\tau$. So, in reality, two
measurements (decisions) are always separated from each other by a finite time
$\tau$ that we can name the separation time. When one talks about a second
measurement (decision) occurring immediately after the first measurement, this
just implies that the separation time $\tau$ is in some sense small.

To give a meaning to the characteristic “small”, we need to compare $\tau$
with a typical time of system variation. This typical time is the relaxation
time $t_{rel}$ during which the system, being perturbed, returns to its
original state. The role of perturbation is played by the process of taking a
decision or accomplishing a measurement.

There can happen two qualitatively different situations, when the separation
time is much smaller than the relaxation time $(\tau\ll t_{rel})$ and the
opposite case, when $\tau\gg t_{rel}$. If the separation time is much smaller
than the relaxation time, then the process of the first measurement (decision)
cannot be separated from the second one. In that case both measurements
(decisions) have to be treated, actually, as one double strongly correlated
measurement consisting of two parts. Exactly this situation has been assumed
in the previous sections when considering the occurrence of a second event
immediately after the first one. The opposite case, when the separation time
is much longer than the relaxation time, can be treated as two separate
measurements.

The case, where $\tau\gg t_{rel}$, can be treated in two ways. One way is by
invoking the exact evolution [18] of the state $\hat{\rho}(t)$, with taking
into account that for a large separation time, the perturbation, imposed on
the subject by the first measurement (decision), fades out. Then the
probability of observing an event $B_{k}$ at the time $t_{0}+\tau$ is not much
influenced by the measurement occurred at $t_{0}$. In that case,

$p(B_{k},t_{0}+\tau\;|\;A_{n},t_{0})\cong p(B_{k},t_{0}+\tau)\;.$ (51)

The other equivalent way is based on the channel-state duality [28, 41, 42]
and can be realized as follows. Suppose at the moment of time $t_{0}$, one
measures (takes a decision) with respect to an observable $\hat{A}$ and, after
a separation time $\tau$, one attempts performing a second measurement with
respect to $\hat{B}$. The separation time is assumed to be larger than the
typical relaxation time, $\tau\gg t_{rel}$. In this situation, we have two
separate measurements. Then the decision space

${\cal H}={\cal H}_{S}\bigotimes{\cal H}_{A}\bigotimes{\cal H}_{B}$ (52)

can be represented as composed of the subject space (39), the space (40) of
alternatives $A_{n}$, and the space of alternatives $B_{k}$,

${\cal H}_{B}={\rm span}\;\\{\;|\;B_{k}\;\rangle\;\\}\;.$ (53)

For $t<t_{0}$, the probability of $A_{n}$ is given by expression (42) acting
on ${\cal H}_{A}$, where

$\hat{\rho}_{A}(t)={\rm Tr}_{SB}\;\hat{\rho}(t)\;,$ (54)

and the trace operation is over ${\cal H}_{S}\bigotimes{\cal H}_{B}$.

Similarly, we can define the state of alternatives $B_{k}$, acting on ${\cal
H}_{B}$,

$\hat{\rho}_{B}(t)={\rm Tr}_{SA}\;\hat{\rho}(t)\;,$ (55)

with the trace over ${\cal H}_{S}\bigotimes{\cal H}_{A}$, and the subject
state acting on ${\cal H}_{S}$,

$\hat{\rho}_{S}(t)={\rm Tr}_{AB}\;\hat{\rho}(t)\;,$ (56)

with the trace over $\mathcal{H_{A}}\bigotimes\mathcal{H_{B}}$.

If at the moment of time $t_{0}$ an event $A_{n}$ has certainly happened, then
the state in the space of alternatives $\mathcal{H_{A}}$ is reduced according
to the rule

$\hat{\rho}_{A}(t_{0}-0)={\rm Tr}_{SB}\;\hat{\rho}(t_{0}-0)\longmapsto{\rm
Tr}_{SB}\;\hat{\rho}(A_{n},t_{0})=\hat{\rho}_{A}(A_{n},t_{0})\;.$ (57)

The conditional probability of observing an event $B_{k}$, after $A_{n}$ has
been observed, becomes

$p(B_{k},t_{0}+\tau\;|\;A_{n},t_{0})={\rm
Tr}\;\hat{\rho}(A_{n},t_{0}+\tau)\;\hat{P}(B_{k})={\rm
Tr}_{B}\;\hat{\rho}_{B}(t_{0}+\tau)\;\hat{P}(B_{k})\;.$ (58)

If the order of events is reversed, then we have the conditional probability

$p(A_{n},t_{0}+\tau\;|\;B_{k},t_{0})={\rm
Tr}\;\hat{\rho}(B_{k},t_{0}+\tau)\;\hat{P}(A_{n})={\rm
Tr}_{A}\;\hat{\rho}_{A}(t_{0}+\tau)\;\hat{P}(A_{n})\;.$ (59)

Equations (58) and (59) can be represented as

$p(B_{k},t_{0}+\tau\;|\;A_{n},t_{0})=p(B_{k},t_{0}+\tau)\;,\qquad
p(A_{n},t_{0}+\tau\;|\;B_{k},t_{0})=p(A_{n},t_{0}+\tau)\;,$ (60)

which have the form of the classical conditional probabilities (33) and (34)
describing the probability of tossing coins or dice. In this way, the
reduction of the quantum conditional probabilities to the related classical
probabilities of tossing is possible, provided the quantum conditional
probabilities describe weakly correlated events. This becomes possible due to
the consideration of the process in the composite space (52) characterizing
the quasi-independence of separate events. Strictly speaking, the
interdependence of the events can enter through the common state
$\hat{\rho}(t)$. The events turn into completely independent when

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

The event independence is caused by a long separation time $\tau$, during
which the correlation between different measurements disappears.

Thus, for a very short separation time $\tau\ll t_{rel}$, two measurements are
strongly correlated and have to be treated as a single twin measurement
realized in the space (41), while for a large separation time $\tau\gg
t_{rel}$, the measurements are weakly correlated becoming quasi-independent
and characterized by the space (52). In the intermediate case, it is necessary
to study the overall evolution of the system state in time [18].

Now we are in a position of summarizing why the Lüders probabilities (21) or
(22) drastically differ from the classical conditional probabilities (25) and
why the Lüders probability (31) contradicts the classical conditional
probabilities (33) and (34). This happens when one forgets that the correct
description of either measurements or of decision processes requires the
consideration of temporal evolution. The evolution can be qualitatively
different depending on the interrelation between the time $\tau$, separating
two different measurements (decisions), and the relaxation time $t_{rel}$,
during which the perturbations, caused by the measurement (decision) process,
attenuate.

The Lüders probability is applicable to the case of two consecutive
measurements (decisions) separated in time by an interval $\tau\ll t_{rel}$,
while in the formulation of classical conditional probabilities one usually
assumes that $\tau$ is either greater or comparable with $t_{rel}$. For
instance, the implementation of two coin tosses requires quite sufficient time
between them, because of which the classical tosses become independent of each
other. Contrary to this, imaginary quantum coins or dice can be tossed one
after another, assumed to be separated by an arbitrarily short time. Actually,
quantum coins and dice are nothing but quantum bits of information that can be
realized with different quantum objects, such as spins, photons, and atoms
[43].

##  8 General Dynamics of Measurements

In the previous section, it is explained that the properties of quantum
probability are different, depending on the relation between the separation
and relaxation times. Now we summarize the general description of the
measurement dynamics for an arbitrary interrelation between these typical
times.

As has been mentioned above, the mathematics of realizing measurement
procedures, event observation, and decision making are practically the same
[44, 45], because of which it is admissible to use interchangeably either the
terminology from quantum measurement theory or from quantum decision theory.
In the interpretation of occurring events, the chance of getting an
alternative $A_{n}$ in a measurement, of course, has a different meaning as
compared to the procedure of deciding to choose an alternative $A_{n}$.
However in both the cases, the mathematics of describing the process are
similar. Thus in measurements, we get an alternative $A_{n}$ with a
probability $p(A_{n})$, and in decision making, an alternative $A_{n}$ is
chosen with a probability $p(A_{n})$. In other words, one can say that in
measurements, an alternative $A_{n}$ is found in the fraction $p(A_{n})$ of
the realized measurements. While in decision theory, $p(A_{n})$ is the
fraction of subjects choosing the alternative $A_{n}$. In that way, the
procedure of describing quantum measurements is mathematically similar to the
process of decision making [18, 28, 37, 44, 46].

A correct definition of quantum probabilities requires, first of all, to
introduce a quantum probability space. Generally, this space can be different
for the varied stages of dynamics, if the latter are interrupted by
measurements. Suppose, at the first stage, lasting from $t=0$ till the time
$t_{0}$, when an event is observed, we are interested in studying the
probabilities related to the operational variable $\hat{A}$. The decision
space

${\cal H}={\cal H}_{S}\bigotimes{\cal H}_{A}$ (61)

is composed of the subject space $\mathcal{H}_{S}$ and the space
$\mathcal{H}_{A}$ of alternatives $\\{A_{n}\\}$. The system state (statistical
operator) evolves according to the rule

$\hat{\rho}(t)=\hat{U}(t,0)\;\hat{\rho}(0)\;\hat{U}^{+}(t,0)\;.$ (62)

The operator measure $\mathcal{P}(\hat{A})$ is formed of the projection
operators $\hat{P}(A_{n})$. The quantum probability space is

$\\{{\cal H},\;\hat{\rho}(t),\;{\cal P}(\hat{A})\\}\qquad(0\leq t\leq
t_{0})\;.$ (63)

The probability measure is defined through the probabilities

$p(A_{n},t)={\rm Tr}\;\hat{\rho}(t)\;\hat{P}(A_{n})={\rm
Tr}_{A}\;\hat{\rho}_{A}(t)\;\hat{P}(A_{n})\;.$ (64)

At the moment of time $t_{0}$, the dynamics is interrupted by a measurement
revealing the occurrence of an event $A_{n}$. This implies the state reduction

$\hat{\rho}(t_{0}-0)\longmapsto\hat{\rho}(A_{n},t_{0})\;,$ (65)

such that the projection of the total decision state to the space of
alternatives

$\hat{\rho}_{A}(t_{0}-0)={\rm
Tr}_{S}\;\hat{\rho}(t_{0}-0)\longmapsto\hat{\rho}_{A}(A_{n},t_{0})$ (66)

acquires the Lüders form (47). The occurrence of the event $A_{n}$ at $t_{0}$
implies that

${\rm Tr}\;\hat{\rho}(A_{n},t_{0})\;\hat{P}(A_{n})={\rm
Tr}_{A}\;\hat{\rho}_{A}(A_{n},t_{0})\;\hat{P}(A_{n})=1\;.$ (67)

This equality defines the state $\hat{\rho}_{A}(A_{n},t_{0})$ as the Lüders
state. The state $\hat{\rho}(A_{n},t_{0})$ serves as an initial state for the
next stage of evolution.

The next stage lasts after the time of the measurement $t>t_{0}$. Now the
evolution of the system state starts from the new initial condition
$\hat{\rho}(A_{n},t_{0})$, so that

$\hat{\rho}(A_{n},t)=\hat{U}(t,t_{0})\;\hat{\rho}(A_{n},t_{0})\;\hat{U}^{+}(t,t_{0})\;.$
(68)

At this stage, we are interested in the a priori probabilities of the events
$B_{k}$. For the case of tossing, an event $B_{k}$ can be the result of
tossing either the same coin or a die or the result of tossing another die.

Then the operator measure $\mathcal{P}(\hat{B})$ is composed of the projection
operators $\hat{P}(B_{k})$. The quantum probability space becomes

$\\{{\cal H},\;\hat{\rho}(A_{n},t),\;{\cal P}(\hat{B})\\}\qquad(t>t_{0})\;.$
(69)

For this probability space, the a priori probability of observing an event
$B_{k}$ at an arbitrary time $t$,

$p(B_{k},t\;|\;A_{n},t_{0})={\rm Tr}\;\hat{\rho}(A_{n},t)\;\hat{P}(B_{k})\;,$
(70)

plays the role of the quantum conditional probability of observing $B_{k}$
after the event $A_{n}$ has certainly been observed at time $t_{0}$.

As is explained in the previous section, if the dependence of the state
$\hat{\rho}(A_{n},t)$ on the measuring perturbation at the time $t_{0}$ fades
out with increasing $t$, so that $t-t_{0}=\tau\gg t_{rel}$, and this state
tends to $\hat{\rho}(t)$, then the events $A_{n}$ and $B_{k}$ become weakly
correlated and we verge to the case of independent events (60).

This consideration shows that, generally, the quantum conditional probability
cannot be represented as in expression (23) for the classical conditional
probability. This can happen only in the limiting case of independent events,
when the time separation between the events is much larger than the relaxation
time.

##  9 Synchronous Measurements and Events

Till now, we have studied conditional probabilities for events occurring at
different times. A natural question arises: Is it possible to accomplish
several measurements or observe several events synchronously at the same time?
Here we keep in mind nonrelativistic events, when the notion of synchronously
occurring events or measurements has sense. In the relativistic case we should
use the notion of spacelike separated measurements or events. If one considers
a realistic system and a realistic measurement procedure, then one has to
accept that the simultaneous measurement of different observables can be
realized only by different measuring devices and, generally, in different
parts of the system. For instance, keeping in mind coin and dice tossing, we
have to agree that a simultaneous toss of either two coins, or a coin and a
die, or two different dice is feasible only in separate spatial locations.
Accepting that two measurements are done in two different parts of a system,
we shall denote these parts as the location $A$ and location $B$.

In this section, we omit, for the simplicity of notations, the subject space.
If necessary, it can be easily included back, which does not influence the
results. In that case, if we measure two observables represented by the
operators $\hat{A}$ and $\hat{B}$, the process should be described in the
composite space of alternatives

${\cal H}_{AB}={\cal H}_{A}\bigotimes{\cal H}_{B}\;.$ (71)

The operators $\hat{A}$ and $\hat{B}$ can be either commuting or not
commuting. The system state $\rho_{AB}(t)$, acting on this space, can be
considered as the state with the traced out subject-space degrees of freedom.

The quantum probability space is

$\\{{\cal H}_{AB},\;\hat{\rho}_{AB}(t),\;{\cal P}(\hat{A})\bigotimes{\cal
P}(\hat{B})\\}\;.$ (72)

Then it is easy to define the probabilities of separate events occurring in
the corresponding locations

$p(A_{n},t)={\rm Tr}_{AB}\;\hat{\rho}_{AB}(t)\;\hat{P}(A_{n})={\rm
Tr}_{A}\;\hat{\rho}_{A}(t)\;\hat{P}(A_{n})\;,$ $p(B_{k},t)={\rm
Tr}_{AB}\;\hat{\rho}_{AB}(t)\;\hat{P}(B_{k})={\rm
Tr}_{B}\;\hat{\rho}_{B}(t)\;\hat{P}(B_{k})\;,$ (73)

where

$\hat{\rho}_{A}(t)={\rm
Tr}_{B}\;\hat{\rho}_{AB}(t)\;,\qquad\hat{\rho}_{B}(t)={\rm
Tr}_{A}\;\hat{\rho}_{AB}(t)\;.$

It is also straightforward to introduce the joint probability of two events
[28, 37] occurring at a time $t$,

$p(A_{n}B_{k},t)={\rm
Tr}_{AB}\;\hat{\rho}_{AB}(t)\;\hat{P}(A_{n})\bigotimes\hat{P}(B_{k})\;.$ (74)

Notice that the joint probability is symmetric with respect to the event
interchange,

$p(A_{n}B_{k},t)=p(B_{k}A_{n},t)\;.$ (75)

The joint probability satisfies the standard properties

$p(A_{n},t)=\sum_{k}p(A_{n}B_{k},t)\;,\qquad
p(B_{k},t)=\sum_{n}p(A_{n}B_{k},t)\;.$ (76)

The conditional probability of two events, occurring at different locations,
can be introduced by the usual formula

$p(A_{n}|B_{k},t)\equiv\frac{p(A_{n}B_{k},t)}{p(B_{k},t)}\;.$ (77)

This conditional probability enjoys the same properties as the classical
conditional probability,

$\sum_{n}p(A_{n}|B_{k},t)=\sum_{k}p(B_{k}|A_{n},t)=1\;.$ (78)

Probability (77) is not symmetric with respect to the event interchange, since

$p(B_{k}|A_{n},t)=\frac{p(B_{k}A_{n},t)}{p(A_{n},t)}\;.$ (79)

The locations $A$ and $B$, in general, are correlated, because of which the
state $\hat{\rho}_{AB}(t)$ does not have the form of a tensor product. In the
particular case, when the locations are not correlated, and the events are
independent, so that the state factorizes,

$\hat{\rho}_{AB}(t)=\hat{\rho}_{A}(t)\bigotimes\hat{\rho}_{B}(t)\;,$ (80)

then

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

and we again get the standard property of conditional probabilities

$p(A_{n}|B_{k},t)=p(A_{n},t)\;,\qquad p(B_{k}|A_{n},t)=p(B_{k},t)\;.$ (81)

In this way, the quantum conditional probability (77) can be accepted as a
generalization of the classical conditional probability.

##  10 Quantum Decision Theory

The main aim of the present paper has been the detailed analysis of the
differences between quantum and classical probabilities. To be more
illustrative, the case of tossing coins and dice in both variants, quantum and
classical, has been treated. The thorough comparison has shown that, strictly
speaking, quantum and classical probabilities are different entities. The
quantum probability, such as the Lüders probability, cannot be considered as a
generalization of the classical conditional probability, since the former
cannot be reduced to the latter. These are just different entities designed
for quite differing systems, either quantum or classical. So to speak: to each
its own.

However for a curious reader, it is difficult to avoid some questions as
follows. It is clear and well known that quantum probabilities are
successfully applied to quantum systems. But could they be applied to real-
world problems of cognitive science? For instance, could quantum probabilities
be used in decision making that is so widely employed in and so important for
many fields of human activity, such as economics, finances, managing, and so
on? When are quantum probabilities especially useful, or needed at all? Which
are the types of problems where they could be applied? The answer to these
questions is given in the present section.

In addition, explaining when and where quantum probabilities could be involved
in cognitive sciences, it will become more clear the importance of considering
the subject space of mind introduced in Sec. 6. There are, of course, quite
simple cases, where there is no any need of invoking quantum techniques for
real-life classical problems. Thus, describing the toss of a classical coin
does not require the use of quantum terminology. Vice versa, dealing with a
quantum analog of coin tossing, e.g. when observing a spin of a particle, the
standard quantum techniques are sufficient. However, the use of quantum
probabilities in cognitive sciences can be useful in analyzing rather
complicated situations having to do with some kind of dual influence on the
decision processes. For this purpose, it is necessary to deviate from the
trivial case of coin tossing and to turn to essentially more complicated and
delicate issues of decision making.

To grasp the feeling why quantum techniques could be useful in decision
making, one has to remember one of the most important points of quantum theory
incorporating the so-called particle-wave duality. Then, intuitively, one
could guess that quantum theory could be of help in describing those decision-
making processes that in some sense are of dual nature. For instance, when one
needs to make a choice between the given alternatives, at the same time
experiencing uncertainty with regard to these alternatives because of the
existence of additional alternative attributes related to behavioral effects
[18, 44, 45, 46].

A well known duality in decision making has to do with that being termed by
different names, although expressing the same essence. This is the cognition-
emotion duality, rational-irrational duality, reasoning-affective duality,  or
conscious-subconscious duality. There exists vast literature on the problem of
studying the influence of emotions on the process of taking decisions, of
which we mention just a few [47, 48, 49, 50, 51, 52, 53, 54]. In the
mathematical language of quantum theory, this duality is described below [18,
44, 45, 46, 55, 56].

In quantum measurement theory, the role of emotions is played by perturbations
from surrounding, which can be called quantum noise. There are arguments [57]
that, actually, all decisions and measurements are accompanied by a kind of
quantum noise due to random quantum fluctuations in the net of human neurons.
The signal triggering the process of decision making travels along human
neurons with an intrinsic temporal uncertainty of $1$ ms [58] accompanied by
fluctuations in the number of open neuron ion channels. These molecular
fluctuations are due to random Brownian motion of polypeptides in their
surrounding fluid. Since the Brownian fluctuations are fundamentally quantum,
the probabilities of events, in the strict sense, are also quantum.

The decision space

${\cal H}={\cal H}_{S}\bigotimes{\cal H}_{A}\;,$

as in Sec. 6, is the tensor product of the subject space of mind and the space
of alternatives. When choosing between the alternatives, the choice of the
decision maker includes two sides. From one side, one has to evaluate the
utility of each alternative. But from the other side, the choice is
accompanied by emotions associated with the given alternative. This is
equivalent to saying that the decision maker considers not just the set of
alternatives, but rather a set of composite prospects

$\pi_{n}\equiv A_{n}z_{n}\;,$ (82)

each being formed by an alternative $A_{n}$ and the associated set of emotions
represented by the notation $z_{n}$.

In quantum measurement theory, each process of measurement of an event $A_{n}$
is complimented by the accompanying quantum noise denoted through $z_{n}$.
Again, from the mathematical point of view, the description is the same,
whether we consider a measurement accompanied by quantum noise or take a
decision decorated by emotions.

Emotions (noise) are set into correspondence to the vector

$|\;z_{n}\;\rangle=\sum_{\alpha}b_{n\alpha}\;|\;\alpha\;\rangle$ (83)

pertaining to the subject space of mind ${\cal H}_{S}$. The coefficients
$b_{n\alpha}$ are random quantities, which reflects the randomness of
emotions. These vectors do not need to be orthonormalized, so that the scalar
product

$\langle\;z_{m}\;|\;z_{n}\;\rangle=\sum_{\alpha}b_{m\alpha}^{*}b_{n\alpha}$

is not necessarily the Kronecker delta. Respectively, the prospect states

$|\;\pi_{n}\;\rangle=|\;A_{n}z_{n}\;\rangle=|\;A_{n}\;\rangle\;\bigotimes\;|\;z_{n}\;\rangle=\sum_{\alpha}b_{n\alpha}\;|\;A_{n}\alpha\;\rangle$
(84)

also are not necessarily orthonormalized,

$\langle\;\pi_{m}\;|\;\pi_{n}\;\rangle=\delta_{mn}\;\langle\;z_{n}\;|\;z_{n}\;\rangle\;.$

The prospect operator

$\hat{P}(\pi_{n})=|\;\pi_{n}\;\rangle\langle\;\pi_{m}\;|=|\;A_{n}z_{n}\;\rangle\langle\;z_{n}A_{n}\;|$
(85)

is not idempotent, since

$\hat{P}(\pi_{m})\;\hat{P}(\pi_{n})=\delta_{mn}\;\langle\;z_{n}\;|\;z_{n}\;\rangle\;\hat{P}(\pi_{n})\;.$

The matrix element

$\langle\;\alpha
A_{m}\;|\;\sum_{k}\hat{P}(\pi_{k})\;|\;A_{n}\beta\;\rangle=\delta_{mn}b_{n\alpha}^{*}b_{n\beta}$

shows that the resolution of unity for the prospect operators cannot be
satisfied in the strong sense, although can be valid in the weak sense of the
average

$\langle\;\sum_{n}\hat{P}(\pi_{n})\;\rangle={\rm
Tr}\;\hat{\rho}(t)\;\sum_{n}\;\hat{P}(\pi_{n})=1\;,$ (86)

where the trace is over the decision space $\mathcal{H}$. Thus the family

${\cal P}_{\pi}\equiv\\{\hat{P}(\pi_{n})\\}$ (87)

of the prospect operators constitutes a positive operator-valued measure [59].

Therefore the triple

$\\{{\cal H},\;\hat{\rho}(t),\;{\cal P}_{\pi}\\}$ (88)

forms the quantum probability space. The prospect probabilities

$p(\pi,t)\equiv{\rm Tr}\;\hat{\rho}(t)\;\hat{P}(\pi_{n})\;,$ (89)

by the above construction, are semi-positive and normalized,

$\sum_{n}\;p(\pi_{n},t)=1\;,\qquad 0\leq p(\pi_{n},t)\leq 1\;.$ (90)

Substituting into the prospect probability (89) the prospect operators (85)
makes it possible to separate in probability (89) the non-negative part

$f(\pi_{n},t)\equiv\sum_{\alpha}\;|\;b_{n\alpha}\;|^{2}\;\langle\;\alpha
A_{n}\;|\;\hat{\rho}(t)\;|\;A_{n}\alpha\;\rangle\;,$ (91)

which corresponds to classical probability, being normalized,

$\sum_{n}\;f(\pi_{n},t)=1\;,\qquad 0\leq f(\pi_{n},t)\leq 1\;.$ (92)

The remaining part

$q(\pi_{n},t)\equiv\sum_{\alpha\neq\beta}\;b^{*}_{n\alpha}b_{n\beta}\;\langle\;\alpha
A_{n}\;|\;\hat{\rho}(t)\;|\;A_{n}\beta\;\rangle\;$ (93)

is not sign-defined, having the property

$\sum_{n}\;q(\pi_{n},t)=1\;,\qquad-1\leq q(\pi_{n},t)\leq 1\;.$ (94)

In this way, the prospect probability (89) is the sum of two terms:

$p(\pi_{n},t)=f(\pi_{n},t)+q(\pi_{n},t)\;.$ (95)

The semi-positive term (91) describes the classical probability or the
fraction of decision makers choosing, between the given alternatives,
following rational rules. It can be connected with the utility of alternatives
by means of the Luce rule [60] as follows. If the alternatives $A_{n}$ are
characterized by the attributes $a_{n}$, then the probabilistic weight of each
alternative is

$f(\pi_{n},t)=\frac{a_{n}}{\sum_{n}a_{n}}\qquad(a_{n}\geq 0)\;.$ (96)

This term can be called utility fraction or rational fraction. For non-
negative expected utilities, the natural attribute $a_{n}$ is the expected
utility itself. Quantitative evaluation of this term has been studied in
detail in Refs. [18, 56, 61, 62, 63, 64, 65, 66, 67].

The sign-undefined term (93) is associated with emotional feelings of a
decision maker inducing the attraction or repulsion towards the alternatives
caused by emotional evaluation of the quality of each alternative. Therefore
this term can be named attraction factor or quality factor. By its definition,
it is a random quantity, being different for different subjects and even for
the same subject at different times. The type of randomness can be prescribed
by a distribution over $q$. As is clear, being random, does not preclude to
find its typical average value that plays the role of a non-informative prior.
The non-informative prior for the attraction factor was found [18, 55, 56, 62,
64, 65, 67] to be $\pm 0.25$, with the sign defined by attractiveness or
repulsiveness of the corresponding prospect.

In the process of quantum measurements, the term $q(\pi_{n},t)$ is caused by
random noise. In that case, one needs to define a distribution over $q$. If
the properties of the noise are characterized by average values of $q$, these
values will define the typical uncertainty of a measurement.

Summarizing, the prospect probability, taking account of both the utility of
alternatives, as well as their attractiveness, can be evaluated by the formula

$p(\pi_{n},t)=f(\pi_{n},t)\pm 0.25\;,$ (97)

provided the normalization condition (90) holds true.

In the references cited above, there are numerous examples of the use of the
described approach to decision making, demonstrating good numerical agreement
with empirical data. Here we give just one example showing how this approach
allows for the explanation of the decoy effect [65].

##  11 Example of Decoy Effect

The decoy effect was described by Huber et al. [68] and observed in many
experimental studies, e.g. [69, 70, 71, 72, 73, 74]. The decoy effect
describes how, when we are choosing between two alternatives, the addition of
a third, less attractive option (the decoy) can influence our perception of
the original two choices. Decoys are completely inferior to one option (the
target) but only partially inferior to the other (the competitor). For this
reason, the decoy effect is sometimes called the asymmetric dominance effect.

The meaning of the decoy effect can be illustrated by the following example.
Suppose a buyer is choosing between two objects, $A$ and $B$. The object $A$
is of better quality, but of higher price, while the object $B$ is of slightly
lower quality, while less expensive. As far as the functional properties of
both objects are close to each other, but $B$ is cheaper, the majority of
buyers prefer the object $B$. But then the sales person slightly changes the
situation presenting a third object $C$, which is of about the same quality as
$A$, but of a much higher price than $A$. This causes the buyer to reconsider
the choice between the objects $A$ and $B$, while the object $C$, having
practically the same quality as $A$ but being much more expensive, is of no
interest. Choosing now between $A$ and $B$, the majority of buyers prefer the
higher quality but more expensive object $A$. The object $C$, being not a
choice alternative, plays the role of a decoy. Experimental studies confirmed
the decoy effect for a variety of objects: cars, microwave ovens, shoes,
computers, bicycles, beer, apartments, and many others.

The decoy effect contradicts the regularity axiom in decision making telling
that if $B$ is preferred to $A$ in the absence of $C$, then this preference
has to remain in the presence of $C$. From the point of view of classical
decision theory, the decoy effect is a paradox. Decoys work subconsciously,
being an example of a behavioral nudge that steers individuals towards making
a certain choice. As other subconscious and emotional influences in decision
making, this effect can be explained in the frame of the approach described in
the previous Sec. 10.

Assume that buyers are choosing between two objects A and B, say between two
ovens, as in the experiment of Ariely and Wallsten [72]. The oven $A$ is of
higher quality but more expensive, while the oven $B$ is of moderate quality
but cheaper. Since the quality of these two ovens is not much different, but
$B$ is cheaper, the majority prefer the oven $B$. In the experiment [72], the
fraction of buyers preferring either $A$ or $B$ was

$f(A)=0.4\;,\qquad f(B)=0.6\;.$

Then the third oven $C$ was displayed costing much more expensive than $A$,
but being of just slightly higher quality. The oven $C$, being of much higher
price than $A$, but of close quality, is not a choice. However, it attracted
attention of buyers to the quality of the ovens. The emotional attraction of
buyers induces the attraction factors $q(A)$ and $q(B)$. In that sense, we
need to consider not simply alternatives $A$ and $B$, but the prospects
$\pi_{A}=Az_{A}$ and $\pi_{B}=Bz_{B}$, although, for the simplicity of
notation, we continue writing just $A$ and $B$. As far as the feature of
quality has become more attractive, the related attraction factors are such
that $q(A)>q(B)$. Estimating them as non-informative priors, we have
$q(A)=0.25$ and $q(B)=-0.25$. Therefore the theory predicts the choice
probabilities

$p(A)=f(A)+q(A)=0.4+0.25=0.65\;,\qquad p(B)=f(B)+q(B)=0.6-0.25=0.35\;.$

In other words, we obtain the estimates predicting the real fraction of buyers
preferring this or that oven. Comparing this prediction with the experimental
data

$p_{exp}(A)=0.61\;,\qquad p_{exp}(B)=0.31\;,$

we see that they are very close to each other.

##  12 Conclusions

The central point of the paper is the problem of defining conditional
probability in quantum theory. In order to clearly elucidate the difficulties
and possible ways of constructing quantum conditional probabilities, the
consideration has been based on the example of tossing quantum coins and dice.
This case is of interest by its own, since quantum coins and dice present a
paradigm of quantum bits that are widely employed in quantum information
processing, quantum computing, and quantum decision making.

There exist two types of quantum conditional probabilities that can be called
a temporal conditional probability and a spatial conditional probability. The
temporal conditional probability considers two events occurring at different
times. It is based on the quantum state reduction after a measurement and uses
the von-Neumann-Lüders state. The latter is the probability
$p(B_{k},t_{0}+0|A_{n},t_{0})$ of an event, say $B_{k}$, at the time
$t_{0}+0$, occurring immediately after another event, say $A_{n}$, has
certainly occurred at the previous time $t_{0}<t_{0}+0$. It is shown that,
strictly speaking, the Lüders probability cannot be reduced to the classical
conditional probability. Therefore it cannot be treated as a generalization of
the latter. The Lüders probability and the classical conditional probability
are different notions valid for different systems, either quantum or
classical. The key difference between these probabilities is caused by the
state reduction taking place for quantum measurements, but being absent for
classical systems. The direct consequence of the state reduction is the
principle of measurement reproducibility in quantum theory [29] according to
which the second measurement, accomplished immediately after the first one,
has to reproduce the result of the latter. No such principle exists for
classical systems.

When one attempts to apply quantum techniques to cognitive sciences, with
using the Lüders probability, one should take into account a decision maker
and the related subject state of mind. Then, although the system state
experiences the reduction in the space of alternatives, but it remains more or
less invariant in the subject space, thus treating the decision maker as a
self-consistent personality.

Another type of probability is the spatial conditional quantum probability
$p(A_{n}|B_{k},t)$ that considers two events occurring simultaneously in time,
but in different spatial locations. The probability $p(A_{n}|B_{k},t)$ is the
probability of an event $A_{n}$ occurring in one location at time $t$, under
the condition that at the same time the other event $B_{k}$ occurs in another
location. The structure and properties of this probability are close to
classical conditional probability.

The dual nature of quantum theory allows to extend the classical decision
making to the problems where behavioral effects are of great importance. In
that case it is necessary to take account of the subject space of mind and to
consider not merely alternatives but composite prospects of alternatives
decorated by emotions accompanying the latter. The suggested approach of
taking into account emotions yields good agreement of predicted and
experimentally observed data.

In the process of quantum measurements, the role of emotions is played by
quantum fluctuations accompanying the measurements. At the same time, the
origin of emotions, to some extent, can be due to quantum fluctuations in the
brain. The existence of such fluctuations has been justified by physiological
studies demonstrating the presence of Brownian fluctuations in the neuron net.

## Acknowledgments

The author is grateful for useful discussions to J. Harding, H. Nguyen, D.
Sornette, and E.P. Yukalova.

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