Probability Vectors within the Classical and Quantum Frameworks

Journal of Russian Laser Research, Volume 35, Number 1, January, 2014

Dedicated to the memory of our friend and colleague Allan Solomon,Emeritus Professor at Sorbonne Unversity, Parisand Emeritus Professor of Mathematical Physics

at the Open University, Milton Keynes, UK.

PROBABILITY VECTORS

WITHIN THE CLASSICAL AND QUANTUM FRAMEWORKS

Margarita A. Man’ko,1 ∗ Vladimir I. Man’ko,1, 2 Giuseppe Marmo,3 Alberto Simoni,3

and Franco Ventriglia3

1P. N. Lebedev Physical Institute, Russian Academy of Sciences

Leninskii Prospect 53, Moscow 119991, Russia2Moscow Institute of Physics and Technology (State University)

Institutskii Per. 9, Dolgoprudnyi, Moscow Region 141700, Russia3Dipartimento di Fisica dell’Universita “Federico II” e Sezione INFN di Napoli

Complesso Universitario di Monte S. Angelo

Via Cintia, Naples 80126, Italy

*Corresponding author e-mail: mmanko@ sci.lebedev.ru

e-mails: manko@ sci.lebedev.ru [email protected] [email protected] ventri@ na.infn.it

Abstract

We consider properties of the probability distributions associated with both classical and quantumsystems. We discuss the notion of distances between the probability vectors and between the densitystates. We study the transforms of the probability vectors by means of stochastic and bistochasticmatrices. We review the concept of positive and completely positive maps from the viewpoint of thetomographic-probability approach for describing the quantum states and their dynamics.

Keywords: probability distribution, classical probability vector, density state, Fisher–Rao metric,

stochastic matrices.

1. Introduction

Classical and quantum pictures of physical phenomena and mutual relations between probabilistic

quantum and deterministic classical descriptions of the states of physical systems [1] are closely con-

nected with various aspects of differential geometry and probability theory. The geometrical approach

to quantum-mechanical processes was developed in [2]. Pure states in quantum mechanics are associated

with wave functions [3] or with vectors in the Hilbert space [4]. Mixed quantum states are identified with

Manuscript submitted by the authors in English on December 12, 2013.

1071-2836/14/3501-0079 c©2014 Springer Science+Business Media New York 79

Journal of Russian Laser Research Volume 35, Number 1, January, 2014

density matrices [5–7], which are Hermitian nonnegative trace-class matrices associated with operators

calculated in some basis in the Hilbert space.

Probability distributions in classical statistical mechanics are used to describe the system states in

the presence of fluctuations associated, e.g., with influences of the environment. Thus, for fluctuating

classical random observables, probability distributions are natural mathematical objects to describe the

states.

In the quantum formalism, fluctuations cannot be removed, even for the states of quantum systems

at zero temperatures. Thus, wave functions and density matrices intrinsically contain information on

quantum fluctuations in the system states and determine the probability distributions, e.g., of the particle

position or the spin projection on a chosen direction in the space. The modulus squared of the wave

function or diagonal elements of the density matrix are just examples of probability distributions, which

naturally appear in the quantum-mechanical picture of the states of physical systems. There exist different

relations between the probability distributions associated with classical and quantum pictures of the

system behavior [8].

A tomographic approach to quantum and classical systems [9–12], including classical fields [13–15], has

recently been developed. In this approach, quantum states are identified with tomographic-probability

distributions containing complete information on the system states (cf [16–21]). The probability-distribu-

tion description of quantum states is connected by integral transforms with the corresponding quasipro-

bability distributions like the Wigner function [22], the Husimi–Kano function [23,24], and the diagonal

representation function [25,26]. The tomographic-probability distribution (symplectic tomogram) is con-

nected with the Wigner function by the Radon transform [27], and this transform has been generalized

recently in [28, 29].

The properties of the probability distributions, which appear in the quantum-mechanical context,

have some peculiarities associated with quantum correlations in the systems.

The aim of this paper is to consider some aspects of probability distributions in both settings, classical

and quantum. We review known properties of the probability vectors, like the entropic characteristics [7,

30–32], and discuss some new elements in the description of quantum-state differences in the probability

representation of quantum mechanics, as well as entropic quantum inequalities [33, 34]. We employ

the relative tomographic entropy as a characteristics of the quantum state difference. We study the

semigroups of transformations of probability vectors by introducing the k-star product of stochastic and

bistochastic matrices.

This paper is organized as follows.

In Sec. 2, we consider the probability vectors and review linear stochastic and bisctochastic maps

of probability vectors in Sec. 3. In Sec. 4, we study metric properties of probability vectors by using

the Fisher–Rao metric. We discuss the distance between quantum states and entropy in Sec. 5 and the

tomographic-probability vectors of qudit states in Sec. 6. We present transforms of density states in

Sec. 7 and give our conclusions and prospectives in Sec. 8.

2. Probability Vectors

Let us consider N nonnegative numbers p1, p2, . . . , pN such that∑N

k=1 pk = 1. These numbers can be

considered as outcomes in experiments where a random variable value, which is fluctuating, e.g., due to

the interaction with an environment, is measured. The numbers pk provide the probability distributions

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Volume 35, Number 1, January, 2014 Journal of Russian Laser Research

for measured random observable and can be considered as components of a vector �p called the probability

vector. The domain in RN associated with the probability vectors is called the simplex.

One can consider a generalization of the probability vector if two random variables characterizing

the system under consideration are measured. Then there exists a joint probability distribution P (j, k),

which can be considered as a matrix with nonnegative matrix elements such that∑N1

j=1

∑N2k=1 P (j, k) = 1.

The two random variables are assumed to be measured simultaneously; the first random variable has N1

values, and the second random variable has N2 values.

The joint probability distributions provide two probability vectors �p1 and �p2. The probability vector

�p1 has N1 components p1j =∑N2

k=1 P (j, k), and the probability vector �p2 has N2 components p2k =∑N1j=1 P (j, k). An extension to introduce the notion of joint probability distribution P (k1, k2, . . . , kn) for

n variables is straightforward.

If the random variables are associated with continuous indices, the probability distribution is a

nonnegative function P (x), which can also be considered as a normalized probability vector, such

that

∫P (x) dx = 1. Analogously, for two random variables one has the joint probability distribu-

tion P (x1, x2) ≥ 0, such that

∫P (x1, x2) dx1 dx2 = 1. These joint distributions determine the marginals

P1(x1) =

∫P (x1, x2) dx2 and P2(x2) =

∫P (x1, x2) dx1, which are analogs of the corresponding proba-

bility N -vectors discussed above for discrete random variables.

The probability vectors �p provide us with the possibility to calculate the moments of measurable

fluctuating variables q, which can take values qn. The highest moments of the variables read

〈qk〉 =N∑

n=1

qkn pn. (1)

For k = 1, one has the mean value of the random variable q, and the value 〈q2〉 determines the dispersion

of the variable q as σqq = 〈q2〉 − 〈q〉2.Analogous relations for highest moments are available for infinite probability vectors (N = ∞) as well

as for the continuous variables associated with distributions P (x) ≥ 0.

If one considers the joint probability distributions P (j, k) for random variables q1 and q2 taking values

q1j and q2k, respectively, the highest moments are defined as

〈qk11 qk22 〉 =N1∑j=1

N2∑k=1

qk11j qk22k P (j, k). (2)

Thus, the means of the random variables q1 and q2 are given by 〈q1〉 and 〈q2〉. The variances of the randomvariables read σq1q1 = 〈q21〉−〈q1〉2 and σq2q2 = 〈q22〉−〈q2〉2, and the covariance is σq1q2 = 〈q1q2〉−〈q1〉〈q2〉.

The convex sum of probability vectors is again a probability vector. Any probability vector can

be presented as the convex sum of the simplex vertices. It is worth pointing out that the probability

vectors can be considered as collections of nonnegative numbers that are the coordinates of a point on

the simplex. These numbers may be associated with different random variables taking different values

qn.

There are probabilistic characteristics that depend on random variables like moments (1) and (2),

but there exist probabilistic characteristics that do not depend on the nature of random variables. The

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latter probabilistic characteristics, like an analog of “the quantum purity”

P =

N∑n=1

p2n ≤ 1, (3)

describe the feature of the probabilistic distribution itself. These characteristics, as well as Ps =∑Nn=1 p

sn ≤ 1, s = 3, 4, . . ., do not depend on values of random variables and describe intrinsic prop-

erties of the probability vectors, i.e., “inner” properties of points on the simplex.

3. Linear Transforms of the Probability Vectors

In this section, we review briefly the properties of linear transforms on the simplex.

We address the following problem: What are the linear transforms that map an arbitrary point in

the simplex (i.e., vector �p0) onto another point in the simplex (vector �p ) ?

The problem is reduced to obtaining and characterizing the matrix M such that

M�p0 = �p. (4)

It is known that, if there are no constraints on the probability vectors, the matrix M turns out to be

a so-called stochastic matrix. This means that all columns of the matrix M must be the probability

N -vectors, i.e.,

M = ‖ �M1, �M2, . . . , �MN‖. (5)

Thus, one has ( �Ms)k ≥ 0 and∑N

k=1(�Ms)k = 1, s = 1, 2, . . . , N . We can easily check that for such

matrix M the vector �p is the probability vector for an arbitrary initial vector �p0.

The stochastic matrices M form a semigroup, i.e., for two stochastic matrices M1 and M2, the product

M1M2 = M is also a stochastic matrix. The unity matrix M1 = 1N is the identity in the semigroup.

The inverse M−1 of the stochastic matrix M , even if detM �= 0, does not belong to the set of stochastic

matrices, since the matrix elements of the inverse matrix can take negative values, and therefore its

columns do not represent probability vectors anymore.

The convex sum of stochastic matrices is a stochastic matrix. The semigroup of stochastic matrices

has the subset of bistochastic matrices, which we denote by B ∈ M . The bistochastic matrix B is

a stochastic matrix such that the transposed matrix B tr is also a stochastic matrix. All bistochastic

matrices have the property

�p trN B = �p tr

N , B�pN = �pN , �p trN = (1/N, 1/N, . . . , 1/N). (6)

This means that “the center” of the simplex is the probability vector (�pN )j = 1/N , j = 1, 2, . . . , N

invariant under the action of the matrices B and B tr.

The bistochastic matrices form a semigroup, i.e., for any bistochastic matrices B1 and B2, their

product B = B1B2 is also a bistochastic matrix. The convex sum of bistochastic matrices is also a

bistochastic matrix. There exists a specific set of N ! bistochastic N×N matrices, which form a group

isomorphic to the permutation group of N elements. These matrices have only one nonzero element,

equal to unity, in each row and column, and the other matrix elements are equal to zero. All other

bistochastic matrices are convex sums of the N ! permutation-group matrices.

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The identity matrix E and theN×N matrix I with all matrix elements equal to 1/N form a semigroup,

which either does not transform the probability vectors or collapses the probability vectors onto the vector

with components pn = 1/N (the simplex center). The rule of multiplication of semigroup elements reads

I2 = I, E2 = E, and EI = IE = I, the multiplication rule for all bistochastic matrices is IB = BI = I,

and for all stochastic matrices M , MI = I.

It is worth noting that the k-product [35] of stochastic matrices M1�M2 = M1 kM2, where k is a fixed

stochastic matrix, provides a stochastic matrix. Thus, the stochastic matrices form a semigroup with

respect to this deformed product, which we may call a star-product rule for matrices determined by the k

product. An analogous structure takes place for bistochastic matrices. For an arbitrary fixed bistochastic

matrix k0, the k product of two bistochastic matrices B1 � B2 = B1k0B2 provides a bistochastic matrix.

A linear evolution of probability vectors �p(t) = M(t)�p0 on the simplex can be described by a semigroup

of stochastic matrices, providing the stochastic map of the initial probability vector �p0. If one considers

the dynamics on a subset σ of the simplex, the linear map of the initial probability vector �p0 ∈ σ onto

the vector �p(t) = m(t)�p0, such that �p(t) ∈ σ, can be realized by the matrices m, which are not stochastic

ones, since they may contain negative matrix elements [36, 37].

4. Distances as Characteristics of the Probability Distributions

To distinguish different probability distributions, one should introduce the notion of distance between

points on the simplex. For the case of two N -dimensional probability vectors �p1 and �p2, the distance can

be given as

s2 = (�p1 − �p2)2 = �p 2

1 + �p 22 − 2 �p1�p2.

This distance has the obvious geometrical meaning of the length of a triangle side connecting the two

vector vertices in Euclidean geometry.

It is convenient to introduce a different metric on the space of probabilities. Let us discuss the metric

on the simplex. The flat metric ds2 =∑

n dx2n can be described in terms of pn, where xn =

√pn. One has∑N

n=1 x2n = 1, which corresponds to the normalization condition of the probability vector

∑Nn=1 pn = 1.

Starting from the flat metric expression, we obtain the Fisher–Rao metric

ds2 =1

4

∑n

dpn dpnpn

. (7)

The Fisher–Rao metric is unique according to the Chentsov theorem [38] due to the important pro-

perty that it is the only monotonic metric possessing the property that the distance D(�p, �q) between

two probability vectors �p and �q, under the action of the stochastic map M , does not increase ,i.e.,

D(M�p,M�q) ≤ D(�p, �q).

5. From Classical to Quantum Probabilities

In this section, we discuss how the probability vectors appear in the quantum setting. The wave

function or state vector | ψ〉 in the Hilbert space is associated with a pure state of the quantum system.

The mixed state of the quantum system is determined by the Hermitian nonnegative density operator ρ

with the property Tr ρ = 1. We call the density operator “the density state.” The wave function may be

given in the complex form ψ = ρeiϕ, and it is normalized.

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For a finite Hilbert space, one has ρn =√pn, where 〈en | ψ〉 = ψn, and vectors | en〉 form an

orthonormal basis in the Hilbert space. Due to the normalization condition 〈ψ | ψ〉 = 1, the nonnegative

numbers pn satisfy the equality∑N

n=1 pn = 1, i.e., they can be considered as components of a probability

vector �p.

If the state is mixed, the probability vector appears as the vector with components pn = 〈en | ρ | en〉,which are diagonal elements of the density operator in the basis | en〉. If the density state has diagonal

form in this basis, i.e., ρ =

⎛⎜⎝

p1 0 · · · 00 p2 · · · 0· · · · · · · · · · · ·0 0 · · · pN

⎞⎟⎠, the components of the probability vector �p are just

the eigenvalues of the state density operator ρ.

We may identify diagonal density matrices with classical states corresponding to the probability

distribution pn. Quantum observables are associated with N×N Hermitian matrices, and quantum

states are a convex body, i.e., for given density states ρ1 and ρ2, the operator ρ = λρ1 + (1− λ)ρ2, where

λ ∈ [0, 1], is also a density state. Classical states, being identified with the diagonal density matrices, are

also a convex body. What does the space of states look like? The space of states is a stratified manifold,

each stratum is given by matrices of the same rank, extremal states are pure states, and they may be

presented by rays in terms of vectors | ψ〉 as follows [39,40]:

ρψ =| ψ〉〈ψ |〈ψ | ψ〉 . (8)

For example, in the case of N = 2, the quantum states are a ball. Points on the surface S2 of the ball

are pure states. The ball center is identified with the maximum mixed (chaotic) density state with two

diagonal matrix elements equal to 1/2 and nondiagonal matrix elements equal to zero.

What about distances in the quantum setting?

We consider pure states. It turns out that the metric on the stratum of rank 1 projectors is related

to the following metric on the space of rays:

g =〈dψ | dψ〉〈ψ | ψ〉 − 〈ψ | dψ〉〈dψ | ψ〉

〈ψ | ψ〉2 . (9)

This metric was discussed in [41].

The quantum observables, for example, the observable A with Hermitian matrix

A =

(A11 A12

A21 A22

), (10)

where Ann = 〈n | A | n〉, n = 1, 2 are the mean values of the observable in the states | n〉〈n |, also define

probability vectors [42]. In fact, it is easy to check that there exists a unitary matrix U , such that

A = U

(A1 00 A2

)U †, (11)

where A1 and A2 are the eigenvalues of the observable A. The unitary matrix U has the eigenvectors of

the observable matrix A as its columns; this means that

U = ‖�U1, �U2‖, A�Uk = Ak�Uk, k = 1, 2, (12)

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and the vector components are expressed in terms of the matrix elements of matrix U as follows:

�U1 =

(U11

U21

), �U2 =

(U12

U22

). (13)

Then we have

A11 = p(1)1 A1 + p

(1)2 A2, A22 = p

(2)1 A1 + p

(2)2 A2. (14)

We can write the equality for the diagonal matrix elements of the matrix A in terms of the unitary matrix

U and eigenvalues of the matrix A in vector form; it reads

�A = |U |2 �Ad, (15)

where �A has two components A11 and A22, and �Ad has two components A1 and A2.

If theN×N matrixA is identified with a nonnegative density matrix ρ with eigenvalues (ρ1, ρ2, . . . , ρN ),

which may be considered as N components of the vector �ρd, the equality obtained reads

�ρ = |U |2�ρd. (16)

There exist two probability vectors

�p (1) =

(|U11|2|U21|2

), �p (2) =

(|U21|2|U22|2

)(17)

determined by the eigenvector components. Thus, relation (14) elucidates the quantum formula for the

observable mean value given by Ann = Tr(A | n〉〈n | ). The formula takes the form known in the classical

probability theory, namely,

Tr(A | n〉〈n | ) = ∑

k

p(n)k Ak, (18)

where Ak are possible outcomes, which are eigenvalues of the operator A, and the probability distributions

p(n)k are determined by the eigenvector components of the matrix A.

Thus, the probability vectors �p (1) and �p (2) (17) are given by rows of the orthostochastic matrix

|U |2jk = |Ujk|2. The orthostochastic matrices |U |2 belong to the set of bistochastic matrices B, but the

product of two orthostochastic matrices does provide a bistochastic matrix, which is not orthostochastic

(for N > 2). For N = 2, the orthostochastic matrices form a one-parameter semigroup. In quantum

mechanics, “classical probability vector components” appear not only as matrix elements of the orthos-

tochastic matrices determined by unitary matrices connecting basis in the Hilbert space.

Probability vectors in the quantum setting appear also as fidelities.

For two observables A and B described by Hermitian operators, one has the spectral decompositions

A =

N∑j=1

Aj | ej〉〈ej |, B =

N∑j=1

Bj | fj〉〈fj |, (19)

where

A | ej〉 = Aj | ej〉, B | fj〉 = Bj | fj〉. (20)

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Then, for a given density state ρ, we have two fidelities

pj = Tr ρ | ej〉〈ej |, qj = Tr ρ | fj〉〈fj | . (21)

Nonnegative numbers pj and qj can be constructed from a normalized matrix ρ (i.e., Tr ρ = 1) by means

of projectors | ej〉〈ej | and | fj〉〈fj |. They can be considered as components of two probability vectors �p

and �q, respectively.

The distance between these probability vectors reads

D(�p, �q) =

N∑j=1

√pjqj , (22)

and it is the Fisher–Rao distance.

6. Some Other Quantum-State Distances

The problem of distances between quantum states is connected with the problem of distinguishability

of the quantum states (how the states differ from each other). A review of the distance problem is

presented, e.g., in [43].

For example, for two pure quantum states | ψ1〉 and | ψ2〉, we can use the Fubini–Study distance [44]

D(ψ1, ψ2) =√2(1− |〈ψ1 | ψ2〉|2

)1/2. (23)

The definition of the distance can be presented in the form [45]

D(ψ1, ψ2) = cos−1 |〈ψ1 | ψ2〉|. (24)

For mixed quantum states, there exists the following definition of the distance [46]:

D(ρ1, ρ2) =1

2‖ρ1 − ρ2‖1. (25)

A norm of the operator ‖A‖1 =∑N

n=1 |An|, where An are eigenvalues of the operator A.

The distance between mixed states with density operators ρ1 and ρ2 can be also given by the Bures–

Uhlman distance [47, 48]

D(ρ1, ρ2) =√2(1− Tr

[(ρ

1/21 ρ

1/22 )(ρ

1/22 ρ

1/21 )

]1/2 )1/2. (26)

If ρ1 = ρ and ρ2 =| ψ〉〈ψ |, the Bures–Uhlman distance reads

D(ρ, | ψ〉〈ψ |) =√2(1−

√〈ψ | ρ | ψ〉 )1/2. (27)

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7. Entropic Characteristics of the Probability Vectors in Classical and

Quantum Settings

For each probability vector �p, one associates the Shannon entropy [30]

H = −N∑

n=1

pn ln pn. (28)

There also exist q-entropies, among them the Renyi entropy

HR(q) =1

1− qln

( N∑n=1

pqn

), (29)

with the property HR(q → 1) = H.

In the quantum setting, one has the entropies expressed in terms of the density operators coinciding

with the above entropies for the diagonal density states. The von Neumann entropy reads [7]

S = −Tr ρ ln ρ. (30)

The Renyi quantum entropy is defined as [31]

SR(q) =1

1− qln Tr ρ q. (31)

The Tsallis q-entropy is defined as [32]

ST(q) =1

1− q

(Tr ρ q − 1

). (32)

In the limit q → 1, the Renyi and Tsallis entropies coincide with the Shannon entropy.

For q = 2, the linear entropy S� =(1−Tr ρ2

)is expressed in terms of the Renyi and Tsallis entropies.

Some properties of the probability vectors appearing in both classical and quantum settings are

characterized by the discussed entropies.

8. Tomographic Probability Vectors

For qudit states, the density operator can be mapped onto the tomographic-probability distribution

called the quantum spin tomogram

w(m,u) = Tr(u† | m〉〈m | uρ), (33)

where u† | m〉 is the rotated basis vector, which is the eigenstate of the spin projection operator Jz, i.e.,

Jz | m〉 = m | m〉, m = −j,−j + 1, . . . , j. The rotation operator is the unitary matrix u†. If matrix

u is chosen as the matrix of a unitary irreducible representation of the group SU(2), the tomogram

w(m,u) becomes the spin tomogram w(m,�n), where �n is the unit vector (�n2 = 1) determining the spin

quantization axes, and m is the spin projection on these axes.

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The map of the density state of qudit onto the tomogram w(m,�n) is invertible. In view of this fact, the

tomogram can be used as an alternative to the density matrix for describing the density states [10,49,50].

Thus, in the tomographic picture of quantum states [12], a family of probability vectors appears naturally

in the quantum setting.

The tomograms of qudit states w(m,u) with the density matrix ρ can be written in the form of

probability N -vectors, where N = 2j + 1, as follows:

�wρ(u) = |uu0|2�ρ, wρ1(u) = w(−j, u), wρ2(u) = w(−j + 1, u), . . . , wρN (u) = w(j, u). (34)

Here, �ρ is the vector with nonnegative eigenvalues of the density matrix ρ1, . . . , ρN , and u0 is the uni-

tary matrix with columns taken as the corresponding eigenvectors of the density matrix, i.e., u0 =

‖�u01 �u02 . . . �u0N‖.Taking into account the known properties, e.g., the nonnegativity of the relative entropy of the

probability vectors �p and �q

H(�p|�q) =N∑

n=1

pn lnpnqn

≥ 0, (35)

which is used to characterize the difference of two probability distributions, we obtain the possibility to

characterize the difference between quantum states ρ1 and ρ2, namely,

H(�wρ1(u)|�wρ2(u)

)=

j∑m=−j

w1(m,u) lnw1(m,u)

w2(m,u). (36)

For each unitary matrix, the relative tomographic entropy of two different states of qudit is nonne-

gative. If u is taken as a matrix of irreducible representation of the group SU(2), the nonnegativity

condition of the relative tomographic entropy reads as inequality for the function on the sphere S2, i.e.,

H12(�n) =

j∑m=−j

w1(m,�n) lnw1(m,�n)

w2(m,�n)≥ 0. (37)

The number characterizing the difference of two quantum states can be taken as the mean relative entropy

on the sphere

D12 =

∫H12(�n)

d�n

4π. (38)

9. Positive Maps

The probability vectors are transformed by stochastic and bistochastic matrices into other probability

vectors. The quantum density states can also be transformed using linear maps ρ → Lρ = ρ ′, where ρ ′

is nonnegative self-adjoint operator and Tr ρ ′ = Tr ρ = 1. The linear transforms L form a semigroup,

i.e., the transform L3 = L1L2 belongs to the set of positive maps, if L1 and L2 are positive maps. The

density matrix ρ can be considered as a vector �ρ; see, e.g., [51]. Then the positive map is associated with

a matrix acting on the vector. The condition of the positivity of the map has been presented in [52].

Among the positive maps, there are specific maps called completely positive maps (in information

terminology, quantum channels). The completely positive maps can be given in the form suggested in [52],

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namely, the matrix form for finite density state; in the literature, it is called the Kraus decomposition [53].

It reads

ρ′ = Lρ =∑s

KsρK†s ,

∑s

K†sKs = 1, (39)

where the index s is an arbitrary discrete or continuous index.

The completely positive maps form a semigroup. The unitary evolution of a density matrix

ρ(t) = U(t)ρ(0)U †(t) (40)

provides a simple example of a completely positive map. If one maps the density N×N matrix ρ onto

the column vector �ρ = (ρ1, ρ2, . . . , ρN2), where the first N components of the vector are the first row of

the matrix ρ, and the next N components are the matrix elements from the second row of the density

matrix, etc., the completely positive map can be presented in the matrix form

�ρ′ = L�ρ =(∑

s

Ks ⊗K∗s

)�ρ. (41)

All complex N2×N2 matrices of the form L =∑

sKs ⊗K∗s , satisfying the condition

∑sK

∗sKs = 1N

for complex N×N matrices, form the semigroup representation of completely positive maps (or semigroup

of quantum channels).

Now we show how the quantum channels transform the tomographic-probability vectors �wρ(u). For

this, we should recall the general scheme of “quantizer–dequantizer formalism” providing an invertible

map from operators A onto functions fA(�X), which are called symbols of the operators [35]. Here, �X is

a point on the manifold.

One has the dequantizer operator U( �X) which gives the symbol fA(�X) = Tr

(U( �X)A

). The quantizer

D( �X) provides the reconstruction of the operator through the symbol A =

∫d �XfA(

�X)D( �X).

We obtain a generic form of the completely positive map action on any operator A → A′ as the actionof the map onto the symbol of the operator as follows:

fA(�X) → fA′( �X) =

∫d �X ′fA( �X

′)K( �X, �X ′), (42)

where the kernel of the map reads

K( �X, �X ′) =∑s

Tr(D( �X ′)K†

s U( �X)Ks

). (43)

We consider the case where the operator A is the density operator ρ and the quantizer–dequantizer

pair[D( �X) and U( �X)

]corresponds to the tomographic map, e.g., providing the spin tomogram of

the quantum state, which is the probability vector of the qudit state �wρ(u) = |uu0|2�ρ. Here, �ρ is the

(N = 2j + 1)-vector with components(ρ1, ρ2, . . . , ρ2j+1

)being the eigenvalues of the density matrix, u0

is the unitary matrix whose columns are corresponding eigenvectors, and u is the matrix of irreducible

representation of the SU(2) group. The map reads

�wρ′(m,�n) =

j∑m′=−j

∫d�n ′wρ

(m′, �n ′)K(

m,m′, �n, �n ′), (44)

89


and the kernel of the map is

K(m,m′, �n, �n ′) = ∑

s

Tr[D(m′, �n ′)K†

s U(m,�n

)Ks

]. (45)

Explicit forms of the spin tomographic quantizer D(m,�n

)and dequantizer U

(m,�n

)are given, e.g.,

in [54, 55].

Thus, the linear completely positive transform of the quantum spin-tomographic probability vector

contains the summation over the spin projection m and integration over d�n ′ on the sphere S2. In

turn, the kernel K(m,m′, �n, �n ′) plays the role of a “stochastic” matrix transforming the probability

vector �wρ(u). The linear dynamics of spin tomograms corresponds to specific kernels, and the matrix

elements of the kernels can take negative values [36, 37]. The kernels are identified with matrices M =

‖ �M1, �M2, . . . , �MN‖ (5), where �Ms have the properties of quasiprobability distributions analogous to the

normalized Wigner function [22], which can take negative values.

10. Conclusions

To conclude, we point out the main results of this paper.

We reviewed the properties of the probability vectors and their entropic characteristics.

In the quantum setting, we studied how the probability vectors appear due to different reasons. The

standard relation of the wave-function modulus squared to the probability distribution is such a possibility

for pure states. For mixed states, the diagonal elements of the density matrix in an arbitrary basis provide

probability vectors. The metric in the probability space like the Fisher–Rao metric naturally appears,

and the Chentsov theorem provides the reason why this metric has the unique property of monotonicity

under the action of stochastic maps. The tomographic approach to the description of quantum states

also gives rise to the appearance of probability vectors in a completely quantum setting.

We discussed stochastic and bistochastic matrices in relation to the problem of linear transforms of

the probability vectors in the classical setting.

We investigated the positive and completely positive maps (quantum channels) and properties of the

transformation matrix [kernels given by Eq. (45)] of quantum tomograms.

Also we reviewed some approaches to the problem of distances between quantum states (both pure

and mixed). Employing the relative entropy properties, we introduced a specific tomographic distance

between qudit states as an integral over the sphere S2 of the relative tomographic entropies depending

on the quantization direction [see Eq. (35)].

The development of the consideration of probability vectors appearing in the quantum setting pro-

posed in this paper will be presented in a future publication.

Acknowledgments

M.A.M. and V.I.M. thank the Dipartimento di Fisica dell’Universita “Federico II” e Sezione INFN di

Napoli, Italy for kind hospitality.

90


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Probability Vectors within the Classical and Quantum Frameworks