Statistical properties of qutrit in probability representation of quantum mechanics

Physica A 533 (2019) 121898

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Statistical properties of qutrit in probability representation of quantum mechanics ∗

Ashot Avanesov a,b , , Vladimir I. Man’ko a,b,c a

Department of General and Applied Physics, Moscow Institute of Physics and Technology (State University), Institutskiy per. 9, Dolgoprudny, Moscow Region 141700, Russia b Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospect 53, Moscow 119991, Russia c Tomsk State University, Department of Physics, Lenin Avenue 36, Tomsk 634050, Russia

highlights • Statistics of quantum observable in probability representation. • Tsallis entropy of classical-like probability distributions of quantum state. • The properties of entropic functions of classical-like probability distributions.

article

info

Article history: Received 15 March 2019 Received in revised form 18 June 2019 Available online 28 June 2019 Keywords: Quantum statistics Tsallis entropy Tomographic probability representation of quantum mechanics

a b s t r a c t Qutrit quantum observables are simulated by the set of classical-like random variables. Statistical properties of the quantum observables are expressed in terms of the statistical moments of classical-like observables. The properties of new information characteristics based on Tsallis entropy of the standard probability distributions are investigated in the case of qubit systems. © 2019 Published by Elsevier B.V.

1. Introduction The pure quantum states in the standard formalism of quantum mechanics are identified with wave functions [1] or vectors in a Hilbert space [2]. The mixed quantum states are associated with density operators acting in the Hilbert space [3,4]. Recently the tomographic probability representation of quantum mechanics was suggested for the system with continuous variables [5] and for spin-systems [6,7]. Here, quantum states are identified with fair probability distributions. In particular, the spin tomogram is the probability of spin projection m on an arbitrary direction in the space given by a ⃗ . Hence, we can describe the state of spin systems by the functions of the unit vector n⃗ and integer number m. unit vector n By giving spin tomogram function, we can reconstruct the density operator of the quantum state. For further information on the tomographic probability representation of quantum mechanics, we refer to review [8]. The attempt to find such construct probability representation of quantum states is related to many works [9–12] as well as works associated with QBism [13–15]. ∗ Corresponding author at: Department of General and Applied Physics, Moscow Institute of Physics and Technology (State University), Institutskiy per. 9, Dolgoprudny, Moscow Region 141700, Russia. E-mail addresses: [email protected] (A. Avanesov), [email protected] (V.I. Man’ko). https://doi.org/10.1016/j.physa.2019.121898 0378-4371/© 2019 Published by Elsevier B.V.

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A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

Recently, quantum suprematism or probability representation of qubit states was suggested [16–18]. In this picture, the two-level atom states are described by three probability distributions of three dichotomic random variables and a generalization to the description of qudit states in terms of the probabilities is proposed [19]. This approach was also used to construct a new geometric interpretation of spin-1/2 (qubit) states in terms of the Triada of Malevich’s squares [20]. The quantum suprematism picture is based on tomographic probability and Bloch representations of qubit states [21]. In the case of the two-level quantum system, the density matrix of its state can be parametrized by three real parameters [22]. Therefore, the probability of some spin projection on an arbitrary direction in the space is expressed via these parameters. Thus, we need only three values of the tomographic function to reconstruct the density matrix. The associated probability distributions can describe the qubit state. They also correspond to classical-like random variables that can be used in the description of quantum observables [17]. Thus, the quantum state is the set of classical-like probability distributions. We are also able to describe the quantum observables by means of the classical-like random variables. Therefore, it is possible to consider our formalism as a method of classical simulation of quantum systems. The exact mappings between classical and quantum processes were discussed in several papers, for instance in [23]. The tomographic probabilities can be combined in one vector. As the sum of its elements is fixed, we can normalize it. However, the conditions imposed on the constructed vector must be stronger than normalization due to the properties of the density matrix of the quantum state. We can consider this object as a non-classical probability distribution. The nonclassical (non-Kolmogorov) probability theories, as well as the difference between classical and quantum probabilities, were studied in [24,25]. The actual quantum statistics theory was developed in classical works [26–28]. The aim of this work is to study in detail statistics of the qubit and qutrit states and introduce new entropic characteristics of the qubit states based on Shannon and Tsallis entropies [29,30] of the standard probability distributions. Some connections between tomography representation of quantum states and entropic functions were discussed in [31,32]. As we describe the quantum observable as a set of classical-like random variables, it is important to find the connection between statistical properties of classical-like random variables and the properties of the quantum observable [33,34]. It is easy to show that the mean value of the qubit observable is the sum of the mean values of the classical-like variables. In the paper [35], authors obtained a recurrent relation between the highest moments of the qubit observable and the mean values of its classical-like random variables by using the notion of generation function. The entropic characteristics of the qubit states in the suprematism picture were also discussed in the papers [33]. In the present work, we study the same problem by using the notion of Tsallis entropy of classical random variables. The article is organized as follows. We provide with introduction of the suprematism picture of qubit states in Section 2. In the same section, we present the expression for the highest moments of the quantum observable in terms of the mean values of the classical-like random variables. In Sections 3 and 4 the statistical properties of qutrit observable in terms of the probability representation are studied. In particular, in Section 3 we obtain the relation for the mean value of qutrit observable and in Section 4 its highest moments is investigated. Finally, in Section 5 we introduce the notions of generalized Tsallis entropy and find its maximal and minimal values in the case of qubit systems. In Section 6 we summarize the main results and make the conclusions. 2. The highest moments of quantum observable In the probability representation of quantum mechanics, the mean value of the qubit state can be expressed as a simple sum of mean values of classical-like variables. Indeed, we can show that by performing the following steps. Firstly, let us consider three classical probability vectors

ξ1 =

[

]

X , −X

ξ2 =

[

]

Y , −Y

ξ3 =

[ ]

Z1 . Z2

(1)

The probability distribution for the introduced classical variable can be presented in the form

[ ⃗1 = P

p1 1 − p1

]

,

[ ⃗2 = P

p2

1 − p2

]

,

[ ⃗3 = P

p3

1 − p3

]

.

(2)

In the case, the mean values of introduced classical variables have the form

⟨ξ1 ⟩ = (2p1 − 1)X ,

⟨ξ2 ⟩ = (2p2 − 1)Y

⟨ξ3 ⟩ = p3 Z1 + (1 − p3 )Z2 .

(3)

Let us construct a Hermitian matrix from the classical variables ξ1 , ξ2 and ξ3 Aˆ =

[

Z1

X + iY

]

X − iY . Z2

(4)

As we do not use any restriction on the values of parameters X , Y and Z , it is possible to conclude that Aˆ is an arbitrary Hermitian matrix. Thus, the parametrization (4) is suitable to describe quantum observables. We must point out that the expression (4) provides with the ordinary way to parametrize Hermitian matrix, but here we interpret the real parameters X , Y and Z as classical variables.

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

3

The parameters of probability distributions (2) can be combined in the matrix

[ ρˆ = (

(

) p3 ( ) p1 − 12 + i p2 − 12

p1 −

1 2

( )] − i p2 − 12 . 1 − p3 )

(5)

If we also require that the parameters p1 , p2 and p3 satisfy the condition

(

1

p1 −

)2

( )2 ( )2 1 1 1 + p2 − + p3 − ≤ ,

2

2

2

(6)

4

then the matrix ρˆ is the density matrix, that describes the state of a quantum two-level system (a qubit system). By introducing the real numbers ri = pi − 12 (i ∈ {1, 2, 3}) we come to Bloch parametrization of qubit density matrix

[

1

]

1 + r3 2 r1 + ir2

ρˆ =

r1 − ir2 . 1 − r3

(7)

The nonnegativity of eigenvalues of the density matrix leads to the inequality The mean value of an arbitrary quantum observable has the form

2 i ri

∑

⟨A⟩ = Trρˆ Aˆ .

= 1 that is equivalent to (6). (8)

Finally, the simple calculations allow us to conclude that the mean value of the qubit observable (4) can be presented as the sum of the mean values of the introduced classical variables ξi

⟨A⟩ = ⟨ξ1 ⟩ + ⟨ξ2 ⟩ + ⟨ξ3 ⟩.

(9)

The next target is the connections between the highest moments of quantum observable and statistical properties of classical variables. In the case of qubit states, it could be found relatively easily. First of all, one can present an arbitrary Hermitian matrix in the form Aˆ = a · Iˆ +

3 ∑

uj σˆ j ,

(a, u1 , u2 , u3 ∈ R) ,

(10)

j=1

where σˆ 1,2,3 are matrices of Pauli

σˆ 1 =

[

0 1

]

1 , 0

σˆ 2 =

[

] −i

0 i

0

,

σˆ 3 =

[

]

1 0

0 . −1

(11)

The real parameters can be expressed in terms of the classical-like variables (1) u1 = X ,

u2 = Y ,

Z 1 = a + u3 ,

Z 2 = a − u3 .

(12)

We need to find the expression for Aˆ n )n ( )n ( )n ( )n ( ∑ ∑ ∑ ∑ uj σˆ j aIˆ + uj σˆ j − aIˆ − uj σˆ j aIˆ + uj σˆ j + aIˆ − + . (13) Aˆ n = 2 2 ∑ There are only even degrees of uj σˆ j in the numerator of the first fraction and the odd degrees in the second fraction. Let us make the designation u= As

(∑

√

u21 + u22 + u23 .

uj σˆ j

Aˆ n =

)2k

(14)

= u2k Iˆ and

(∑

(a + u)n + (a − u)n 2

uj σˆ j

Iˆ +

)2k+1

= u2k

∑

uj σˆ j we obtain

3 (a + u)n − (a − u)n ∑

2u

uj σˆ j .

(15)

j=1

Using the expressions (10) and (8) we finally come to the result

⟨An ⟩ =

(a + u)n (u − a) + (a − u)n (u + a) 2u

+

(a + u)n − (a − u)n 2u

⟨A⟩.

(16)

Here, the parameters a and u are the functions of classical-like variables X , Y and Z1,2 which expression can be found from (12) a=

Z1 + Z2 2

√ ,

u=

X2 + Y 2 +

(Z1 − Z2 )2 4

.

(17)

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A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

ˆ they depend on Thus, we find the interesting property of the highest moments of an arbitrary quantum observable A: mean value ⟨A⟩ and on classical-like variables. It means that the dependence on state parameters is contained only in mean value term ⟨A⟩. 3. Statistical properties of quantum observable in case of qutrit systems. Mean value As we introduce classical variables to determine quantum observable for qubit systems and describe its statistical properties, we come to the problem of developing the same approach for more complicated systems. In the present paper, we consider the case of qutrit systems. In standard formalism, the state of the qutrit system is described by the 3 × 3 density matrix

[ ρ11 ρˆ = ρ21 ρ31

ρ12 ρ22 ρ32

] ρ13 ρ23 . ρ33

(18)

Due to hermiticity of density matrix and nonnegativity of its eigenvalues the state of qutrit is determined by 8 real parameters. In [19] the matrix elements were expressed in terms of probabilities of positive spin-1/2 projection for three artificial qubits. We follow this approach and introduce the probability description of the qutrit states. Let us construct the 4 × 4 density matrix from the qutrit density matrix ρˆ by the following way

[

]

ρˆ ⃗0T

ρˆ →

⃗ 0

,

0

(19)

⃗ is zero-filled vector of size 3. It can be considered that this 4 × 4 density matrix describes the state of a composite where 0 system of two qubits. By using partial trace procedure, we obtain the states of subsystems ρˆ

(1)

ρ + ρ33 = 11 ρ21

] ρ12 , ρ22

[

ρˆ

(2)

[ ρ + ρ22 = 11 ρ31

] ρ13 . ρ33

(20)

However, it is possible to construct the 4 × 4 matrix by another way

[

ρˆ →

⃗T 0

0 ⃗ 0

]

ρˆ

.

(21)

Here, we need the subsystem’s density matrix that contains the element ρ23

ρˆ

(3)

ρ = 22 ρ32

] ρ23 . ρ11 + ρ33

[

(22)

In the previous section, we show that the qubit density matrix can be expressed in terms of the fair probabilities. Let us utilize the parametrization for the density matrices (20) and (22)

⎡ ρˆ (k) = ⎣(

(k)

p1 −

1 2

)

(k)

Here, the parameters pi (k) pi

(

(k)

p3

(k)

p1 −

( ) 1 − i p(k) 2 − 2

( )⎤ 1 − i p(k) 2 − 2 ⎦. (k) 1 − p3

1 2

)

(23)

(i = 1, 2, 3) are the fair probabilities of positive spin-1/2 projection for artificial qubit k, i.e.

= Trρˆ (k) Πi ,

(24)

where

[

1

]

1 · 1 2

ˆ1 = Π

1 , 1

ˆ2 = Π

1 2

[ ·

1

−i

]

i , 1

[ 1 ˆ Π3 = 0

0 0

] (25) (3)

(1)

are projectors corresponding to the positive eigenvalues of operators σˆ 1 , σˆ 2 and σˆ 3 relatively. Note, that p3 = 1 − p3 . (k) Thus, we connect the qutrit state with 8 probability parameters pi that must obey the inequalities 3 ( ∑

(k) pi

−

i=1

1

)2

2

≤

1 4

,

k ∈ {1, 2, 3}.

(26)

The elements of density matrix ρˆ are expressed in terms of probability parameters. However, the validity of (26) does not guarantee the nonnegativity of eigenvalues of the qutrit matrix ρˆ . Indeed, let us consider the matrix 1 ρˆ = · 2 6 0 1

[

2 1 0

0 0 . 4

]

(27)

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

5

It is possible to see that the matrix has negative eigenvalue − 61 , but constructed density matrices of artificial subsystems are positive semidefinite ones, and consequently, the relation (26) are satisfied. Let us use the following notation (1)

p1 = p5 ,

(1)

p1 = p3 , p2 = p4 ,

(2)

p1 = p7 ,

p2 = p6 ,

(2)

p2 = p8 ,

(1) p3

(2) p3

(3)

(3)

= p1 ,

= p2 ,

(28)

then the density matrix of qutrit state has the following expression in terms of the probability parameters

⎡

p1 + p2 − 1

(

1 2

(

1 2

p3 −

) ( ) ⎢( ρˆ = ⎣ p3 − 12 + i p4 − 12 ( ) ( ) p5 − 12 + i p6 − 12

)

( ) − i p4 − 12

( )⎤ − i p6 − 21 ( ) ( )⎥ p7 − − i p8 − 21 ⎦ . 1 − p2 (

p5 −

1 − p1 p7 −

)

(

+ i p8 −

1 2

)

1 2 1 2

)

(29)

We decide that p1 + p2 − 1, 1 − p1 and 1 − p2 determine the one three-dimensional classical distribution. Indeed, the diagonal elements of the density matrix are true probabilities. It is a natural way to combine from them the probability distribution. The other parameters form the dichotomic distributions. Thus, we have 7 probability distributions to describe the state of a qutrit system. In the case of qutrit systems, an arbitrary quantum observable is described by Hermitian matrix 3 × 3. It means that we need 8 real parameters to determine the quantum observable. However, here we introduce 6 two-dimensional classical-like variables and one three-dimensional Z1 Z2 , Z3

[ ] η=

ξj =

[

]

Xj

− Xj

, j ∈ {3, . . . , 8} .

(30)

The corresponding probability distributions are determined by the following expressions Prob (Z1 ) = p1 + p2 − 1,

Prob (Z2 ) = 1 − p1 ,

Prob Xj = pj ,

)

Prob −Xj = 1 − pj ,

( )

(

Prob (Z3 ) = 1 − p2 ,

(31)

j ∈ {3, . . . , 8} .

(32)

These classical-like variables (30) are combined in the matrix Z1

[

X3 − iX4 Z2 X7 + iX8

Aˆ = X3 + iX4 X5 + iX6

X5 − iX6 X7 − iX8 . Z3

]

(33)

Using (8) we obtain

⟨A⟩ = ⟨η⟩ +

8 ∑ ⟨ξj ⟩.

(34)

j=3

4. Statistical properties of quantum observable in case of qutrit systems. The highest moments The derivation of the expressions of the highest moments of quantum observable in the case of qutrit system appears to be not so obvious as the same problem in the case of qubit systems because of the properties of three-dimensional hermitian matrices. We can parametrize the Hermitian matrix by the way Aˆ = aIˆ +

8 ∑

uj χˆ j ,

(35)

j=1

where χˆ j are Gell-Mann matrices 1

χˆ 1 = √

0 0 1

[ χˆ 5 =

3

1 0 0

0 1 0

0 0 0

1 0 , 0

[

0 0 , −2

]

]

[ χˆ 2 = 0 0 i

[ χˆ 6 =

0 0 0

1 0 0

0 −1 0

0 0 , 0

]

] −i 0 0

χˆ 3 = 0 0 0

[ ,

χˆ 7 =

0 1 0

1 0 0

[

0 0 1

0 1 , 0

0 0 , 0

]

]

[ χˆ 4 =

[ χˆ 8 =

0 0 0

0 0 i

0 i 0

−i 0 0 0

0 0 , 0

]

]

−i .

(36)

0

The introduced parametrization can be expressed in terms of classical-like variables (30) by using the Eq. (33) 1 Z1 = a + √ u1 + u2 , 3

1 Z 2 = a + √ u1 − u2 , 3

2 Z3 = a − √ u1 , 3

Xj = uj , j ∈ {3, . . . , 8}.

(37)

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A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

Table 1 Anticommutator relations of Gell-Mann matrices.

{}

χˆ 1

χˆ 1

4 I 3 √2 3 √2 3 √2 3

χˆ 2 χˆ 3 χˆ 4 χˆ 5 χˆ 6

ˆ−

χˆ 1 3

√2

χˆ 2

χˆ 2

χˆ 3

χˆ 2 3 4ˆ I + √2 χˆ 1 3 3

√2

χˆ 4

χˆ 5

χˆ 6

χˆ 7

χˆ 8

− √13 χˆ 5

− √13 χˆ 7

− √13 χˆ 8

0

χˆ 5

− √13 χˆ 6

0

χˆ 7

χˆ 8

χˆ 5

χˆ 6

−χˆ 8 4ˆ I−

χˆ 7

χˆ 6

−χˆ 5

χˆ 3

−χˆ 4

χˆ 3 3

√2

χˆ 4 3

√2

0

0

4 I 3

χˆ 4

0

0

4 I 3

− √13 χˆ 5

χˆ 5

χˆ 7

−χˆ 8

χˆ 6

χˆ 6

χˆ 8

χˆ 7

0

4 I 3

χˆ 7

−χˆ 7

χˆ 5

χˆ 6

χˆ 3

χˆ 4

χˆ 8

−χˆ 8

χˆ 6

−χˆ 5

−χˆ 4

χˆ 3

−

χˆ 7

−

χˆ 8

−

√1 3 √1 3 √1 3

ˆ+

√2 3

χˆ 1

χˆ 6

χˆ 3

ˆ+

χˆ 1 3

√2

3

√1 3

χˆ 1

0

ˆ−

χˆ 1 3

√1

−χˆ 7

χˆ 4 4ˆ I− 3

0

−χˆ 8

χˆ 3 χˆ 1 3

√1

0

ˆ−

4 I 3

√1 3

χˆ 1

Thus, parameters a, u1 and u2 are functions of classical variable η a=

Z1 + Z3 + Z3 3

1

,

(

u1 = √ 3

Z1 + Z2 2

)

− Z3 ,

u2 =

Z1 − Z2 2

.

(38)

Let us start from deriving the expressions for ⟨A2 ⟩ Aˆ 2 = a2 Iˆ + 2

∑

uj χˆ j +

∑

ui uj χˆ i χˆ j .

(39)

i,j

j

Here, we need to discuss the properties of matrices χˆ j . Let us recall the anticommutator relation for Gell-Mann matrices

∑ { } 4 dijk χˆ k , χˆ i , χˆ j = δij Iˆ + 3

(40)

k

where in our notation d331 = d441 = d221 = −d111 = −2d551 = −2d661 = −2d771 = −2d881 = √2 and 3 d357 = d368 = −d458 = d467 = d255 = d266 = −d277 = −d288 = 1. The rest dijk can be found in consideration that permutation of indexes does not change the structure constant. All possible anticommutators of Gell-Mann matrices are presented in Table 1. We can present the expression of Aˆ 2 in terms of anticommutators of matrices χj Aˆ 2 = a2 Iˆ + 2a

∑

uj χˆ j +

j

1∑ 2

ui uj χˆ i , χˆ j .

}

{

(41)

i,j

So, the main difficulty here is that the matrices χˆ j do not anticommute with each other, unlike the matrices of Pauli. By using anticommutator’s relations and parametrization (35) of quantum observable Aˆ we present Aˆ 2 in the form

⎛ Aˆ 2 = 2aAˆ + ⎝

⎞ 2∑ 3

u2j − a2 ⎠ Iˆ +

j

1∑ 2

dijk ui uj χˆ k .

(42)

i,j,k

For the last term, we have

∑ i,j,k

) 1 ( 2 dijk ui uj χˆ k = − √ 2u21 − 2u22 − 2u23 − 2u24 + u25 + u26 + u27 + u28 χˆ 1 + √ u1 u2 χˆ 2 + 3 3 (

) ( ) 2 2 √ u1 u3 + u5 u7 + u6 u8 χˆ 3 + √ u1 u4 + u6 u7 − u5 u8 χˆ 4 + 3 3 ( ) ( ) 1 1 + − √ u1 u5 + u2 u5 + u3 u7 − u4 u8 χˆ 5 + − √ u1 u6 + u2 u6 + u3 u8 + u4 u7 χˆ 6 + ( 3 ) ( 3 ) 1 1 + − √ u1 u7 − u2 u7 + u3 u5 + u4 u6 χˆ 7 + − √ u1 u8 − u2 u8 + u3 u6 − u4 u5 χˆ 8 . 3

3

(43)

From Eq. (34) we can conclude that

⟨aIˆ + u1 χˆ 1 + u2 χˆ 2 ⟩ = ⟨η⟩,

⟨uj χˆ j ⟩ = ⟨ξj ⟩, j ∈ {1, . . . , 8}.

Then, for the term (43) we obtain

⟨

⟩ ∑ i,j,k

dijk ui uj χˆ k

2

2

= − √ au1 + √ u1 (⟨η⟩ + ⟨ξ3 ⟩ + ⟨ξ4 ⟩) + 3

3

(44)

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

(

)

1

+ u2 − √ u1 (⟨ξ5 ⟩ + ⟨ξ6 ⟩) −

(

3

7

)

1

√ u1 + u2 (⟨ξ7 ⟩ + ⟨ξ8 ⟩) + 2L(p1 , . . . , p8 ),

(45)

3

where L(p1 , . . . , p8 ) is linear function of the state probabilities

) 1 ( 2L(p1 , . . . , p8 ) = √ 4u21 − 2u22 − 2u23 − 2u24 + u25 + u26 + u27 + u28 (p1 + 2p2 − 2)+ 3 + (u5 u7 + u6 u8 ) (2p3 + 1) + (u6 u7 − u5 u8 )(2p4 + 1) + (u3 u7 − u4 u8 )(2p5 − 1)+ +(u3 u8 + u4 u7 )(2p6 − 1) + (u4 u5 + u4 u6 )(2p7 − 1) + (u3 u6 − u4 u5 )(2p8 − 1).

(46)

Finally, we come to the expression for the second moment of qutrit observable Aˆ

√

u1 ⟨Aˆ 2 ⟩ = 2a⟨Aˆ ⟩ + √ (⟨η⟩ + ⟨ξ3 ⟩ + ⟨ξ4 ⟩) + 3

√

3u2 − u1

√

2 3

(⟨ξ5 ⟩ + ⟨ξ6 ⟩) −

3u2

u1 +

(⟨ξ7 ⟩ + ⟨ξ8 ⟩) + ⎛2 3 ⎞ 2∑ 2 au 1 +L(p1 , . . . , p8 ) + ⎝ uj − a2 − √ ⎠ . √

3

j

3

(47)

Let us discuss the case of the n-moment of qutrit observable. We can present the matrix Aˆ n in the form 8 ∑

Aˆ n = bIˆ +

vj χˆ j .

(48)

j=1

We also can rewrite the terms of the sum

(

)

bIˆ + v1 χˆ 1 + v2 χˆ 2 = aIˆ + u1 χˆ 1 + u2 χˆ 2 + (b − a)Iˆ + (v1 − u1 )χˆ 1 + (v2 − u2 )χˆ 2

(49)

8 ∑ [uj = 0]vj χˆ j ,

(50)

8 ∑

vj χˆ j =

j=3

8 ∑ vj [uj ̸= 0]

uj

j=3

uj χˆ j +

j=3

where [Ω ] is indicator of the set Ω , i.e. [Ω ] = 1 if Ω is true and [Ω ] = 0 otherwise. Note, that ⟨uj χˆ j ⟩ = ⟨ξj ⟩, ⟨aIˆ + u1 χˆ 1 + u2 χˆ 2 ⟩ = ⟨η⟩ and ⟨χˆ j ⟩ = 2pj − 1. Coefficients vi and b are functions of ui and a, so we can present the expression of nth moment of observable Aˆ in the form

⟨Aˆ n ⟩ = ⟨η⟩ +

∑

ˆ ⟨ξj ⟩ + Cj (A)

j=2

ˆ = where Cj (A)

vj uj

8 ∑ [

] ˆ ˆ , ⟨ξj ⟩ = 0 Bj (A)(2p ˆ A) j − 1) + R(ρ,

(51)

j=3

ˆ = vj (for j > 2) and R(ρ, ˆ is the function of parameters p1 , p2 and all uj (for j > 2 and uj ̸ = 0), Bj (A) ˆ A)

ˆ = b − a + √1 (v1 − u1 )(3p2 − 2) + (v2 − u2 )(2p1 + p2 − 2) R(ρ, ˆ A) 3

(52)

5. Information characteristics of classical-like probabilities The entropic functions are widely used as the standard tools to characterize informational properties of the physical systems. There are several types of entropic functions that have been introduced so far. We can describe the state of the quantum system as the set of genuine probability distributions. Therefore, it is also possible to consider the information properties of the state as a function of these distributions. This approach was developed in [33]. Here, we can introduce the entropic function of the state of the quantum system being based on the notion[ of Tsallis entropy that is the generalization ]T ∑m of Shannon entropy. We remind that for a given probability distribution P⃗ = p1 ... pm (here i=1 pi = 1) Shannon entropy reads

⃗ =− S(P)

m ∑

pi ln pi ,

(53)

i=1

and Tsallis entropy has the form

⃗ = Sq (P)

1 1−q

(

m ∑

) q pi

−1 ,

q > 0,

(54)

i=1

and

⃗ = S(P) ⃗. lim Sq (P)

q→1

(55)

8

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

Let us consider the quantum n-level system. The set of n2 − 1 probability parameters determines its state. From these probabilities we can construct n2 − n dichotomic distributions and one distribution of size equal to n. Thus, the quantum state is associated with N = n2 − n + 1 probability distributions. For example, in the case of qubit systems we have p1 p2 p3

[ ] ⃗= P

Ξ (2) =

↔

{[

] [ ] [ ]} p2 p3 , , . 1 − p1 1 − p2 1 − p3 p1

(56)

For the set of probability distributions Ξ (n) = {P⃗1 , . . . , P⃗N } we introduce the following function N ∑

Wq (Ξ (n) ) =

Sq (P⃗i ).

(57)

i=1

In the case of q → 1 we obtain the entropic function based on Shannon entropy W (Ξ (n) ) =

N ∑

S(P⃗i ).

(58)

i=1

In the present paper, the functions Wq and W are called generalized Tsallis entropy and generalized Shannon entropy relatively. For the maximums of functions Wq and W in the case of qubit systems we have max W (Ξ (2) ) = 3 ln 2,

max Wq (Ξ (2) ) =

( )q

(

1 1−q

1

3·

2

) −1 ,

(59)

here Ξ (2) describes the absolutely mixed state, i.e.

Ξ (2) =

{

1 2

[ ] ·

[ ]

[ ]}

1 1 1 1 1 , · , · 1 1 1 2 2

.

(60)

Due to the nonnegativity of the density matrix, the probability parameters pi must obey some relations that can be derived by using Sylvester’s criterion. In the case of qubit system the probability parameters p1 , p2 and p3 satisfy the condition (6). That also imposes the restriction on the minimal value of the functions Wq and W . In the paper, we search the minimum of the function Wq (Ξ (2) ). 5.1. Qubit system The generalized Tsallis entropy in the case of the qubit system can be expressed in the form Wq (Ξ (2) ) = Wq (p1 , p2 , p3 ) =

3 q ∑ p + (1 − pi )q − 1 i

1−q

i=1

.

(61)

We can see that Wq ≥ 0 and Wq = 0

⇔

pi = 0, 1 ∀i.

(62)

However, due to (6) the minimum of function Wq must be positive. All possible quantum states form a convex set. Therefore, the minimum of the generalized entropy is located on the border of the set. Hence, we should solve the convex optimization problem to find the conditional minimum of Wq min Wq (p1 , p2 , p3 ) :

3 ( ∑

pi −

i=1

1

)2

2

=

1 4

.

(63)

Lagrangian of the problem has the form L(p1 , p2 , p3 ) =

3 q ∑ p + (1 − pi )q − 1 i

i=1

1−q

( +µ·

3 ( ∑

pi −

i=1

1 2

)2 −

1 4

) .

(64)

The solution of the problem (63) must satisfy the equations

( ) ( ) · pqi −1 − (1 − pi )q−1 + 2µ · pi − 21 = 0 ∀i = 1, 2, 3 )2 ∑ ( pi − 12 − 14 = 0,

{

q 1−q 3 i=1

or be a singular point of Lagrangian (64). According to the last equation of the system (65) there is at least one parameter pi that is not equal to

(65)

1 . 2

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

9

Let us suppose that pi1 = pi2 = 12 . For the third parameter we have that pi3 = 1 or pi3 = 0. If q > 1, then these values of parameters pi satisfy the Eqs. (65) with q (66) µ= 1−q If 0 < q ≤ 1, the chosen point p1

[

p2

p3

]T

is singular one. The corresponding value of generalized entropy does not (2)

depend on what pi is equal to 1 or 0. Thus, we have six possible states Ξ1 generalized Tsallis entropy and 2

(2)

Wq (Ξ1 ) =

1−q

(

1

·

2q−1

where, for instance p2 = p3 = (2)

Ξ1 =

{[

1 2

] l

that could correspond to the minimum of

) −1 ,

(67)

and

[ ]

[ ]}

1 1 1 1 1 + (−1) , , 1 1 − (−1)l 2 2 1 1 , 2

Let us consider the case of pi1 = system of equation takes the form

pi2 ̸ =

,

l ∈ Z.

1 2

and pi3 ̸ =

(68) 1 . 2

This condition also means that pi2,3 ̸ = 0 and pi2,3 ̸ = 1. The

⎧ ⎪ ⎨µ = fq (pi2 ) fq (pi2 ) = fq (pi3 ) ⎪ ⎩(p − 1 )2 + (p − 1 )2 = 1 , i2 i3 2 2 4

(69)

where fq (x) =

q 1−q

·

xq − (1 − x)q

.

2x − 1

(70)

We can see that fq (x) = fq (y) only if x = y or x = 1 − y. It means

( pi2 −

1

)2

2

( )2 1 1 = pi3 − = . 2

(71)

8

We obtain the following value of function Wq (2) Wq (Ξ2 ) (2)

Here Ξ2

=

2

[(

1

1−q

)q

1

+ √

2

( +

1

] −1 +

− √

2

2 2

)q

1

2 2

(

1 1−q

·

1 2q−1

−1 .

is determined by the parameters pi where only one of them is equal to

(2)

Ξ2 =

{ [ ]

1 1 1 , 2 1 2

[

1+ 1−

] (−1)m √

2 (−1)m √ 2

,

1

[

1+

2

1−

)

1 , 2

for instance p1 =

(72) 1 2

and

]} (−1)n √

,

2 (−1)n √ 2

Finally, let us suppose that none of pi is equal to

1 . 2

m, n ∈ Z.

(73)

The Eqs. (65) take the form

⎧ ⎪ µ = fq (p1 ) ⎪ ⎪ ⎨f (p ) = f (p ) q 1 q 2 f (p ) = f ⎪ q 2 q (p3 ) ⎪ ⎪ )2 ( )2 ( )2 ⎩( p1 − 21 + p2 − 12 + p3 − 12 = 14 ,

(74)

From the properties of function fq (x) we obtain

( p1 −

1

)2

2

( )2 ( )2 1 1 1 = p2 − = p3 − = . 2

2

(75)

12

The corresponding value of function Wq is (2) Wq (Ξ2 ) (2)

Here Ξ2

(2)

Ξ2

=

3 1−q

[( ·

1 2

1

)q

+ √

( +

2 3

1 2

1

)q

− √ 2 3

]

−1 .

(76)

is the set of probability distributions determined by the probability parameters that satisfy (75), i.e.

⎧ ⎡ ⎨1 1 + ⎣ = ⎩2 1 −

(−1)l √ 3 (−1)l √ 3

⎤ ⎦,

1 2

[

1+ 1−

(−1)m √ 3 (−1)m √ 3

] ,

1 2

[

1+ 1−

(−1)n √ 3 (−1)n √ 3

]⎫ ⎬ ⎭

,

l, m, n ∈ Z.

(77)

10

A. Avanesov and V.I. Man’ko / Physica A 533 (2019) 121898

Fig. 1. The dependence of the minimum of the generalized Tsallis entropy on its parameter q in the case of qubit systems. The vertical lines split (2) the space into the regions of values of parameter q where the minimum of function Wq is achieved in particular state Ξi , i ∈ {1, 2, 3}.

(

(2)

Finally, we should compare the values of (67), (72) and (76). The corresponding dependence of function min Wq (Ξ1 ), (2)

(2)

)

Wq (Ξ2 ), Wq (Ξ3 ) on the value of parameter q is presented in Fig. 1. 6. Summary In conclusion, we formulate the main results of our work. Firstly, we studied the statistical properties of quantum observable in the probability representation. As it was shown in the paper [17] quantum observable can be considered as the set of classical-like random variables in the sense that its mean value is simply the sum of the mean values of its classical-random variables. However, the highest moments do not possess such a property. In the present paper, we derived the expression for statistical moments of quantum observable in terms of the classical-like random variables in case of qubit and qutrit states. The recurrent relation for qubit states was obtained in the paper [35]. In the qubit case, we also confirm the dependence of the highest moments on the mean values of the whole quantum observable. However, in the qutrit system case, we only managed to express the dependence of the highest moments on the mean values of the classical-like variables that determine the quantum observable. We also briefly reviewed the notion of generalized Tsallis entropy and derived its possible values in case of qubit systems. In the probabilistic approach to describe quantum states, the function naturally arises as the sum of entropies of the classical-like probabilities that determine the quantum state. Note, the common information characteristics are chosen such that they depend only on eigenvalues of the density matrix of the quantum state. In contrast, the generalized entropy takes different values for different density matrices with equal eigenvalues. It carries more information about possible measurement procedures and their outcomes and statistics. Actually, the same function we can introduce in the case of classical physics as a sum of entropies of classical random variables. However, in the quantum case, there are additional restrictions on the minimum of that function. In the present paper, we solved the optimization problem and obtained the minimum of the generalized Tsallis entropy in case of qubit states. Besides that, it was shown that the state where the function reached its minimum depends on the parameter q. Acknowledgments The work of Vladimir I. Man’ko was supported by the Russian Science Foundation under Project No. 1611-00084. In addition, Vladimir I. Man’ko acknowledges the partial support of the Tomsk State University Competitiveness Improvement Program. References [1] [2] [3] [4]

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