Non-extensive statistical entropy, quantum groups and quantum entanglement

Physica A 391 (2012) 3413–3416

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Non-extensive statistical entropy, quantum groups and quantum entanglement M.A.S. Trindade a , J.D.M. Vianna b,c,∗ a

Departamento de Ciências Exatas e da Terra, Universidade do Estado da Bahia, Rodovia Alagoinhas/Salvador, BR 110, Km 03, 48040-210, Alagoinhas, Bahia, Brazil b Instituto de Física, Universidade Federal da Bahia, Campus Ondina, 40210-340 Salvador, Bahia, Brazil c

Instituto de Física, Universidade de Brasília, Campus Darcy Ribeiro, 70910-900 Brasília, DF, Brazil

article

info

Article history: Received 20 August 2011 Received in revised form 19 December 2011 Available online 6 February 2012 Keywords: Tsallis entropy Quantum groups Entanglement

abstract In this work, we explore a new connection between quantum groups and Tsallis entropy through the energy spectrum of a Hamiltonian with SUq (2) symmetry. Identifying the deformation parameter of the entropy with the parameter of deformation of the associated quantum group, we deduce Tsallis entropy for states related to such a system with SUq (2) symmetry and conducted an investigation of quantum entanglement. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Non-extensive Tsallis statistics [1] is a generalization of the Boltzmann–Gibbs formulation of statistical mechanics; it violates the additive property, a part of the Callen third-postulate. In fact, in the non-extensive statistics, for a composed system (A + B) formed from two independent systems A and B, the total entropy does not correspond to the sum of the entropies of A and B, but there is an extra term (1 − q)/kSqA SqB , where k is the Boltzmann constant, q is an entropic parameter and in the limit q → 1 the extensive property is recovered. The non-extensive entropy of Tsallis has been applied successfully to a myriad of physical [2], biological [3], economical [4] and social [5] problems. Concurrently with applications in various contexts, mathematical aspects of the theory [6–8] have also been developed. One of the most exploited aspects refers to quantum groups [9], a mathematical structure related to deformations of Lie algebras, where one has also a parameter q whose limit q → 1 recovers the original symmetry [10,11]. Besides there are mathematical developments related to quantum entanglement [12–15]. The quantum groups are quasi-triangular Hopf algebras [9] and can be obtained from deformations of Lie algebras. In the context of quantum optics, quantum groups were introduced producing a generalization of the Jaynes–Cummings model of a two-level atom in a cavity [16]. An interesting connection with the squeezed coherent states was established by Celleghini et al. [17]. In this work it has been shown that the complex parameter q gives rise to an additional degree of freedom in the space of parameters controlling the squeezing; an expression for the energy spectrum suggesting an experimental test to observe the q-effects has also been proposed. In Ref. [18] a realization of the quantum Heisenberg–Weyl algebra in terms of operators of finite differences has been presented and it was showed that whenever a finite scale is involved in a selfcontained physical theory, a q-deformation of the observable algebras occurs, with the deformation parameter related to the finite spacing. Such a development can be applied to mathematical understanding of lattice quantum mechanics and Bloch

∗

Corresponding author at: Instituto de Física, Universidade Federal da Bahia, Campus Ondina, 40210-340 Salvador, Bahia, Brazil. E-mail addresses: [email protected] (M.A.S. Trindade), [email protected] (J.D.M. Vianna).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.01.022

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M.A.S. Trindade, J.D.M. Vianna / Physica A 391 (2012) 3413–3416

functions. Other authors exploited the quantum groups in the treatment of many issues related to quantum gravity [19] and molecular spectroscopy [20,21]. The q-parameter has also been related to temperature in the study of thermal field theory [22]. The first connection of quantum groups with non-extensive entropy was performed by Tsallis [23]. In this work, starting from the bosonic oscillator algebra and the average values of observables, Tsallis proposed a relationship between the parameter q of his statistics and the parameter qG of the harmonic oscillator algebra. In 1997, Abe [24] shows using the q-calculus that Tsallis entropy contains intrinsically the deformation concept in its structure; based on this idea, he presents a new generalized entropy which represents a modification of the symmetric Tsallis entropy. In 2002, Lavagno and Narayana Swamy [25] present a realization of thermostatistics, also based on the formalism of q-calculus and using the Jackson derivative instead of the ordinary thermodynamic derivative. Then they discuss a connection between their q-deformed expressions and those obtained by Tsallis, noting that despite the existence of certain similarities, such as the non-additivity property, the new expressions represent different deformations. Also in 2002, Ubriaco [26] discusses the implications of the introduction of the invariance group SU q (2) in non-extensive quantum statistical mechanics and shows that invariance of the matrix density leads to a thermodynamics which is equivalent to that of the Bose–Einstein formulation. Quantum entanglement is a key ingredient in the theory of quantum computation and quantum information where entropy plays a fundamental role [27]. In this context the quantum Tsallis entropy has been investigated, allowing the characterization of entanglement in non-extensive formulation: Hamadou-Ibrahim et al. [15], for example, presented a sufficient criterion for the entanglement based on Tsallis entropy, and investigated the dynamics of entanglement in a system of two qubits interacting with the environment. In the present work we explore a relationship between non-extensive quantum statistical mechanics and quantum group SU q (2) through the energy spectrum of a Hamiltonian invariant under this group. As it is known symmetries of physical systems are related to the Hamiltonian and consequently to its spectrum; so, it is possible to explore these symmetries using the canonical ensemble and physical quantities obtained from this ensemble, for example. Following such a construction we consider a Hamiltonian with SU q (2) symmetry to determine Tsallis entropy where the q-parameter of the deformed algebra suq (2) is identified to the entropic parameter. Our development differs from the Ubriaco approach [26] since our operators are a realization of the SU q (2) group and not covariant operators by the action of this group; it is close to proposals that seek experimental evidence of the symmetries of quantum groups in the vibrational and rotational molecular motions, but our motivation is fully different. This communication is organized as follows. Section 2 is devoted to present the Hamiltonian invariant under SU q (2) and the analysis of its spectrum, the corresponding density operator and hence the quantum Tsallis entropy. As a physical application of our result we consider in Section 3 the Werner state and we obtain a sufficient criterion for the entanglement of this state identified as a state associated to a Fock subspace of the system. Final remarks are presented in Section 4. 2. Quantum group and Tsallis entropy Consider the algebra of q-deformed harmonic oscillator: aaĎ − q−1 aĎ a = qN ;

aaĎ − qaĎ a = q−N , qN aĎ q−N = qaĎ ;

qN aq−N = q−1 a;

q±N q∓N = 1,

(1)

where a and aĎ are the creation and annihilation operators, respectively, and q is an arbitrary real parameter. In the Fock space, for real q, we have aĎ |n⟩ =



[n + 1]q |n + 1⟩;

qN |n⟩ = qn |n⟩; where [A]q =

qA −q−A q−q−1

Ď

J + = a1 a2 ;

a| n⟩ =



[n]q |n − 1⟩;

with N |n⟩ = n|n⟩; n = 0, 1, 2, . . . ,

(2)

and N is the number operator. We introduce operators J+ , J− and J0 as Ď

J − = a2 a1 ;

J0 =

1 2

(N1 − N2 ),

(3)

where we consider two deformed bosonic oscillators, which we call the subsystem 1 and subsystem 2, each obeying relations (1) and (2). Then we obtain the deformed algebra suq (2) [J0 , J± ] = J± ,

[J+ , J− ] = [2J0 ]q .

(4)

Consider now the Hamiltonian with symmetry SU q (2) [10,11]

Hq = γ C = γ

 J0 −

1

2

2

q

 + J+ J− ,

(5)

M.A.S. Trindade, J.D.M. Vianna / Physica A 391 (2012) 3413–3416

3415

 1 2

with C = J0 − 2 + J+ J− , an Casimir operator and γ a constant to be determined. Noting that in the Fock space basis set {|n1 , n2 ⟩} we have 2  2  1 1 |n1 , n2 ⟩ = (n1 − n2 − 1) |n1 , n2 ⟩, J0 −



2

2

q

q

J+ J− |n1 , n2 ⟩ = [n1 ]q [n2 + 1]q |n1 , n2 ⟩,

(6)

where n1 and n2 are the number of bosons in modes 1 and 2, respectively, we obtain to the spectrum of the Hamiltonian (5) Eq = γ



1 2

(n1 − n2 − 1)



2

+ [n1 ]q [n2 + 1]q .

(7)

q

The corresponding density operator in the canonical ensemble becomes

ρ=

1 Z





exp −βγ

J0 −

1

2

2

q

 ,

+ J+ J−

(8)

and the partition function Z can be written as ∞ 

Z =





exp −βγ

1 2

n1 ,n2 =0

 2 (n1 − n2 − 1) + [n1 ]q [n2 + 1]q .

(9)

q

As Tsallis entropy is defined by Ref. [15] Sq =

1 q−1

(1 − Trρ q ),

(10)

in this case, with Eq. (8), we have Sq =



1

∞ 

1−

q−1





1

exp −βγ

Z

n1 ,n2 =0



1 2

q  2 (n1 − n2 − 1) + [n1 ]q [n2 + 1]q .

(11)

q

We note that in this expression of Tsallis entropy the deformed Lie algebra parameter q corresponds to the parameter q which characterizes non-extensive systems. It is interesting to note that in Tsallis construction [23] the q-parameter of the non-extensive statistics is different from the deformed oscillator parameter qG and the relationship between q- and qG -parameter depends on the temperature. 3. Entanglement condition According to Hamadou-Ibrahim et al. [15] a sufficient condition for entanglement is given by Dq ≡ Sq1 − Sq > 0,

(12)

where Sq1 is the entropy corresponding to a subsystem 1. Let us consider in Fock space a subspace whose basis set is given by B = {|00⟩, |01⟩, |10⟩, |11⟩}. Consider then a Werner state [28], i.e.,

ρ = p|ψ⟩ ⟨ψ| +

( 1 − p) 4

I,

(13)

√

where 1/3 < p ≤ 1, I is the identity operator on this subspace and |ψ⟩ = 1/ 2(|00⟩ + |11⟩). In this case, we have from (10) Sq =



1

  1− 3

q−1

1−p

q

 +

4

1 + 3p

q 

4

,

(14)

and taking the partial trace over subsystem 1, we have Sq1 =

1

 q 



q−1

1−2

1 2

,

(15)

so that the condition (12) results 1 q−1

  3

1−p 4

q

 +

1 + 3p 4

q 

 q  −2

1

2

> 0.

(16)

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M.A.S. Trindade, J.D.M. Vianna / Physica A 391 (2012) 3413–3416

In the limit q → 1, the inequality (16) becomes

 3

1−p 4



 ln

1−p



4

 +

1 + 3p 4



 ln

1 + 3p 4



  − ln

1 2

> 0,

(17)

and we can ensure that for p greater than 0.75 (approximately) the Werner state is entangled.  The case q = 2 must be observed. In this situation, we can guarantee the existence of entanglement in the interval

1 3

< p ≤ 1. In Ref. [15]

Hamadou-Ibrahim et al. note that D2 is an experimentally accessible entanglement indicator based on Ref. [29]. We conjecture that the Tsallis q-parameter, which here appears explicitly in the Hamiltonian that describes the physical system, can have a deeper connection with the quantum non-locality and experimental tests can confirm this conjecture based on the spectrum of the deformed Hamiltonian and in the observation of non-classical correlations through D2 . 4. Conclusions and perspectives In summary, we have deduced Tsallis entropy in a canonical ensemble using a Hamiltonian Hq invariant under the quantum group SU q (2). Notice that our expressions identify the entropic parameter q with the parameter of deformation of the quantum group SU q (2). In fact, considering that symmetry properties of a system appear in the thermodynamic quantities associated with the spectrum of the Hamiltonian in question, we have shown as it is possible to calculate a q-entropy from Hq . As an application of our result we have obtained a sufficient condition for entanglement to the Werner state obtained from a subspace  of the system under consideration; in particular for q = 2 we have found the existence of entanglement in the interval

1 3

< p ≤ 1, a result related to experimentally measurable entanglement indicator D2 . Our

result indicates that we can determine statistical quantities in the non-extensive theory by exploring symmetry properties and the correspondent deformed Hamiltonians and suggest a relationship between the q-parameter and the quantum non-locality. The application of our development to other deformed symmetries is in progress and will be published in forthcoming paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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