DYNAMICS OF ENTANGLEMENT OF THREE-MODE GAUSSIAN STATES IN THE THREE-RESERVOIR MODEL

Transcription

1 DYNAMICS OF ENTANGLEMENT OF THREE-MODE GAUSSIAN STATES IN THE THREE-RESERVOIR MODEL HODA ALIJANZADEH BOURA 1,,a, AURELIAN ISAR,b, YAHYA AKBARI KOURBOLAGH 1,c 1 Department of Physics, Azarbaijan Shahid Madani University, Tabriz , Iran a : h.alijanzadeh@gmail.com, c : yakbari@azaruniv.edu Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, Reactorului 30, RO-07715, P.O.B. MG-6, Măgurele-Bucharest, Romania b : isar@theory.nipne.ro Received May 1, 016 We describe the dynamics of entanglement of three-mode Gaussian states of a system composed of three bosonic modes, each one immersed in its own thermal reservoir, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. By using the criteria for the separability of three-mode systems, we classify the states by different values of the parameters characterizing the system and the thermal reservoirs. We consider a fully inseparable state as an initial state and show that for definite values of temperature and dissipation constants, the class of entanglement to which the state of the system belongs is changing during its time evolution. For all non-zero values of temperatures of the thermal baths, suppression of entanglement of the initial state always takes place, and in the limit of large times the state is fully separable, corresponding to an asymptotic product state. Key words: Three-mode Gaussian states, open systems, separable states, quantum entanglement. PACS: Yz, Bg, Mn. 1. INTRODUCTION Characterizing and quantifying entanglement represent one of the most crucial problems in quantum information theory [1, ]. In the existing literature on quantum information and communication great attention has been paid to bipartite and multipartite systems of continuous variables. There are many achievements in this direction, mainly in the case of bipartite systems. In particular, there were intensively studied the quantum correlations, including quantum entanglement and quantum discord. In the previous papers we have studied the time evolution of quantum correlations in systems consisting of two-bosonic modes interacting with a thermal environment [3 11]. At the same time the issue of entanglement for multipartite states possesses an even greater challenge, and there exist fewer achievements in this direction [1 18]. The concept of separability originates from the observation by Peres [19] according to which a partial transpose of a density matrix for a separable state is still a Rom. Journ. Phys., Vol. 61, Nos. 7-8, P , Bucharest, 016 v.1.4* #3fae80f8

2 116 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh valid positive definite density matrix. Horodecki [1] proved that this is a necessary and sufficient condition for a state to be separable if the dimension of the Hilbert space does not exceed 6. The Peres criterion leads to a natural entanglement measure called negativity, determined by the negative eigenvalues of the partial transpose of the covariance matrix of the state. Afterwards Vidal and Werner proved that the negativity is an entanglement monotone, and therefore a proper entanglement measure [0]. Furthermore, the logarithmic negativity gives an upper bound to the distillable entanglement [1]. In addition, purity and entanglement of the state are uniquely determined by its covariance matrix [, 3]. Tripartite three mode Gaussian state undergoing parametric amplification and amplitude damping as well as thermal noise was studied in Ref. [4], where the conditions for full separability and full entanglement of the state are worked out. In Ref. [5] there were studied the separability properties and the dynamics of tripartite entanglement under the influence of a dissipative Agarwal bath in a three-mode system of continuous variables. It was shown that if two symmetric modes are propagated in separate baths, the bipartite entanglement between one of the two and the third mode vanishes in a finite time. For fully symmetric tripartite states it was found that entanglement vanishes in a finite time in the presence of separate baths, while it persists for a long time in the presence of a common bath. In this paper we describe the time evolution of the entanglement for a system composed of three bosonic modes coupled to three independent thermal environments. We work in the framework of the theory of open quantum systems and take a fully inseparable state as an initial state of the considered system. We show that for definite non-zero values of the temperatures, entanglement suppression of the initial state takes place. Only in the special case of a fully symmetric system and identical thermal reservoirs, for zero tempertures of the thermal baths the initial fully inseparable state remains fully inseparable for all finite times. However, in the limit of large times for all temperatures the state is always fully separable, corresponding to an asymptotic product state. The structure of the paper is as follows. In Sec. we write the evolution equation for the covariance matrix of the considered open system interacting with a thermal environment. Then in Sec. 3 we investigate the separability of the system of three bosonic modes, each one interacting with its own reservoir, by using the criterion of separability introduced in Ref. [6]. We classify the states in the threereservoir model by various values of the parameters characterizing the system and the thermal reservoirs. A summary is given in the last section.

3 3 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model EVOLUTION OF COVARIANCE MATRIX The covariance matrix σ of an n-mode bosonic system is a real, symmetric and positive n n matrix. We remind that the first moments of the canonical variables of a system, which can always be shifted to zero by using local unitary operations, are irrelevant for the study of entanglement. Therefore, without losing generality, we may take them to be zero. We work only with Gaussian states of zero displacement vectors and such Gaussian states are completely characterized by their covariance matrix, which for a system of three bosonic modes is given by: σ = σ x1 x 1 σ x1 p 1 σ x1 x σ x1 p σ x1 x 3 σ x1 p 3 σ x1 p 1 σ p1 p 1 σ p1 x σ p1 p σ p1 x 3 σ p1 p 3 σ x1 x σ p1 x σ x x σ x p σ x x 3 σ x p 3 σ x1 p σ p1 p σ x p σ p p σ p x 3 σ p p 3 σ x1 x 3 σ p1 x 3 σ x x 3 σ p x 3 σ x3 x 3 σ x3 p 3 σ x1 p 3 σ p1 p 3 σ x p 3 σ p p 3 σ x3 p 3 σ p3 p 3 where the matrix elements are defined as:, (1) σ ξi ξ j = 1 Tr[(ξ iξ j + ξ j ξ i )ρ], i,j = 1,,3, () ξ = (x 1, p 1, x, p, x 3, p 3 ) are the canonical variables (coordinates and momenta) of the three-mode bosonic system and ρ denotes its density operator. The matrix σ is a bona fide covariance matrix iff it satisfies the uncertainty relation [13, 3] where J = 3 i=1 det(σ + i J) 0, (3) [ 0 1 J i, with J i = 1 0 Covariance matrix (1) has the following block structure: A D 1 D 13 σ = D1 T B D 3, (4) D13 T D3 T C where A, B and C are Hermitian matrices which denote the symmetric covariance matrices for the individual reduced one-mode states, while matrices D contain the cross-correlations between modes (T denotes the transposed matrix). The time evolution of the initial covariance matrix σ(0) of the system, under the action of a general Gaussian channel, can be characterized by two matrices X and Y [7, 8]: σ = Xσ(0)X T + Y, (5) ].

4 1164 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh 4 where Y is a positive operator. These two matrices depend on the parameters characterizing the environment. Eq. (5) guarantees that σ is a physical covariance matrix for all finite times t. In the case of one bosonic mode (harmonic oscillator) with the general Hamiltonian H = 1 m p 1 + mω x 1 + µ (x 1p 1 + p 1 x 1 ), (6) where the parameter µ determines the damping ratio with ω > µ, Ω ω µ, the following expressions have been obtained in the framework of the theory of open systems based on quantum dynamical semigroups, where the Markovian time evolution of the density operator is given by the Kossakowski-Lindblad master equation [8 31]: [ X 1 = e λ 1t cosωt + µ Ω sinωt 1 Ω sinωt 1 Ω sinωt cosωt µ Ω sinωt and Y 1 = X 1 s( )X1 T + s( ). (8) Here λ 1 is the dissipation constant and, considering that the asymptotic state of the open system is a Gibbs state [3 34], we obtain s( ) = 1 coth 1 [ ] 1 0, (9) kt where T 1 is the temperature of the thermal reservoir (we take = 1,ω = 1). Consider a system of three identical bosonic modes, each one coupled to its own thermal bath. If the initial three-mode 6 6 covariance matrix is σ(0), then its subsequent evolution is given by σ = (X 1 X X 3 )σ(0)(x 1 X X 3 ) T ] (7) + (Y 1 Y Y 3 ), (10) where X 1,,3 and Y 1,,3 are given by Eqs. (7), (8) and similar ones, corresponding to each bosonic mode immersed in its own thermal reservoir, characterized by temperatures T 1, T and T 3 and dissipation constants λ 1, λ and λ 3, respectively. Fully symmetric tripartite states are invariant under the exchange of any two modes. The standard covariance matrix to describe a fully symmetric state has the fully symmetric form A D D σ = D T A D. (11) D T D T A As an initial state of the considered system we take an entangled three-mode state (formal generalization to the case of three-mode state of the two-mode squeezed

5 5 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model 1165 vacuum state with squeezing parameter r), with the covariance matrix of the form: cosh r 0 sinh r 0 sinh r 0 σ(0) = 1 0 coshr 0 sinhr 0 sinhr sinh r 0 cosh r 0 sinh r 0 0 sinhr 0 coshr 0 sinhr. (1) sinh r 0 sinh r 0 cosh r 0 0 sinhr 0 sinhr 0 coshr 3. SEPARABILITY IN THE SYSTEM OF THREE BOSONIC MODES The separability problem of three-mode Gaussian states was completely solved [6], namely the separability of a three-mode system can be determined by the positivity of the partially-transposed density matrices. Based on the study [6], threemode Gaussian states can be classified into five different entanglement classes, from fully inseparable states to fully separable states: Class 1. Fully inseparable states are those which are not separable for any grouping of the parties. Class. One-mode biseparable states are those which are separable if two of the parties are grouped together, but inseparable with respect to the other groupings. Class 3. Two-mode biseparable states are separable with respect to two of the three bipartite splits but inseparable with respect to the third. Class 4. Three-mode biseparable states are separable with respect to all three bipartite splits but cannot be written as a mixture of tripartite product states. Class 5. The fully separable states can be written as a mixture of tripartite product states. We will see that the states in the present model can be classified as follows: a) for thermal reservoirs with different temperatures, an initial fully inseparable state (class 1) has an evolution in time through all other four entangled classes -5; b) for a fully symmetric system and identical thermal reservoirs, an initial fully inseparable state (class 1) evolves in time to a three-mode biseparable state (class 4) and finally becomes a fully separable state (class 5). Denoting by Λ j,j = 1,,3, the partial transposition matrices, Λ 1 = diag(1, 1,1,1,1,1), Λ 3 = diag(1,1,1,1,1, 1), Λ = diag(1,1,1, 1,1,1), the partially transposed covariance matrices are given by σ j = Λ j σλ j. Then the criterion for the fully inseparable states is σ j + i J < 0, for all j = 1,,3. For a fully symmetric three-mode state, this criterion can be simplified to, for example σ 1 + i J < 0. (13)

6 1166 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh 6 For σ j + i J 0, the jth mode is separable from the subsystem spanned by the other two modes at time t. The positivity of this expression requires its minimum eigenvalue γ 0. If γ is negative, the j-th mode is inseparable from the tripartite system. In the case of a fully symmetric state, we only concern the positivity of σ 1 + i J for three mode separability. For σ j + i J 0, for all j = 1,,3, the state will be a positive partial transpose (PPT) three-mode state, and it can be biseparable or fully separable. There exists a criterion to distinguish the biseparable and fully separable states [6]. It follows that in order to determine the separability properties of the considered system we have to find γ, which denotes the smallest eigenvalue of Λ j σλ j + i J, j = 1,,3. We consider the separability of two types of tripartite states: fully symmetric states, which are invariant under the exchange of any two modes, and nonsymmetric states. For simplicity, we take first λ j = λ and µ = 0 and also the same temperature T j = T for the three thermal reservoirs. In this case, we obtain a fully symmetric system of three bosonic modes immersed in identical thermal reservoirs, and the eigenvalues are symmetric with respect to all three modes. We obtain the following eigenvalues: a ± (b + d ) ± (b + d )(8 + 9(b + d )), a ± (b + d ), (14) where a = e λt coshr + C (1 e λt ), and C coth 1 kt γ = a 1 b = e λt costsinhr, d = e λt sintsinhr. It can easily be seen that the smallest one is (b + d ) + (b + d )(8 + 9(b + d )). (15) Since σ( ) depends on temperature only, we can affirm that r and λ do not affect the separability at infinity of time. The eigenvalues of Λ j σ( )Λ j + i J are given by C j±1, and since C j 1 (T j 0), we can re-confirm that for all temperatures of the reservoirs, we have a fully separable state (class 5) in the limit of large times, corresponding to the chosen asymptotic Gibbs product state. The evolution of the smallest eigenvalue γ (15) as function of time t and temperature T, for an entangled initial state, is illustrated in Fig. 1, where we consider identical temperatures T and dissipation constants λ. Likewise, in Figs. and 3 we represent the dependence of the set of all three (j = 1,,3) smallest eigenvalues γ

7 7 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model 1167 on time for different values of temperatures and dissipation constants characterizing the thermal reservoirs. Fig. 1 Evolution of smallest eigenvalue γ (15) on time t and environment temperature T (through coth kt 1 ) for a fully inseparable initial state with squeezing parameter r = 1 and dissipation λ = 0.1. For identical temperatures T and dissipation constants λ, the system is symmetric and the eigenvalues display a permutational symmetry among the three modes. It follows that the states can belong to classes 1, 4 and 5 only. Indeed, from Fig. 1 it can be seen that initially the state is a fully inseparable state (class 1), for temperatures T > 0 the suppression of entanglement of the initial state takes place at some finite moment of time, when the state becomes a three-mode biseparable state (class 4), and in the limit of large times the state becomes a fully separable state, corresponding to an asymptotic product state (class 5). For zero temperature the initial state remains fully inseparable for all finite times and becomes fully separable in the limit of infinite time. If the three temperatures T j > 0 are different one from each other, it would be possible to reach also classes and 3 for some finite time, but always the states belong finally to class 5. Indeed, from Figs. and 3 we can notice that for different temperatures of the reservoirs, the state is initially a fully inseparable state (class 1) and during its evolution it becomes first an one-mode biseparable state (class ), then a two-mode biseparable state (class 3), and after another finite time a threemode biseparable state (class 4). We remind that always, for large times, the state is fully separable and belongs to class 5. If we take equal temperatures only for two reservoirs and equal dissipation constants for the corresponding two modes, then the initial fully inseparable state can belong during its evolution to only one of classes or 3 for some finite time, since these two mode manifest the same behaviour and

8 1168 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh 8 therefore their corresponding smallest eigenvalues γ coincide and after another finite time to class 4. Fig. Evolution of the set of all three (j = 1,,3) smallest eigenvalues on time t for a fully inseparable initial state, for r = 1, coth kt 1 1 =.5, coth kt 1 = 3.5, coth kt 1 3 = 1.5 and λ = 0.1. To demonstrate the influence of dissipation, we also consider in Fig. 4 identical temperatures of the three thermal reservoirs, and only two identical dissipation constants. The evolution of the smallest eigenvalues γ of the three modes on time and dissipation shows that the values of λ which change the sign of γ are different for modes interacting with environments with different dissipation constants. From Fig. 5 we notice again that the initial entangled state belongs to class 1, during its evolution it moves temporarily to class, then to class 3, after another finite time to class 4, and finally the state becomes fully separable (class 5). From this figure we can also conclude that the influence of dissipation on the evolution of the eigenvalues is relatively stronger than the influence of the temperature, in the sense that in the considered case the class 3 (two-mode biseparable) is less stable than classes 1 (fully inseparable) and (one-mode biseparable). As a general conclusion, we can say that an initial fully inseparable state can have such an evolution that, depending on the parameters of the system and the temperatures and dissipation constants characterizing the environments, it can belong temporarily to different classes of entanglement. As a result of the interaction with the environment, the system gradually becomes more and more separable. As ex-

9 9 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model 1169 Fig. 3 Evolution of the set of all three (j = 1,,3) smallest eigenvalues on time t for a fully inseparable initial state, for r = 1, coth kt 1 1 =, coth kt 1 = 3, coth kt 1 3 = 1 and λ = 0.1. Fig. 4 Evolution of smallest eigenvalues on time t and dissipation λ for a fully inseparable state, for r = 1, coth kt 1 1 = 4, coth kt 1 = 4, coth kt 1 3 = 4, λ 1 = 0.1 and λ = λ 3 = λ.

10 1170 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh 10 pected, the temperature and dissipation strongly influence the dynamics of the separability properties, namely their increasing favorizes the destruction of the entanglement. Fig. 5 Evolution of smallest eigenvalue on time t for a fully inseparable initial state, for r = 1, λ 1 = λ 3 = 0.0, λ = 0.1, coth kt 1 1 = 3, coth kt 1 = coth kt 1 3 = (red line for mode 1, green line for mode, blue line for mode 3). 4. SUMMARY We have determined the dynamics of entanglement of three-mode Gaussian states by using the symplectic formalism of covariance matrices, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. By using the well-known criteria of Ref. [6] for the separability of the three-mode systems, we have classified the states for different values of the parameters characterizing the system and the thermal reservoirs. We considered a fully inseparable state as an initial state and have shown that for definite values of temperature and dissipation constants the classes of entanglement to which the state of the system belongs is changing during its time evolution. For all non-zero values of temperatures of the thermal baths, suppression of entanglement of the initial state always takes place, and in the limit of large times the state is fully separable, corresponding to an asymptotic product state. Temperatures and dissipation of the thermal reservoirs strongly influence the dynamics of entanglement of the three-mode bosonic system.

11 11 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model 1171 REFERENCES 1. A. Ekert, Phys. Rev. Lett. 67, 661 (1991).. C. H. Bennet, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). 3. A. Isar, Phys. Scr. 8, (010). 4. A. Isar, Phys. Scr. T 147, (01). 5. A. Isar, Rom. Rep. Phys. 65, 711 (013). 6. A. Isar, Rom. J. Phys. 58, 599 (013). 7. A. Isar, Rom. J. Phys. 58, 1355 (013). 8. T. Mihaescu and A. Isar, Rom. J. Phys. 60, 853 (015). 9. S. Suciu and A. Isar, Rom. J. Phys. 60, 859 (015). 10. H. A. Boura and A. Isar, Rom. J. Phys. 60, 178 (015). 11. H. A. Boura, A. Isar, and Y. A. Kourbolagh, Rom. Rep. Phys. 68, 19 (016). 1. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 3, 1 (1996). 13. R. Simon, Phys. Rev. Lett. 84, 76 (000). 14. P. V. Loock and A. Furusawa, Phys. Rev. A 67, (003). 15. T. C. Wei and P. M. Goldbart, Phys. Rev. A 68, (003). 16. G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 73, (006). 17. O. Gühne and M. Seevinc, New J. Phys. 1, (010). 18. S. Olivares, Eur. Phys. J. Special Topics 03, 3 (01). 19. A. Peres, Phys. Rev. Lett. 76, 1413 (1996). 0. G. Vidal and R. F. Werner, Phys. Rev. A 65, (00). 1. F. Verstraete, K. Audenaert, J. Dehaene, and B. D. Moor, J. Phys. A: Math. Gen. 34, 1037 (001).. X. Shao-Hua, S. Bin, and S. Ke-Hui, Chin. Phys. Lett. 6, (009). 3. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 7 (000). 4. X. Chen, J. Phys. A: Math. Theor. 41, (008). 5. Y. Zhao, F. Zheng, J. Liu, and Y. Yao, Physica A 43, 80 (015). 6. G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev. A 64, (001). 7. T. Heinosaari, A. S. Holevo, and M. M. Wolf, Quantum Inform. Comput. 10, 0619 (010). 8. A. Isar, A. Sandulescu, H. Scutaru, E. Stefanescu, and W. Scheid, Int. J. Mod. Phys. E 3, 635 (1994). 9. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 81 (1976). 30. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 31. A. Sandulescu and H. Scutaru, Ann. Phys. 173, 77 (1987). 3. A. Isar, Helv. Phys. Acta 67, 436 (1994). 33. A. Isar, Phys. Scr. T160, (014). 34. A. Isar, J. Russ. Laser Res. 35, 1 (014).