LOGARITHMIC NEGATIVITY OF TWO BOSONIC MODES IN THE TWO THERMAL RESERVOIR MODEL

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1 LOGARITHMIC NEGATIVITY OF TWO BOSONIC MODES IN THE TWO THERMAL RESERVOIR MODEL HODA ALIJANZADEH BOURA 1,2, AURELIAN ISAR 2 1 Department of Physics, Azarbaijan Shahid Madani University, Tabriz , Iran 2 Department of Theoretical Physics Horia Hulubei National Institute for Physics and Nuclear Engineering Reactorului 30, RO , POB-MG6, Măgurele-Bucharest, Romania h.alijanzadeh@gmail.com, isar@theory.nipne.ro Received January 23, 2015 We describe the time evolution of logarithmic negativity, which gives the entanglement strength of two-mode Gaussian states in terms of the covariance matrix, for a system of two bosonic modes each one interacting with its thermal reservoir. We take the squeezed thermal states as initial states and show that at a definite moment of time the suppression of entanglement (entanglement sudden death) takes place for all nonzero temperatures of the thermal baths. For zero temperature of both thermal reservoirs, an initial entangled squeezed thermal state, in particular a squeezed vacuum state, remains entangled for all finite times, but in the limit of infinite time the state becomes separable. Key words: Gaussian states, open systems, logarithmic negativity, squeezed thermal states. PACS: Yz, Bg, Mn. 1. INTRODUCTION Entanglement or inseparability is one of the main concepts in all branches of quantum information theory. Different entanglement measures have been introduced for mixed states. One of such measures is the logarithmic negativity of the state, which is easily computable and gives an upper bound to the distillable entanglement. Vidal and Werner proved that the logarithmic negativity is computable for general Gaussian states [1]. In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, for a system consisting of two bosonic modes immersed in two independent thermal reservoirs, it was described the time evolution of the Simon separability function [2 4] and of the quantum entropic discord and classical correlations of the two bosonic modes [5]. Gaussian states of infinite dimensional quantum systems play an important role in quantum information: they can be created relatively easily and can be used in quantum cryptography and quantum teleportation [6 8]. In this paper, we describe RJP Rom. 60(Nos. Journ. Phys., 9-10), Vol , Nos. 9-10, (2015) P , (c) 2015 Bucharest, - v.1.3a*

2 2 Logarithmic negativity of two bosonic modes in the two thermal reservoir model 1279 the evolution of the logarithmic negativity in terms of time, temperatures, squeezing parameter and average number of thermal photons for a system composed of two bosonic modes coupled to two independent thermal environments. We employ the same theory of open systems and take entangled squeezed thermal states as initial states of the considered system. We show that entanglement suppression of the initial state takes place at some definite moment of time, for all non-zero temperatures of the thermal baths, in agreement with the previously obtained results [4, 9 11]. For zero temperature of both thermal reservoirs, an initial entangled squeezed thermal state, in particular a squeezed vacuum state, remains entangled for all finite times, but in the limit of infinite time the state becomes separable. 2. MASTER EQUATIONS We study the dynamics of a composed system consisting of two identical noninteraction bosonic modes (harmonic oscillators), each one being in weak interaction with its local thermal environment. In the axiomatic theory of open systems based on completely positive quantum dynamical semigroups, the irreversible time evolution of an open system is described by the following Kossakowski-Lindblad quantum Markovian master equation for the density operator ρ(t) in the Schrödinger representation [12 14]: dρ(t) dt = i [H,ρ(t)] ([V j ρ(t),v j ] + [V j,ρ(t)v j ]), (1) j here means Hermitian conjugation, H denotes the Hamiltonian of the open system and the operators V j, V j, defined on the Hilbert space of H, represent the interaction of the open system with the environment. We are interested in the set of Gaussian states, therefore we introduce quantum dynamical semigroups that preserve this set during time evolution of the system. So H is taken as a polynomial of second degree in the coordinates x, y and momenta p x, p y of the two bosonic modes and V j, V j are taken as polynomials of the first degree in these canonical observables. This evolution represents a Gaussian channel that maps Gaussian states to Gaussian states. By definition, the covariance matrix (CM) σ is a real, symmetric and positive 4 4 matrix. Note that the first moments can always be shifted to zero by a sequence of local unitary operations. Hence, they are irrelevant for the study of entanglement and without lose of generality we may take them to be zero and work only with Gaussian states (GS) of zero displacement vectors. Such GS are completely characterized

3 1280 Hoda Alijanzadeh Boura, Aurelian Isar 3 by their CM: σ(t) = where the matrix elements are defined as: σ xx (t) σ xpx (t) σ xy (t) σ xpy (t) σ xpx (t) σ pxpx (t) σ ypx (t) σ pxpy (t) σ xy (t) σ ypx (t) σ yy (t) σ ypy (t) σ xpy (t) σ pxpy (t) σ ypy (t) σ pypy (t) (2) σ ij = 1 2 Tr[(ξ iξ j + ξ j ξ i )ρ] Tr(ξ i ρ)tr(ξ j ρ), (3) where ξ = (x, p x, y, p y ). It turns out that the matrix σ is a bona fide CM iff it satisfies the uncertainty relation [15, 16] where Ω = 2 i=1 det(σ + i Ω) 0, (4) 2 [ 0 1 J with J = 1 0 CM has the following block structure: [ A C σ(t) = C T B ]. ], (5) where A, B and C are 2 2 Hermitian matrices (T denotes the transposed matrix). A and B denote the symmetric covariance matrices for the individual reduced onemode states, while the matrix C contains the cross-correlations between modes. The 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(t) and Y (t) [17]: σ(t) = X(t)σ(0)X T (t) + Y (t), (6) where Y (t) is a positive operator. These two matrices are environmental functions. Eq. (6) guarantees that σ(t) is a physical covariance matrix for all finite times t. For a bosonic mode (harmonic oscillator) we choose the general Hamiltonian H = 1 2m p2 x + mω2 2 x2 + µ 2 (xp x + p x x), (7) where the parameter µ determines the damping ratio. Then from Eq. (1) we obtain the following system of equations for the elements of the covariance matrix [14, 18] [ ] σxx (t) σ s(t) = xpx (t), (8) σ xpx (t) σ pxpx (t)

4 4 Logarithmic negativity of two bosonic modes in the two thermal reservoir model 1281 representing the quantum correlations of the canonical observables of a single-mode system: ds(t) = Z 1 s(t) + s(t)z1 T + 2D 1, (9) dt where (we take from now on, for simplicity, m = ω = = 1) [ ] (λ1 µ) 1 Z 1 = 1 (λ 1 + µ) [ Dxx D, D 1 = xpx D xpx D pxpx ]. (10) Diffusion coefficients D xx,d pxpx,d xpx and the dissipation constant λ 1 satisfy the fundamental constraint D xx D pxp x D 2 xp x λ (11) The solution of Eq. (9) is given by s(t) = X 1 (t)[s(0) s( )]X T 1 (t) + s( ), (12) where the matrix X 1 (t) = exp(z 1 t) has to fulfil the condition lim t X 1 (t) = 0. In order for this limit to exist, Z 1 must only have eigenvalues with negative real parts. In the underdamped case ω > µ, Ω ω 2 µ 2, Eq. (12) takes the form s(t) = X 1 (t)s(0)x T 1 (t) + Y 1 (t), (13) where [ X 1 (t) = e λ 1t cosωt + µ Ω sinωt 1 Ω sinωt ] 1 Ω sinωt cosωt µ Ω sinωt (14) and Y 1 (t) = X 1 (t)s( )X1 T (t) + s( ). (15) By taking as asymptotic state the Gibbs state, which corresponds to a harmonic oscillator in thermal equilibrium at temperature T 1, the values at infinity of the covariance matrix are obtained from the equation Z 1 s( ) + s( )Z T 1 = 2D 1, (16) where the matrix of diffusion coefficients becomes D 1 = 1 [ (λ1 µ)coth 1 2kT (λ 1 + µ)coth 1 2kT 1 The restriction (11) is satisfied only if λ 1 > µ and ]. (17) (λ 2 1 µ 2 )coth 2 1 2kT 1 λ 2 1. (18) By using Eqs. (10), (16) and (17), s( ) takes the following form: s( ) = 1 2 coth 1 [ ] 1 0. (19) 2kT 1 0 1

5 1282 Hoda Alijanzadeh Boura, Aurelian Isar 5 3. LOGARITHMIC NEGATIVITY FOR TWO BOSONIC MODES IN INDEPENDENT THERMAL ENVIRONMENTS Consider a system of two identical bosonic modes, each one coupled to its own thermal bath. If the initial two-mode 4 4 covariance matrix is σ(0), then its subsequent evolution is given by σ(t) = (X 1 (t) X 2 (t))σ(0)(x 1 (t) X 2 (t)) T + (Y 1 (t) Y 2 (t)), (20) where X 1,2 (t) and Y 1,2 (t) are given by Eqs. (14), (15) and similar ones, corresponding to each bosonic mode immersed in its own thermal reservoir, characterized by temperatures T 1 and T 2 and dissipation constants λ 1 and λ 2, respectively. As initial state of the considered system we take an entangled two-mode nonsymmetric squeezed thermal state, with the covariance matrix of the form: σ ST S (0) = with the matrix elements given by a 0 c 0 0 a 0 c c 0 b 0 0 c 0 b a = n 1 cosh 2 r + n 2 sinh 2 r cosh2r,, (21) b = n 1 sinh 2 r + n 2 cosh 2 r + 1 cosh2r, (22) 2 c = 1 2 (n 1 + n 2 + 1)sinh2r. Here n 1, n 2 are the average number of thermal photons associated with the two modes and r is the squeezing parameter. A two-mode squeezed thermal state is entangled when r satisfies the inequality r > r s, with cosh 2 r s = (n 1+1)(n 2 +1) n 1 +n 2 +1 [19]. For two-mode Gaussian states the logarithmic negativity, which characterizes the degree of entanglement, is given by E N (ρ) = max(0, log 2 2ν s ), (23) where ν s is the smallest symplectic eigenvalue of the partially transposed covariance matrix of the two-mode Gaussian state [20]. More concretely, in our case the logarithmic negativity can be calculated using the formula where E N (t) = max(0, 1 2 log 2[4f(σ(t))]), (24) f(σ(t)) = 1 2 (deta+detb 2detC) ([ 1 2 (deta+detb 2detC)]2 detσ(t) ) 1 2 (25).

6 6 Logarithmic negativity of two bosonic modes in the two thermal reservoir model 1283 For E N > 0, it determines the strength of entanglement, and if E N = 0 then the state is separable. The evolution of logarithmic negativity for an entangled initial squeezed thermal state is illustrated in Figs. 1 and 2, where we represent the dependence of E N on time t and temperature T 2. Likewise, in Figs. 3 and 4 we represent the dependence of E N on time and, respectively, average photon number n and squeezing parameter r. It can be seen that for all parameters characterizing the initial state and for nonzero temperatures of the two thermal reservoirs, the phenomenon of entanglement sudden death takes place at some definite moment of time. After that, the state remains separable. The survival time of entanglement is decreasing with increasing the temperatures of the thermal reservoirs and the dissipation coefficients. The obtained results are in agreement with the ones previously obtained in Ref. [4]. From Fig. 3, where we assume n 1 = n 2 = n and T 1 = T 2 = 0 and plot E N (t) as a function of mean thermal photon number (n), we can notice that E N is decreasing in time. An initial entangled squeezed thermal state, in particular a squeezed vacuum state (n = 0), remains entangled for all finite times, but in the limit of infinite time the state becomes separable. This case is also emphasized in Fig. 4, where it is shown that for an initial squeezed vacuum state E N (t) is strictly positive for all r > 0 at any moment of time. Therefore, the parameters and coefficients characterizing the system and the two thermal environments strongly influence the dynamics of logarithmic negativity. 4. CONCLUSIONS We have determined the evolution of the logarithmic negativity of two-mode Gaussian states for a system of two bosonic modes each one interacting with its thermal reservoir, by using the symplectic formalism of covariance matrices, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. We have analyzed the behaviour of the logarithmic negativity as a function of parameters and coefficients characterizing the two bosonic modes and the two thermal environments. We have shown that at a definite moment of time the suppression of entanglement (entanglement sudden death) takes place for all nonzero temperatures of the thermal baths. After the suppression of the entanglement, the states remain always separable. For zero temperature of both thermal reservoirs, an initial entangled squeezed thermal state, in particular a squeezed vacuum state, remains entangled for all finite times, but in the limit of infinite time the state becomes separable.

7 1284 Hoda Alijanzadeh Boura, Aurelian Isar 7 Fig. 1 E N (t) versus time t and temperature T 2 for an entangled initial squeezed thermal state with squeezing parameter r = 1 and n 1 = 1, n 2 = 2, λ 1 = 0.1, λ 2 = 0.01, µ = and coth(1/2kt 1 ) = 1.2 (m = ω = = 1).

8 8 Logarithmic negativity of two bosonic modes in the two thermal reservoir model 1285 Fig. 2 E N (t) versus time t and temperature T 2 for an entangled initial squeezed thermal state with squeezing parameter r = 1 and n 1 = 1, n 2 = 1, λ 1 = 0.4, λ 2 = 0.3, µ = 0 and coth(1/2kt 1 ) = 1 (m = ω = = 1).

9 1286 Hoda Alijanzadeh Boura, Aurelian Isar 9 Fig. 3 E N (t) versus time t and average thermal photon number n for an initial squeezed thermal state with squeezing parameter r = 0.5 and n 1 = n 2 = n, λ 1 = 0.2, λ 2 = 0.3, µ = 0 and coth(1/2kt 1 ) = coth(1/2kt 2 ) = 1 (m = ω = = 1).

10 10 Logarithmic negativity of two bosonic modes in the two thermal reservoir model 1287 Fig. 4 E N (t) versus time t and squeezing parameter r for an initial entangled squeezed vacuum state with n 1 = n 2 = 0, λ 1 = 0.2, λ 2 = 0.3, µ = 0 and coth(1/2kt 1 ) = coth(1/2kt 2 ) = 1 (m = ω = = 1).

11 1288 Hoda Alijanzadeh Boura, Aurelian Isar 11 Acknowledgements. H. A. B. acknowledges the financial support by the research fund No from Azarbaijan Shahid Madani University and A. I. acknowledges the financial support received from the Romanian Ministry of Education and Research, through the Projects CNCS-UEFISCDI PN-II-ID-PCE and PN /2014. REFERENCES 1. G. Vidal and R. F. Werner, Phys. Rev. A 65, (2002). 2. A. Isar, Romanian J. Phys. 58, 599 (2013). 3. A. Isar, Romanian Rep. Phys. 65, 711 (2013). 4. A. Isar, Phys. Scr. T160, (2014). 5. A. Isar, J. Russ. Laser Res. 35, 1 (2014). 6. A. Ferraro, S. Olivares and M. G. A. Paris,Gaussian States in Quantum Information (Bibliopolis, Napoli, 2005). 7. J. Eisert and M. B. Plenio, Int. J. Quantum Inform. 1, 479 (2003). 8. S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005). 9. A. Isar, Open Sys. Inform. Dyn 18, 175 (2011). 10. A. Isar, Romanian J. Phys. 58, 1355 (2013). 11. A. Isar, Phys. Scr. T153, (2013). 12. V. Gorini, A. Kossakowski and E. C. G. Sudarshan, J. Math. Phys (1976). 13. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 14. A. Isar, A. Sandulescu, H. Scutaru, E. Stefanescu and W. Scheid, Int. J. Mod. Phys. E 3, 635 (1994). 15. R. Simon, Phys. Rev. Lett. 84, 2726 (2000). 16. L. M. Duan, G. Giedke, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). 17. T. Heinosaari, A. S. Holevo and M. M. Wolf, Quantum Inform. Comput. 10, 0619 (2010). 18. A. Sandulescu and H. Scutaru, Ann. Phys. 173, 277 (1987). 19. P. Marian, T. A. Marian and H. Scutaru, Phys. Rev. A 68, (2003). 20. K. L. Liu and H. S. Goan, Phys. Rev. A 76, (2007).