dynamics of entanglement of three-mode gaussian states in ...dynamics of entanglement of three-mode...

DYNAMICS OF ENTANGLEMENT OF THREE-MODE GAUSSIAN STATESIN THE THREE-RESERVOIR MODEL

HODA ALIJANZADEH BOURA1,2,a, AURELIAN ISAR2,b, YAHYA AKBARI KOURBOLAGH1,c

1Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53741-161, IranE-maila: [email protected], E-mailc: [email protected]

2Department of Theoretical Physics, Horia Hulubei National Institute for Physics and NuclearEngineering, Reactorului 30, RO-077125, P.O.B. MG-6, Magurele-Bucharest, Romania

E-mailb: [email protected]

Received May 12, 2016

We describe the dynamics of entanglement of three-mode Gaussian states ofa system composed of three bosonic modes, each one immersed in its own thermalreservoir, in the framework of the theory of open systems based on completely positivequantum dynamical semigroups. By using the criteria for the separability of three-modesystems, we classify the states by different values of the parameters characterizing thesystem and the thermal reservoirs. We consider a fully inseparable state as an initialstate and show that for definite values of temperature and dissipation constants, the classof entanglement to which the state of the system belongs is changing during its timeevolution. For all non-zero values of temperatures of the thermal baths, suppression ofentanglement of the initial state always takes place, and in the limit of large times thestate is fully separable, corresponding to an asymptotic product state.

Key words: Three-mode Gaussian states, open systems, separable states, quan-tum entanglement.

PACS: 03.65.Yz, 03.67.Bg, 03.67.Mn.

1. INTRODUCTION

Characterizing and quantifying entanglement represent one of the most crucialproblems in quantum information theory [1, 2]. In the existing literature on quan-tum information and communication great attention has been paid to bipartite andmultipartite systems of continuous variables. There are many achievements in thisdirection, mainly in the case of bipartite systems. In particular, there were inten-sively studied the quantum correlations, including quantum entanglement and quan-tum discord. In the previous papers we have studied the time evolution of quantumcorrelations in systems consisting of two-bosonic modes interacting with a thermalenvironment [3–11]. At the same time the issue of entanglement for multipartitestates possesses an even greater challenge, and there exist fewer achievements in thisdirection [12–18].

The concept of separability originates from the observation by Peres [19] ac-cording to which a partial transpose of a density matrix for a separable state is still a

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1162 Hoda Alijanzadeh Boura, Aurelian Isar, Yahya Akbari Kourbolagh 2

valid positive definite density matrix. Horodecki [12] proved that this is a necessaryand sufficient condition for a state to be separable if the dimension of the Hilbertspace does not exceed 6. The Peres criterion leads to a natural entanglement measurecalled negativity, determined by the negative eigenvalues of the partial transpose ofthe covariance matrix of the state. Afterwards Vidal and Werner proved that the neg-ativity is an entanglement monotone, and therefore a proper entanglement measure[20]. Furthermore, the logarithmic negativity gives an upper bound to the distillableentanglement [21]. In addition, purity and entanglement of the state are uniquelydetermined by its covariance matrix [22, 23].

Tripartite three mode Gaussian state undergoing parametric amplification andamplitude damping as well as thermal noise was studied in Ref. [24], where the con-ditions for full separability and full entanglement of the state are worked out. In Ref.[25] there were studied the separability properties and the dynamics of tripartite en-tanglement under the influence of a dissipative Agarwal bath in a three-mode systemof continuous variables. It was shown that if two symmetric modes are propagated inseparate baths, the bipartite entanglement between one of the two and the third modevanishes in a finite time. For fully symmetric tripartite states it was found that entan-glement vanishes in a finite time in the presence of separate baths, while it persistsfor a long time in the presence of a common bath.

In this paper we describe the time evolution of the entanglement for a systemcomposed of three bosonic modes coupled to three independent thermal environ-ments. We work in the framework of the theory of open quantum systems and take afully inseparable state as an initial state of the considered system. We show that fordefinite non-zero values of the temperatures, entanglement suppression of the initialstate takes place. Only in the special case of a fully symmetric system and identicalthermal reservoirs, for zero tempertures of the thermal baths the initial fully insep-arable state remains fully inseparable for all finite times. However, in the limit oflarge times for all temperatures the state is always fully separable, corresponding toan asymptotic product state.

The structure of the paper is as follows. In Sec. 2 we write the evolutionequation for the covariance matrix of the considered open system interacting with athermal environment. Then in Sec. 3 we investigate the separability of the systemof three bosonic modes, each one interacting with its own reservoir, by using thecriterion of separability introduced in Ref. [26]. We classify the states in the three-reservoir model by various values of the parameters characterizing the system andthe thermal reservoirs. A summary is given in the last section.

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3 Dynamics of entanglement of three-mode Gaussian states in the three-reservoir model 1163

2. EVOLUTION OF COVARIANCE MATRIX

The covariance matrix σ of an n-mode bosonic system is a real, symmetric andpositive 2n×2n matrix. We remind that the first moments of the canonical variablesof 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, wemay take them to be zero. We work only with Gaussian states of zero displacementvectors and such Gaussian states are completely characterized by their covariancematrix, which for a system of three bosonic modes is given by:

σ(t) =

σx1x1(t) σx1p1(t) σx1x2(t) σx1p2(t) σx1x3(t) σx1p3(t)σx1p1(t) σp1p1(t) σp1x2(t) σp1p2(t) σp1x3(t) σp1p3(t)σx1x2(t) σp1x2(t) σx2x2(t) σx2p2(t) σx2x3(t) σx2p3(t)σx1p2(t) σp1p2(t) σx2p2(t) σp2p2(t) σp2x3(t) σp2p3(t)σx1x3(t) σp1x3(t) σx2x3(t) σp2x3(t) σx3x3(t) σx3p3(t)σx1p3(t) σp1p3(t) σx2p3(t) σp2p3(t) σx3p3(t) σp3p3(t)

, (1)

where the matrix elements are defined as:

σξiξj =1

2Tr[(ξiξj + ξjξi)ρ], i, j = 1,2,3, (2)

ξ = (x1, p1, x2, p2, x3, p3) are the canonical variables (coordinates and momenta) ofthe three-mode bosonic system and ρ denotes its density operator. The matrix σ is abona fide covariance matrix iff it satisfies the uncertainty relation [13, 23]

det(σ+i

2J)≥ 0, (3)

where

J =3⊕i=1

Ji, with Ji =

[0 1−1 0

].

Covariance matrix (1) has the following block structure:

σ(t) =

A D12 D13

DT12 B D23

DT13 DT

23 C

, (4)

where A, B and C are 2×2 Hermitian matrices which denote the symmetric covari-ance matrices for the individual reduced one-mode states, while 2× 2 matrices Dcontain the cross-correlations between modes (T denotes the transposed matrix).

The time evolution of the initial covariance matrix σ(0) of the system, underthe action of a general Gaussian channel, can be characterized by two matrices X(t)and Y (t) [27, 28]:

σ(t) =X(t)σ(0)XT(t) +Y (t), (5)

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where Y (t) is a positive operator. These two matrices depend on the parameterscharacterizing the environment. Eq. (5) guarantees that σ(t) is a physical covariancematrix for all finite times t.

In the case of one bosonic mode (harmonic oscillator) with the general Hamil-tonian

H =1

2mp2

1 +mω2

2x2

1 +µ

2(x1p1 +p1x1), (6)

where the parameter µ determines the damping ratio with ω > µ, Ω ≡ ω2−µ2, thefollowing expressions have been obtained in the framework of the theory of opensystems based on quantum dynamical semigroups, where the Markovian time evolu-tion of the density operator is given by the Kossakowski-Lindblad master equation[28–31]:

X1(t) = e−λ1t[

cosΩt+ µΩ sinΩt 1

Ω sinΩt− 1

Ω sinΩt cosΩt− µΩ sinΩt

](7)

andY1(t) =−X1(t)s(∞)XT

1 (t) +s(∞). (8)Here λ1 is the dissipation constant and, considering that the asymptotic state of theopen system is a Gibbs state [32–34], we obtain

s(∞) =1

2coth

1

2kT1

[1 00 1

], (9)

where T1 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 itssubsequent evolution is given by

σ(t) = (X1(t)⊕X2(t)⊕X3(t))σ(0)(X1(t)⊕X2(t)⊕X3(t))T

+ (Y1(t)⊕Y2(t)⊕Y3(t)), (10)

where X1,2,3(t) and Y1,2,3(t) are given by Eqs. (7), (8) and similar ones, correspond-ing to each bosonic mode immersed in its own thermal reservoir, characterized bytemperatures T1, T2 and T3 and dissipation constants λ1, λ2 and λ3, respectively.

Fully symmetric tripartite states are invariant under the exchange of any twomodes. The standard covariance matrix to describe a fully symmetric state has thefully symmetric form

σ(t) =

A D DDT A DDT DT A

. (11)

As an initial state of the considered system we take an entangled three-modestate (formal generalization to the case of three-mode state of the two-mode squeezed

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vacuum state with squeezing parameter r), with the covariance matrix of the form:

σ(0) =1

2

cosh2r 0 sinh2r 0 sinh2r 00 cosh2r 0 −sinh2r 0 −sinh2r

sinh2r 0 cosh2r 0 sinh2r 00 −sinh2r 0 cosh2r 0 −sinh2r

sinh2r 0 sinh2r 0 cosh2r 00 −sinh2r 0 −sinh2r 0 cosh2r

. (12)

3. SEPARABILITY IN THE SYSTEM OF THREE BOSONIC MODES

The separability problem of three-mode Gaussian states was completely solved[26], namely the separability of a three-mode system can be determined by the pos-itivity of the partially-transposed density matrices. Based on the study [26], three-mode Gaussian states can be classified into five different entanglement classes, fromfully inseparable states to fully separable states:

Class 1. Fully inseparable states are those which are not separable for anygrouping of the parties.

Class 2. One-mode biseparable states are those which are separable if two ofthe 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 thethree bipartite splits but inseparable with respect to the third.

Class 4. Three-mode biseparable states are separable with respect to all threebipartite 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 tripartiteproduct 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 2-5; b) fora fully symmetric system and identical thermal reservoirs, an initial fully inseparablestate (class 1) evolves in time to a three-mode biseparable state (class 4) and finallybecomes a fully separable state (class 5).

Denoting by Λj , j = 1,2,3, the partial transposition matrices,

Λ1 = diag(1,−1,1,1,1,1), Λ2 = diag(1,1,1,−1,1,1),

Λ3 = diag(1,1,1,1,1,−1),(13)

the partially transposed covariance matrices are given by σ(t)j = Λjσ(t)Λj . Thenthe criterion for the fully inseparable states is σ(t)j + i

2J < 0, for all j = 1,2,3. Fora fully symmetric three-mode state, this criterion can be simplified to, for exampleσ(t)1 + i

2J < 0.

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For σ(t)j + i2J ≥ 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 minimumeigenvalue γ− ≥ 0. If γ− is negative, the j-th mode is inseparable from the tripartitesystem. In the case of a fully symmetric state, we only concern the positivity ofσ(t)1 + i

2J for three mode separability.For σ(t)j + i

2J ≥ 0, for all j = 1,2,3, the state will be a positive partial trans-pose (PPT) three-mode state, and it can be biseparable or fully separable. Thereexists a criterion to distinguish the biseparable and fully separable states [26].

It follows that in order to determine the separability properties of the consideredsystem we have to find γ−, which denotes the smallest eigenvalue of Λjσ(t)Λj +i2J, j = 1,2,3. We consider the separability of two types of tripartite states: fullysymmetric states, which are invariant under the exchange of any two modes, and non-symmetric states. For simplicity, we take first λj = λ and µ = 0 and also the sametemperature Tj = T for the three thermal reservoirs. In this case, we obtain a fullysymmetric system of three bosonic modes immersed in identical thermal reservoirs,and the eigenvalues are symmetric with respect to all three modes. We obtain thefollowing eigenvalues:

a± 1

2

√1 + 10(b2 +d2)±2

√(b2 +d2)(8 + 9(b2 +d2)),

a± 1

2

√1 + 4(b2 +d2),

(14)

where

a= e−2λt cosh2r+C

2(1−e−2λt),

b= e−2λt cos2tsinh2r,

d= e−2λt sin2tsinh2r

and C ≡ coth 12kT . It can easily be seen that the smallest one is

γ− = a− 1

2

√1 + 10(b2 +d2) + 2

√(b2 +d2)(8 + 9(b2 +d2)). (15)

Since σ(∞) depends on temperature only, we can affirm that r and λ do notaffect the separability at infinity of time. The eigenvalues of Λjσ(∞)Λj + i

2J aregiven by Cj±1

2 , and sinceCj ≥ 1 (Tj ≥ 0), we can re-confirm that for all temperaturesof 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 tem-perature T , for an entangled initial state, is illustrated in Fig. 1, where we consideridentical temperatures T and dissipation constants λ. Likewise, in Figs. 2 and 3 werepresent the dependence of the set of all three (j = 1,2,3) smallest eigenvalues γ−

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on time for different values of temperatures and dissipation constants characterizingthe thermal reservoirs.

Fig. 1 – Evolution of smallest eigenvalue γ− (15) on time t and environment temperature T (throughcoth 1

2kT ) 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 symme-tric and the eigenvalues display a permutational symmetry among the three modes. Itfollows that the states can belong to classes 1, 4 and 5 only. Indeed, from Fig. 1 it canbe seen that initially the state is a fully inseparable state (class 1), for temperaturesT > 0 the suppression of entanglement of the initial state takes place at some finitemoment 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, correspond-ing to an asymptotic product state (class 5). For zero temperature the initial stateremains fully inseparable for all finite times and becomes fully separable in the limitof infinite time.

If the three temperatures Tj > 0 are different one from each other, it wouldbe possible to reach also classes 2 and 3 for some finite time, but always the statesbelong finally to class 5. Indeed, from Figs. 2 and 3 we can notice that for differenttemperatures of the reservoirs, the state is initially a fully inseparable state (class1) and during its evolution it becomes first an one-mode biseparable state (class 2),then a two-mode biseparable state (class 3), and after another finite time a three-mode biseparable state (class 4). We remind that always, for large times, the state isfully separable and belongs to class 5. If we take equal temperatures only for tworeservoirs and equal dissipation constants for the corresponding two modes, then theinitial fully inseparable state can belong during its evolution to only one of classes 2or 3 for some finite time, since these two mode manifest the same behaviour – and

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therefore their corresponding smallest eigenvalues γ− coincide – and after anotherfinite time to class 4.

Fig. 2 – Evolution of the set of all three (j = 1,2,3) smallest eigenvalues on time t for a fully insepa-rable initial state, for r = 1, coth 1

2kT1= 2.5, coth 1

2kT2= 3.5, coth 1

2kT3= 1.5 and λ= 0.1.

To demonstrate the influence of dissipation, we also consider in Fig. 4 iden-tical temperatures of the three thermal reservoirs, and only two identical dissipationconstants. The evolution of the smallest eigenvalues γ− of the three modes on timeand dissipation shows that the values of λ which change the sign of γ− are differentfor modes interacting with environments with different dissipation constants.

From Fig. 5 we notice again that the initial entangled state belongs to class1, during its evolution it moves temporarily to class 2, then to class 3, after anotherfinite time to class 4, and finally the state becomes fully separable (class 5). Fromthis figure we can also conclude that the influence of dissipation on the evolutionof the eigenvalues is relatively stronger than the influence of the temperature, in thesense that in the considered case the class 3 (two-mode biseparable) is less stablethan classes 1 (fully inseparable) and 2 (one-mode biseparable).

As a general conclusion, we can say that an initial fully inseparable state canhave such an evolution that, depending on the parameters of the system and the tem-peratures and dissipation constants characterizing the environments, it can belongtemporarily to different classes of entanglement. As a result of the interaction withthe environment, the system gradually becomes more and more separable. As ex-

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Fig. 3 – Evolution of the set of all three (j = 1,2,3) smallest eigenvalues on time t for a fully insepa-rable initial state, for r = 1, coth 1

2kT1= 2, coth 1

2kT2= 3, coth 1

2kT3= 1 and λ= 0.1.

Fig. 4 – Evolution of smallest eigenvalues on time t and dissipation λ for a fully inseparable state, forr = 1, coth 1

2kT1= 4, coth 1

2kT2= 4, coth 1

2kT3= 4, λ1 = 0.1 and λ2 = λ3 = λ.

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pected, the temperature and dissipation strongly influence the dynamics of the sepa-rability properties, namely their increasing favorizes the destruction of the entangle-ment.

Fig. 5 – Evolution of smallest eigenvalue on time t for a fully inseparable initial state, for r = 1,λ1 = λ3 = 0.02, λ2 = 0.1, coth 1

2kT1= 3, coth 1

2kT2= coth 1

2kT3= 2 (red line for mode 1, green

line for mode 2, blue line for mode 3).

4. SUMMARY

We have determined the dynamics of entanglement of three-mode Gaussianstates by using the symplectic formalism of covariance matrices, in the framework ofthe theory of open systems based on completely positive quantum dynamical semi-groups. By using the well-known criteria of Ref. [26] for the separability of thethree-mode systems, we have classified the states for different values of the para-meters characterizing the system and the thermal reservoirs. We considered a fullyinseparable state as an initial state and have shown that for definite values of temper-ature and dissipation constants the classes of entanglement to which the state of thesystem belongs is changing during its time evolution. For all non-zero values of tem-peratures of the thermal baths, suppression of entanglement of the initial state alwaystakes place, and in the limit of large times the state is fully separable, correspondingto an asymptotic product state. Temperatures and dissipation of the thermal reser-voirs strongly influence the dynamics of entanglement of the three-mode bosonicsystem.

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