Multipartite entanglement dynamics and decoherence from a quantum-critical environment

Optics Communications 281 (2008) 5633–5638

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Multipartite entanglement dynamics and decoherence from a quantum-critical environment Xiao San Ma School of Electric Engineering and Information, Anhui University of Technology, Ma’anshan, 243002, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 22 May 2008 Received in revised form 11 July 2008 Accepted 5 August 2008

PACS: 03.67.Mn 03.65.Yz

a b s t r a c t We investigate the entanglement dynamics and decoherence of a multipartite system under an environment which can exhibit a quantum phase transition. Our result implies that the entanglement evolution depends not only on the size of the system and the quantum states of concern but also on the environment. In the sense of the linear entropy to measure decoherence induced by the environment, the decoherence-free subspaces have been identified for our model. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: Entanglement dynamics Decoherence

1. Introduction As a valuable resource in quantum information and quantum computation [1,2], quantum entanglement has attracted much attention from physicists both in theory and in experiment. Much progress concerning entanglement has been achieved [3–8]. One of the basic problems is how to measure entanglement. In 1998, Wootters [6] proposed one quantity of concurrence to measure entanglement. Concurrence is justified as a good entanglement measure, but it is limited in that it is necessary and sufficient only for 2  2 quantum states. With regards to a system with high dimensions, concurrence can not be employed as entanglement measure. Fortunately, Zyczkowski et al. [9] proposed another quantity labeled as negativity by Vidal and Werner [10] to measure entanglement based on the Peres’ criterion for separability [7]. Negativity is an entanglement monotone for quantum states either pure or mixed. As is known that the Peres’ criterion is necessary and sufficient only for 2  2 and 2  3 quantum states, negativity is limited in measuring entanglement of multipartite quantum states with high dimensions. However, due to its operational and calculational properties, negativity has been employed to measure quantum entanglement of multipartite quantum states with high dimensions extensively [11–13].

E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.018

In quantum information processing, real quantum systems will unavoidably be affected by the influence of the surrounding environment. The interaction between the system and the environment can lead to decoherence [14,15]. The extent to which the decoherence affects quantum entanglement is an interesting problem and much works have been done [16–19]. Most of these works for entanglement dynamics are concentrated on the effect of an uncorrelated environment on the system. However, the parties of the environment are always interacting with each other. That’s to say, a real environment should be a correlated environment. The effect of a correlated environment on quantum states of systems is worth studying and researchers have studied Loschmidt echo [20], universal decoherence [21], and bipartite disentanglement [22], respectively, along the line. It is well-known that multipartite systems are of significance for quantum information processing in a large scale and much progress concerning multipartite entanglement has been achieved [23–26]. The effect of a correlated environment on multipartite entanglement is an interesting problem and should be studied. As an extension of the work [22], the effect of a correlated environment on multipartite entanglement will be investigated in this paper. The content is arranged as follows. In Section 2, we give the model and calculate the Hamiltonian evolution. In Section 3, we analyze the effect of the quantum-critical environment on the multipartite entanglement. By using the linear entropy, we identify the decoherence-free subspaces for our model in Section 4. Finally, we conclude our results.

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X.S. Ma / Optics Communications 281 (2008) 5633–5638

2. Hamiltonian evolution





Yn fn g fn g 2 2 2 ag 1  sin Ekfna g t sin Ek b t sin ðhfn  hk b Þ k

jM ab ðtÞj ¼

k>0

h







fn g  sin Ekfna g t cos Ek b t sin hkfna g  cos Ekfna g t

i2 1=2

fn g fn g ð8Þ  sin Ek b t sin hk b

In this paper, we study the entanglement dynamics and decoherence of a multipartite system under a quantum environment consisting of a transverse Ising model which can exhibit a quantum phase transition. The corresponding Hamiltonian reads

H¼

N0 X

rxi rxiþ1 þ x þ g

n X j¼1

i¼N0

! szj

N0 X i¼N0

rzi

ð1Þ

2

where x represents the strength of the transverse magnetic field applied to the quantum environment, g denotes the coupling strength between the parties of system sj ; j ¼ ð1; 2; . . . ; nÞ and the quantum environment, rxi ; rzi are the familiar Pauli matrices for the ith site in the lattice of environment, and the number of the total sites of the quantum environment is N t ¼ 2N0 þ 1. Obviously, the size of the system n is finite and far smaller than that of the environment N t . To get the time evolution operator, we need to follow the standard procedure of Hamiltonian diagonalization [27] with Jordan–Wigner transformation and Fourier transformation to the momentum space. With following pseudospin operators rkx ; rky ; rkz

rkx ¼ cyk cyk þ ck ck ; ðk ¼ 1; 2; . . . ; N0 Þ; r0z ¼ 2cy0 c0  1 rky ¼ icyk cyk þ ick ck ; rkz ¼ cyk ck þ cyk ck  1

ð2Þ

y k;

where Ekfna g ; hkfna g can be calculated by replacement D with Dna in Eq. (4) and Dna is defined by the following expression.

Dna ¼ x þ

n gX g ð1Þau ¼ x þ ðn  2na Þ 2 u¼1 2

In Eq. (9), we have defined the number na ¼ get the time evolution of the basis Mab .

ð9Þ Pn

u¼1

au . Thus, we can

Mab ðtÞ ¼ jMab ðtÞjei/ab ðtÞ

ð10Þ

where /ab ðtÞ is a phase factor. 3. Entanglement dynamics Having got the reduced density matrix, we can evaluate the entanglement dynamics of any n-qubit entangled state. Here, we employ negativity [9,10] to investigate the effect of the correlated environment on the multipartite entanglement. Given a density matrix qs , the negativity is defined by

Tj qs  1

where c ck fk ¼ 0; 1; 2; . . . ; N0 g denote, respectively the creation and annihilation operators of the new fermions in the momentum space. With the above transformation, we get the new Hamiltonian in the momentum space similarly as done in [22,28].

Nðqs Þ ¼

  X hk hk D H¼ ei 2 rkx ðEk rkz Þei 2 rkx þ  þ 1 r0z 2 k>0

respect to part j. According to the reference [10], we can classify entanglement properties of a multipartite quantum state by considering different bipartition of the system. We use Nm;nm to measure the strength of quantum correlation between one group with m parties and the other group with n  m parties. Similarly, we can incorporate this into the reduced density matrix obtained by tracing over some subsystems. In practice, we are interested in some explicit examples to understand the effect of the environment on the entanglement of quantum states. Here, the well-known GHZ and W states which bear incompatible multipartite correlations, in the sense that they can not be transformed into each other by local operation and classical communication[24], are employed to illustrate the effect of the environment on multipartite entanglement of pure quantum states. With regards to the mixed states, we will consider the entanglement dynamics of the Werner state [29] as an example of mixed states. Case 1: As the first example, the n-qubit GHZ state is considered and it reads jwiGHZ ¼ p1ffiffi2 ðj00; . . . ; 0i þ j11; . . . ; 1iÞ. Due to the interaction between the system and the quantum-critical environment, the n-qubit GHZ state perceives the presence of the environment and the time evolution of the density matrix of the GHZ state is

ð3Þ

P where D ¼ x þ g nj¼1 szj and the parameters Ek ; hk take the following expressions, respectively,

Ek ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ D2  4D cosðkaÞ;

 hk ¼ arctan

 2 sinðkaÞ D  2 cosðkaÞ

ð4Þ

where a denotes the lattice spacing and D is used as a cnumber. With these analytical expressions, we can go straightforwardly to get the time evolution of any density matrix of the system.



qs ðtÞ ¼ TrE eiHt qs ð0Þ  jwE ihwE jeiHt



ð5Þ

Here, the initial state of the system is separable with the initial state of the environment which is assumed to be the vacuum state with following expression jwE i ¼ j0ik¼0 kP0 j0ik j0ik satisfying the relation ck j0ik ¼ 0. As is well-known that the initial state of qs ð0Þ for a n-qubit system can always be expanded by a set of natural basis M ¼ fMab g, where M ab ¼ jaihbj ¼ ja1 a2 ; . . . ; an ihb1 b2 ; . . . ; bn j with every au and bv ; ðu; v ¼ 1; 2; . . . ; nÞtaking values of 0 or 1. The bit-string a1 a2 ; . . . ; an ; b1 b2 ; . . . ; bn takes over all the possible permutations of 0 and 1.

qs ð0Þ ¼

X

C ab Mab

ð6Þ

a;b

where C ab are complex coefficients and vary with ða; bÞ for the density matrix qs ð0Þ. It should be noted that the reduced density matrix of the system can come down to the evolution of Mab with time, i.e.

qs ðtÞ ¼

X a;b

C ab M ab ðtÞ ¼

X

  C ab TrE eiHt Mab  jwE ihwE jeiHt

ð7Þ

a;b

After a troublesome calculation, we can get the modula of the time evolution of the natural basis M ab .

2

ð11Þ

T where qs j is the sum of the absolute values of the eigenvalues of T

T

qs j and qs j means the partial transpose of the density matrix qs with

1 2D

 11; . . . ; 1jM ab ðtÞ þ j11; . . . ; 1ih00; . . . ; 0jM ab ðtÞ

qGHZ ðtÞ ¼ ðj00; . . . ; 0ih00; . . . ; 0j þ j11; . . . ; 1ih11; . . . ; 1j þ j00; . . . ; 0i ð12Þ

where * denotes complex conjugation and Dna ; Dnb in the Mab of Eq. (12) correspond to x þ ng ; x  ng , respectively. With the reduced 2 2 density matrix qGHZ ðtÞ, we can obtain the time evolution of the entanglement. Due to highly symmetry of the GHZ state, the quantum correlation between one group consisting of m parties and the other group with n  m parties takes a same expression of negativity for any m ranging from 0 to n. Furthermore, there’s no entanglement of the reduced matrices obtained by tracing the density

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X.S. Ma / Optics Communications 281 (2008) 5633–5638 g=0.01

g=0.1

0.5

0.5 n=3 n=5 n=8 n=10 n=15

0.45 0.4

0.4 0.35

0.3

GHZ

Nm,n–m(t)

GHZ

Nm,n–m(t)

0.35

0.25 0.2

0.3 0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0.1

0.2

0.3

0.4

n=3 n=5 n=8 n=10 n=15

0.45

0

0.5

0

0.02

0.04

t g=10

0.4

0.45 0.4

0.35

0.35

0.3

0.3

GHZ

Nm,n–m(t)

GHZ

0.1

0.5 n=3 n=5 n=8 n=10 n=15

0.45

Nm,n–m(t)

0.08

g=50

0.5

0.25 0.2

0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0.06

t

0

0.5

1

1.5

2

2.5

3

t

0

n=3 n=5 n=8 n=10 n=15

0

0.5

1

1.5

2

2.5

3

t

Fig. 1. Negativity versus time t and x with a critical value of 2 is demonstrated, respectively for g ¼ 0:01; 0:1; 10; 50, where N t ¼ 6001.

matrix qGHZ ðtÞ over some subsystems. Therefore, we employ N GHZ m;nm ðtÞ to represent the quantum entanglement of the density matrix qGHZ ðtÞ and it can be expressed by the following equation.

NGHZ m;nm ðtÞ ¼

1 jM ab ðtÞj; 2

ng ng

Dn a ¼ x þ ; Dn b ¼ x  2 2

ð13Þ

Now, we examine the effect of the size of the system of concern on multipartite entanglement dynamics and the numerical result is given in Fig. 1. We consider two cases of weak coupling and strong coupling. It is easy to find that the entanglement evolution depends on the size of the system. Under weak couplings such as g ¼ 0:01; 0:1, the larger the size of the system is, the faster the entanglement of the GHZ state is destroyed by the environment. When the time goes long enough, the entanglement of the GHZ state will be completely destroyed by the decoherence induced by the quantum-critical environment. While under strong coupling such as g ¼ 10; 50, the entanglement will be decreased to a stable value. The larger the size of the system is, the larger the stable value of the entanglement is. From the cases g ¼ 10 and g ¼ 50, we find that the stable value for the entanglement of a certain-sized state in the case g ¼ 50 is larger than that in the case g ¼ 10. The occurrence of the stable value is owed to the fact that numerical calculation is based on a finite-sized environment. In fact, a real environment usually contains an infinite degrees of freedom and the entanglement of the GHZ state will be completely destroyed by an infinite-sized environment even when the system is under strong coupling. Such a claim can be supported by following results.

As the second observation, the effect of the size of the environment on quantum entanglement evolution is illustrated in Fig. 2 where the entanglement evolution of the GHZ state under the environment with different sizes is plotted for both cases of weak coupling g ¼ 0:1 and strong coupling g ¼ 10. For weak coupling, the larger the size of environment is, the faster the entanglement is vanished. For strong coupling, we can find that the larger the size of environment is, the smaller the stable value of entanglement takes. The entanglement of the GHZ state will be completely destroyed by the infinite-sized environment even under strong coupling. Finally, we should consider the role of the transverse magnetic field of the environment in entanglement evolution and the numerical result is demonstrated in Fig. 3. For g ¼ 0:01, we find that the entanglement shows an oscillating behavior for x ¼ 0:1; 1. When the transverse magnetic field takes a value larger than 2 which is the critical value at which the environment exhibits a quantum phase transition, the entanglement is decreased monotonically to zero. It should be pointed out that the entanglement evolves faster for x ¼ 2 than that for x ¼ 5; 10. The entanglement decay is enhanced by quantum phase transition under weak coupling such as g ¼ 0:01. For g ¼ 0:1, the entanglement is vanished quickly for x ¼ 0:1; 1; 2; 5, while for x ¼ 10, the entanglement revives after its complete disappearance and such a phenomenon has been found recently by the authors in Ref. [30]. For g ¼ 1, we can find that the entanglement decreases faster for the cases of x ¼ 0:1; 1; 2 than that for the cases of x ¼ 5; 10. Here, the strong transverse magnetic field can postpones the death of

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X.S. Ma / Optics Communications 281 (2008) 5633–5638 0.5

0.5 N =2001 t

0.45

Nt=2001

0.45

Nt=6001

Nt=6001

0.35

0.3

0.3

m,n–m

(t)

0.35

0.25 0.2

0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0.02

0.04

0.06

0.08

Nt=20001

0.4

Nt=20001

N

Nm,n–m(t)

0.4

0

0.1

0

0.5

1

1.5

t

2

2.5

3

t

Fig. 2. Negativity versus time t and x with a critical value of 2 is demonstrated, respectively for g ¼ 0:1; 10, where x ¼ 2 and n ¼ 10.

g=0.01

g=0.1

0.5

0.5 ω =0.1 ω =1 ω =2 ω =5 ω =10

0.45 0.4

0.4 0.35

0.3

m,n–m

NGHZ (t)

N

GHZ (t) m,n–m

0.35

0.25 0.2

0.3 0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0.5

1

1.5

2

2.5

ω =0.1 ω =1 ω =2 ω =5 ω =10

0.45

0

3

0

2

4

t g=1 ω =0.1 ω =1 ω =2 ω =5 ω =10

0.4 0.35

0.4 0.35

0.3

m,n–m

0.25 0.2

0.3 0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05 0

0.02

0.04

0.06

0.08

ω =0.1 ω =1 ω =2 ω =5 ω =10

0.45

NGHZ (t)

GHZ (t) m,n–m

10

0.5

0.45

N

8

g=10

0.5

0

6

t

0.1

t

0

0

1

2

3

4

5

t

Fig. 3. Negativity versus time t and x with a critical value of 2 is demonstrated, respectively for g ¼ 0:01; 0:1; 1; 10, where n ¼ 10 and N t ¼ 4001.

entanglement. The stronger the transverse magnetic field is, the longer the time for the complete disentanglement is. For g ¼ 10, the entanglement of the GHZ state under environment with different transverse magnetic field is destroyed to a stable value. It is well-known that a real environment contains an infinite degrees of the freedom. In above numerical calculation and the fol-

lowing content, we make numerical simulation with the assumption that the size of the environment be finite. Even though a finite-sized environment does not exhibit a quantum phase transition, yet our numerical results will shed some light on the entanglement dynamics of multipartite states under a quantum-critical environment.

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X.S. Ma / Optics Communications 281 (2008) 5633–5638

Case 2 : Usually, the n-qubit W state is expressed as jWi ¼ p1ffiffin ðj0; 0; . . . ; 0; 1i þ j0; 0; . . . ; 1; 0i þ    þ j1; 0 . . . ; 0; 0iÞ. The density matrix of W state qw ð0Þ can be expanded by the basis P Mab with following expression qw ð0Þ ¼ 1n a;b M ab ð0Þ. The time evolution of the density matrix qw ð0Þ is

qw ðtÞ ¼

1X M ab ðtÞ n a;b

ð14Þ

For the W state of concern, we find that Mab ðtÞ ¼ Mab ð0Þ for all the a; b as that all the basis has a same na with a value of 1 and get P qw ðtÞ ¼ 1n a;b Mab ð0Þ ¼ qð0Þ. We can conclude that the W state does not perceive the presence of the quantum environment and it can be called a decoherence-free entangled state of our model. In this sense, the entanglement of the W state is more robust than that of the GHZ state. Taking further consideration, we can find that the W class states belong to the decoherence-free entangled states. The W class states usually refer to the class states whose density matrix can be expanded by a set of basis fMab g with all the elements M ab satisfying the relation na ¼ nb . Case 3 Here, we consider the entanglement evolution of the Werner state as a mixed state. The n-qubit Werner state reads

qWer ¼

1p I þ pjwiGHZ hwj 2n

ð15Þ

where p is a parameter characterizing the extent to which the white noise is applied to the pure quantum state jwiGHZ and I is an identity operator with rank of 2n . Similar to that of the GHZ state, the quantum correlation (entanglement) between one group with m parties and the other group with n  m parties for the Werner state takes the same expression

Nm;nm ¼

p 1p jM ab jðtÞ  n ; 2 2

ng ng

Dn a ¼ x  ; Dn b ¼ x þ 2 2

yet the life time of the entanglement of the Werner state is finite and such a phenomenon is called as entanglement sudden death (ESD) which was first observed by Yu and Eberly in [31]. In the right sub-figure, we can find that the entanglement evolution of the Werner state varies with the different couplings. For g ¼ 0:1; 1, the entanglement disappears quickly in a short time; while for g ¼ 10, the entanglement revives after its complete disappearance; for g ¼ 0:01, the time for complete disentanglement is longer than that for the cases g ¼ 0:1; 1; 10; for g ¼ 10, the entanglement evolves with small fluctuations to a stable value. From above three cases, we find that the entanglement evolution depends not only on the system and the quantum states of concern but also on the environment. In principle, we can consider the entanglement dynamics of any n-qubit quantum state. For convenience, we omit other examples. 4. Decoherence-free subspaces Decoherence-free subspaces (DFS) are of significance for the coding in quantum information processing and much works on DFS have been done [32,33]. However, almost of the works concerning DFS are based on the context of a system under an uncorrelated environment. Here, we will examine the DFS of our model consisting of a multipartite system coupled to a correlated environment. Usually, the linear entropy has been employed to measure the extent to which decoherence affects quantum states and it is defined by

SðqÞ ¼ 1  Trðq2 Þ

ð17Þ

Given a general reduced matrix of our model qs ðtÞ as in Eq. (7), the linear entropy of qs ðtÞ reads

ð16Þ Tracing the density matrix over some subsystems, we obtain the reduced density matrix of the Werner state and the residual entanglement is zero. Having got the time evolution of the Werner state, we can numerically calculate the entanglement evolution under different conditions. From the left sub-figure in Fig. 4, we can find that the more the noise is applied to the pure quantum state jwiGHZ , the shorter the time for complete disentanglement is needed. Even though a very small amount of noise is applied to the pure quantum state jwiGHZ ,

Sðqs ðtÞÞ ¼ 1 

X

jC ab j2 jMab ðtÞj2

ð18Þ

a;b

Taking a look at the expression of jMab j, we can easily find that jMab jðtÞ ¼ 1 if na ¼ nb . Therefore, the linear entropy will remain unchangeable for any state that can only be expanded by the set of basis fMab g whose all elements satisfying the relation with na ¼ nb , where na can take n; n  1; n  2; . . . ; 0 and the state is called as a decoherence-free state. Specifically, we can write all the basis only by which can the decoherence-free states be expanded.

0.5

0.25 p=1 p=0.8 p=0.5 p=0.3 p=0.1

0.45 0.4

0.2

0.3

Nm,n–m(t)

0.25

0.15 g=0.01 g=0.1 g=1 g=10 g=100

Wer

Wer

Nm,n–m(t)

0.35

0.2

0.1

0.15 0.1

0.05

0.05 0

0

0.05

0.1

t

0.15

0

0

0.1

0.2

0.3

0.4

0.5

0.6

t

Fig. 4. Negativity versus time t and x with a critical value of 2 is demonstrated for g ¼ 0:1 in the left sub-figure and for p ¼ 0:5 in the right sub-figure, respectively, where N t ¼ 4001 and n ¼ 5.

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na ¼ n;

X.S. Ma / Optics Communications 281 (2008) 5633–5638

j1; 1; . . . ; 1i

na ¼ n  1; ......... na ¼ 0;

j1; 1; . . . ; 1; 0i; j1; 1; . . . ; 0; 1i; . . . ; j0; 1; . . . ; 1; 1i ............

j00; . . . ; 0; 0i ð19Þ

As such, the decoherence-free subspaces are defined as Sna constructed by orthogonal quantum states with the same value of na , n! and the dimension of Sna is na !ðnn . For example, Sn is constructed a !Þ by one element j11; :::; 1i. Any quantum state of the subspace Sna can be expanded by the basis of Sna . Therefore, the pure states belonging to Sna do not perceive the presence of the environment. Any entangled state that can not be expanded by the above basis will perceive the presence of the environment and lose coherence as well as entanglement. Our result as a particular instance is consistent with the well established theory on decoherence-free subspaces [32,33] and it is important for us to deal with error correction. Information is encoded in subspaces Sna (codes) of the total Hilbert space in a way that errors induced by the interaction with the bath can be detected and corrected. The important point is that the detection of errors, if they belong to the class of errors correctable by the given code, should be performed without gaining any information about the actual state of the computing system prior to decoherence. 5. Conclusion To conclude, we have investigated the entanglement dynamics and decoherence of a multipartite system coupled to a quantumcritical environment which can exhibit a quantum phase transition. The entanglement evolution depends not only on the size of the system and the quantum states of concern but also on the system-environment coupling, the strength of the transverse magnetic field, and the size of the environment. Specifically, we analyze the entanglement dynamics of the GHZ state and the Werner state in detail. For the GHZ state, the entanglement evolution varies with the size of the system. Under weak coupling such as g ¼ 0:01; 0:1, the larger the size of the system is, the faster the entanglement is destroyed by decoherence. While under strong coupling such as g ¼ 10; 50, the entanglement shows a quick decay to a stable value with time going long; the larger the size of the system, the larger the stable value is. Beyond the effect of system, we also consider the effect of the environment on the entanglement dynamics. Under weak couplings g ¼ 0:01; 0:1; 1, the entanglement decay is enhanced by the quantum phase transition of the environment. Specially, we find that the entanglement revives after it complete disappearance under certain conditions. The size of the environment affects the entanglement evolution too. The larger the size of the environment is, the faster the entanglement is vanished. An infinite-sized environment can completely destroyed entanglement in a short time. The entanglement evolution of the Werner state also has been analyzed. We find that the noise contained by the Werner state plays an important role in the life time of the entanglement. The more the noise is, the shorter the life time for the entanglement of the Werner state is. As the other

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