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Robustness investigations on quantum entanglement and quantum discord of coupled qubits in squeezed vacuum reservoir YingHua Ji a,c,∗ , YongMei Liu b a b c

Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China College of Mathematics and Information Science of Jiangxi Normal University, Nanchang, Jiangxi 330022, China Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China

a r t i c l e

i n f o

Article history: Received 5 November 2012 Accepted 1 April 2013

PACS: 03.65.Ta 03.67.−a Keywords: Quantum discord Concurrence Squeezed vacuum reservoir Coupled qubits

a b s t r a c t We investigate the quantum discord of coupled qubits in squeezed vacuum reservoir and compare it with the quantum entanglement of system. We ﬁnd that the quantum discord and entanglement perform completely oppositely with the change of squeezed parameters. Quantum discord survives longer with the increase of squeezed amplitude parameter and entanglement death faster on the contrary. Under high squeezed amplitude parameter, the quantum discord can keep nonzero which indicate that the quantum discord is more robust than entanglement. We also ﬁnd that the purity reduction of the initial quantum state will lead to the decay of concurrence or quantum discord. However, the quantum discord damps remarkably more slowly and survives longer than concurrence. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Entanglement reﬂects a nonlocal quantum correlation between each subsystems of a quantum multi-system [1–3]. It is the core resource to realize quantum information and quantum computing. The prepared entangled state is very weak and difﬁcult to preserve because of the decoherence inﬂuence of the interaction between the system and environment. Therefore, based on the existing experiment condition and technology, how to prepare the entangled source with strong ability to environment disturbance, stability and high entanglement is one of the problems people are interested in [4–8]. Recently, Datta proposed a model that can realize the quantum speedup algorithm without depending on the quantum entanglement. In their model, no entanglement resource is used and the state is throughout separable state. People cannot help asking that what play the role of speedup the algorithm. People come to realize that quantum entanglement is not the resource that is indispensible for some quantum computing tasks. With the development of research, people found that quantum correlations can measure the quantum properties better [9,10]. Differing from entanglement, quantum correlation can still possess quantum characteristics even

∗ Corresponding author at: Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China. E-mail address: [email protected] (Y. Ji). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.04.039

for a complete separable state. How to measure the feature is a research direction and hotspot currently. At present, the quantum discord introduced by Olliver and Zurek is used for measuring the quantum correlations [11]. However, it is controversial that the quantum discord acts in quantum computing, especially that in different quantum algorithms, the function of the quantum correlations is still unknown. In the article, we make a comparison between quantum entanglement and quantum discord of coupled interactive qubits in non-Markovian process in squeezed vacuum reservoir. We ﬁnd that, for a certain initial state, the quantum discord and entanglement perform completely oppositely with the change of squeezed parameters in squeezed vacuum states. The characteristic provides a feasible way to enhance the quantum discord of a system. 2. Model Squeezed vacuum state is formed by the squeezing operator S acting on vacuum ﬁelds. It is a kind of non-classical light ﬁeld that has been prepared in experiment. In the squeezed vacuum state, it will make the quantum noise of one quadrature component decrease and the other component increase. Thus, the squeezed vacuum state plays an important role in optical communication and is deﬁned as

0 = S() 0 g

(1)

Y. Ji, Y. Liu / Optik 124 (2013) 5620–5623

S = exp

1 2

∗ b2 −

5621

1 +2 b , 2

(2)

where = rei , r the squeezed amplitude parameter and the squeezed angle (in the following discussion, = 0), and S + =

S − = S − . Here, we model a system of coupled interactive qubits in the squeezed vacuum reservoir. We suppose that the two qubits interact with the environment independently. In interaction picture, the Hamiltonian of the model is [12]

HI (t) = (1+ + 2+ )

gn bn ei(ε−ωn )t + (1− + 2− )

n

−i(ε−ωn )t gn∗ b+ ne

Fig. 1. Concurrence and quantum discord as a function of t with ϕ = /4 in the vacuum state. (a) d = 1; (b) d = 0.8; (c) d = 0.6.

n

(3) gn is the interaction constant between the coupled qubits and the reservoir. bn and b+ n represent the annihilation and creation operator of the squeezed vacuum reservoir. j± (j = 1, 2) is the Pauli operator which is used to describe the qubits. They obey to the commutation and anticommutation algebra relation of Pauli operators. Based on the quantum reservoir theory, the quantum master equation of the reduced density operator satisﬁes [13] d (t) = dt

−

k

2 cosh (r) +k −k − 2−k +k + +k − 2

k=A,B

k

−

−i 2 e sinh(2r)−k −k − sinh (r) −k +k − 2+k −k + −k + 2 2

−

i e sinh(2r)+k +k 2

⎜ ⎜

0

0

14 (t)

0

22 (t)

23 (t)

0

0

32 (t)

33 (t)

0

41 (t)

0

0

44 (t)

AB (t) = ⎜ ⎜

⎝

⎞

⎟ ⎟ ⎟. ⎟ ⎠

−

11 (t)44 (t)}.

(5)

22 (t)33 (t), 23 (t) (6)

A bipartite state may include not only classical correlation but also include quantum correlation. Recently, Ollivier and Zurek introduced the notion, called quantum discord (QD), as a capture of all nonclassical correlations between bipartite quantum systems [16]. QD is deﬁned as the difference between the quantum versions of two classically equivalent deﬁnitions of mutual information and can be given as QD (AB ) = I (AB ) − CD (AB ) ,

=

pk S (k ) is the quantum conditional entropy,

k

k = (IA ⊗ Bk ) (IA ⊗ Bk ) /Tr (IA ⊗ Bk ) (IA ⊗ Bk ) is the conditional density operator corresponding to the outcome labeled by k, and pk = Tr (IA ⊗ Bk ) (IA ⊗ Bk ). Here IA is the identity operator performed on subsystem A. 3. Discussions Without losing the generality, in the following discussion, we consider two classes of initial sates [19],

For the given initial state, the solutions of the quantum master equation see in Appendix A. We can see that the system can always keep the X-state unchanged in the dynamic evolution process. We adopt the concurrence entanglement deﬁned by Wootters to measure the system entanglement. Under X-state [15], C(t) = 2 max{0, 14 (t) −

(4)

11 (t)

S AB {Bk }

is the decay ratio of the reservoir. The solutions of the quantum master equation depend on the initial state of the system. We assume the system is initially at Xstate as follows [14]

⎛

Fig. 2. Concurrence and quantum discord as a function of t with r = 0.5 and ϕ = /4 in the squeezed vacuum state. (a) d = 1; (b) d = 0.8; (c) d = 0.6.

d, = d (ϕ)

(ϕ) +

1−d I4 , 4

(8)

where d is the purity of the initial states which ranges from 0 for maximally mixed I4 is the 4 × 4 idenstates to 1 for pure states, tity matrix, (ϕ) = cos ϕ 00 + sin ϕ 11 and are the Bell-like pure states and the parameter ϕ is sometimes called the degree of entanglement. In this section, we emphasize on discussing how the squeezed amplitude parameter inﬂuences the entanglement and quantum discord. We also compare the impact of vacuum reservoir and squeezed vacuum reservoir on the quantum discord. Figs. 1–5 is the most representative evolution curve we choose. Figs. 1 and 2 respectively reﬂects the effect of vacuum reservoir and squeezed vacuum reservoir on the quantum entanglement and quantum discord. First, it is shown that the dynamical evolution behaviors of entanglement and quantum discord are similar no matter in vacuum reservoir or in squeezed vacuum reservoir, and their survival time decreases with the purity reduction of

(7)

is where I (AB ) = S (A ) + S (B ) − S (AB ) the quantum mutual information and measures the total correlations, CD (AB ) is the classical correlation between the two subsysin Refs. tems. As discussed [17,18], the classical correlation CD (AB ) = max S (A ) − S AB B , where {Bk } is a set of von {Bk }

Neumann measurements performed on subsystem B locally,

Fig. 3. Concurrence and quantum discord as a function of t with d = 1 and ϕ = /4 in the squeezed vacuum state. (a) r = 1; (b) r = 0.5; (c) r = 0.1.

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Y. Ji, Y. Liu / Optik 124 (2013) 5620–5623

Fig. 4. Concurrence as a function of ϕ with t = 1 in the squeezed vacuum state. (a) r = 0.01, d = 1; (b) r = 0.1, d = 1; (c) r = 0.1, d = 0.8.

the initial quantum state. Second, quantum discord survives much longer than entanglement especially in the squeezed vacuum reservoir, indicating the quantum discord adapts to survive in squeezed vacuum reservoir. In addition, the purity reduction of the initial quantum state will lead to the decay of concurrence or quantum discord. However, the quantum discord damps remarkably more slowly than concurrence, which reveals that the quantum discord is more robust than concurrence. Fig. 3 presents different impacts of squeezed parameter on the entanglement and quantum discord when the coupled qubits are in squeezed vacuum reservoir. Obviously in squeezed vacuum reservoir, with the increase of the squeezed amplitude parameter, quantum discord and entanglement change much differently although the evolution trends of entanglement and quantum discord are basically the same. The increase of the squeezed amplitude parameter will shorten the survival time of the entanglement whereas extend that of the quantum discord, which indicates that putting the coupled qubits in the squeezed vacuum reservoir and increasing the squeezed amplitude parameter is beneﬁcial for greatly prolonging the surviving time of the quantum discord. In other words, the quantum discord is more robust than entanglement in squeezed vacuum reservoir. It is extremely strange that the quantum discord and entanglement behave oppositely with the change of squeezed parameter. It seems because that whether the quantum discord or the entanglement can be used to measure the quantum correlation, factually however, there are many different interior causes. Although both quantum discord and entanglement are the measures of certain quantum correlation, entanglement merely reﬂects the non-local correlation while quantum discord represents the quantum correlation of the whole bipartite system.

Above results indicate that there is indeed other quantum correlation than entanglement which can be characterized by quantum discord instead of entanglement (such as concurrence). The angle ϕ can describe the system initial concurrence. Figs. 4 and 5 show the concurrence and quantum discord change with ϕ under squeezed vacuum state. The concurrence appears deaths and survivals periodically with the change of ϕ. Combining with Fig. 3, we know that concurrence is difﬁcult to survive in squeezed vacuum reservoir with high squeezed amplitude. Inversely, it can be seen by combining Fig. 5 with Fig. 3 that, quantum discord is not only suitable to survive in squeezed vacuum reservoir with high squeezed amplitude, but also can be nonzero under appropriate squeezed amplitude although it exhibits periodic ﬂuctuations with the variation of angle ϕ. This characteristic will be helpful for quantum discord to be used as an available quantum resource. Therefore, it is worth further investigating and utilizing the squeezing properties of quantum discord. 4. Conclusions Based on the Markovian process of open quantum system, we discussed the concurrence and quantum discord of coupled interactive qubits in squeezed vacuum reservoir. We investigated the impact of squeezed amplitude parameter on the quantum discord and concurrence. We found that the entanglement and quantum discord have opposite dynamical behaviors with the change of squeezed parameter. The entanglement dead is faster with the increase of squeezed amplitude parameter. On the contrary, the quantum discord survives longer. It indicates that quantum discord can better adopt to survive in squeezed vacuum reservoir and is more robust than entanglement in environment. Namely, the quantum discord can keep nonzero in the higher squeezed amplitude parameter. Meanwhile, we investigated and analyzed the dependent relations between concurrence or quantum discord and the initial concurrence and the purity of initial quantum state. The results show that quantum discord decays with the purity of quantum state obviously more slowly than concurrence. Controlling the initial concurrence of coupled qubits or the purity of quantum state will efﬁciently suppress the damping of quantum correlation between coupled qubits. Therefore, it needs further comparisons and investigations on selecting quantum discord or concurrence as the useful information source in quantum information processing. Acknowledgement This work is supported by the National Natural Science Foundation of China under Grant no. 11164009. Appendix A.

0.7

Quantum discord

0.6

When the system is initially at X-state, the solution of Eq. (4) under the initially state (5) is direct, the density matrix elements of the time evolution are given by

a

0.5 0.4

b

0.3

c

0.2 0.1 0.0

0

1

2

3

ϕ

4

5

6

Fig. 5. Quantum discord as a function of ϕ with t = 1 in the squeezed vacuum state. (a) r = 1, d = 1; (b) r = 0.5, d = 1; (c) r = 0.5, d = 0.8.

11 (t) = A1 tanh4 r − A2 e−f (t) tanh2 r + A4 e−2f (t) ,

(A1)

22 (t) = A1 tanh2 r − A2 e−f (t) sec h2 r − A3 e−f (t) − A4 e−2f (t)

(A2)

33 (t) = A1 tanh2 r + A3 e−f (t) − A4 e−2f (t)

(A3)

44 (t) = 1 − 11 (t) − 22 (t) − 33 (t),

(A4)

where A1 =

cosh4 r cosh2 (2r)

,

(A5)

Y. Ji, Y. Liu / Optik 124 (2013) 5620–5623

A2 =

A3 =

−

1 2

cosh (2r)

cosh2 r 2 (44 (0) − 11 (0)) − cosh r , sec h(2r)

1 cosh2 (2r)

1 2 sec h2 (2r)

(33 (0) − 22 (0))

1 1 , (44 (0) − 11 (0)) + 2 2 sec h(2r)

A4 =

1 cosh2 (2r)

(A6)

(A7)

sinh2 r cosh2 r 1 44 (0) + 11 (0) − sinh(2r) , 4 sec h(2r) sec h(2r) (A8)

f (t) = t cosh(2r).

(A9)

Appendix B. When the system is initially at X-state, the solution of Eq. (4) under the initially state (5) is direct, the density matrix elements of the time evolution are given by 23 (t) = e−f (t) 32 (t) = e

−f (t)

14 (t) = e−f (t) 14 (t) = e−f (t)

B1 e−y(t) − B2 e+y(t) + (B3 − B4 ) ,

(B1)

+y(t)

(B2)

B1 e−y(t) − B2 e

− (B3 − B4 ) ,

B1 e−y(t) + B2 e+y(t) − (B3 + B4 ) , B1 e−y(t) + B2 e+y(t) + (B3 + B4 ) ,

(B3) (B4)

where 4B1 = 23 (0) + 32 (0) + 14 (0) + 41 (0),

(B5)

2B2 = 2B1 − 23 (0) − 32 (0),

(B6)

2B3 = 2B1 − 32 (0) − 14 (0),

(B7)

2B4 = 2B1 − 23 (0) − 14 (0),

(B8)

y(t) = tei sinh(2r).

(B9)

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References [1] J.A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Geometric quantum computation using nuclear magnetic resonance, Nature (London) 403 (2000) 869. [2] L.M. Duan, J.I. Cirac, P. Zoller, Geometric manipulation of trapped ions for quantum computation, Science 292 (2001) 1695. [3] Y.H. Ji, Y.M. Liu, Z.S. Wang, Avoiding the decay of entanglement for coupling two-qubit system interacting with a non-Markov environment, Chin. Phys. B 20 (2011) 070304. [4] M.P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, Environment-induced sudden death of entanglement, Science 316 (2007) 579. [5] C.E. Lopez, G. Romero, F. Lastra, E. Solano, J.C. Retamal, Sudden birth versus sudden death of entanglement in multipartite systems, Phys. Rev. Lett. 101 (2008) 080503. [6] P. Li, Q. Zhang, J.Q. You, Dividing two-qubit Hilbert space via abrupt and asymptotic disentanglement, Phys. Rev. A 79 (2009) 014303. [7] T. Yu, J.H. Eberly, Sudden death of entanglement, Science 323 (2009) 598. [8] F. Mintert, A.R.R. Carvalho, M. Kus, A. Buchleitner, Measures and dynamics of entangled states, Phys. Rep. 415 (2005) 207. [9] A. Datta, A. Shaji, C.M. Caves, Quantum discord and the power of one qubit, Phys. Rev. Lett. 100 (2008) 050502. [10] J.S. Xu, X.Y. Xu, C.F. Li, C.J. Zhang, X.B. Zou, G.C. Guo, Experimental investigation of classical and quantum correlations under decoherence, Nat. Commun. (2010), http://dx.doi.org/10.1038/ncomms1005. [11] H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett. 88 (2001) 017901. [12] M. Ikram, F.L. Li, M.S. Zubairy, Disentanglement in a two-qubit system subjected to dissipation environments, Phys. Rev. A 75 (2007) 062336. [13] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, London, 1997. [14] L. Mazzola, J. Piilo, S. Maniscalco, Sudden transition between classical and quantum decoherence, Phys. Rev. Lett. 104 (2010) 200401. [15] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245. [16] M. Ali, A.R.P. Rau, G. Alber, Quantum discord for two-qubit X states, Phys. Rev. A 81 (2010) 042105. [17] H. Katiyar, S.S. Roy, T.S. Mahesh, A. Patel, Evolution of quantum discord and its stability in two-qubit NMR systems, Phys. Rev. A 86 (2012) 012309. [18] L.A. Correa, A.A. Valido, D. Alonso, Asymptotic discord and entanglement of nonresonant harmonic oscillators under weak and strong dissipation, Phys. Rev. A 86 (2012) 012110. [19] T. Yu, J.H. Eberly, Quantum open system theory: bipartite aspects, Phys. Rev. Lett. 97 (2006) 140403.

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