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Physics Letters A www.elsevier.com/locate/pla

Quantum and classical correlations in a classical dephasing environment Jun-Qi Li ∗ , J.-Q. Liang Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, Shanxi, China

a r t i c l e

i n f o

Article history: Received 11 June 2010 Received in revised form 6 December 2010 Accepted 24 January 2011 Available online 22 February 2011 Communicated by P.R. Holland

a b s t r a c t We calculate the non-Markovian dynamics of quantum correlations, including entanglement and discord, and classical correlation in a two-qubit system with Ornstein–Uhlenbeck noise and discuss the relations in these correlations. It’s found that the initial-state parameters and the non-Markovian properties play an important role in these dynamics behaviors. © 2011 Elsevier B.V. All rights reserved.

Keywords: Entanglement Noise Discord

1. Introduction Entanglement, as a quantum correlation, is widely believed to play a key role in many-body systems, such as quantum information [1,2], condensed matter physics [3], and so forth. Due to its fundamental importance, many efforts have been devoted to the related study of entanglement [4–7]. However, it was found that some quantum tasks, for example, Grover search [8] and deterministic quantum computation with one pure qubit [9], may also be implemented without entanglement. This implies that entanglement is not the only type of correlation useful for quantum technology, which is veriﬁed both theoretically [10] and experimentally [11]. That is to say, there exists other kind of nonclassical correlation other than entanglement that is responsible for the computational speedup over the best known classical approach. Such correlation, may be presented even in mixed separable states, is called quantum discord by Ollivier and Zurek [12] with a deﬁnition based on the distinction between the quantum and classical information theory. Overall, quantum discord quantiﬁes the total amount of nonclassical correlations of a state, while the measure of entanglement only quantiﬁes the nonlocal correlations. It was shown that the quantum discord is exactly equal to both the entanglement and classical correlation for pure states, however, it is not true for general two-qubit mixed states [13]. So far, the relation among quantum discord, entanglement, and classical correlation is not yet clear. Recently, it has been argued that quantum discord is more practical than entanglement to describe quantum correlation [10,11]

*

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and could be considered as a new resource of quantum computation. As a result the concept of quantum discord has received a great deal of interest [14–23]. Because of the unavoidable interaction between a quantum system and its environment, understanding the dynamics of quantum and classical correlations is an interesting line of research [24–28]. Recently, Ref. [24] and Refs. [25,26], respectively, studied the inﬂuence of Markovian and non-Markovian environment on the dynamic behaviors of asymmetric quantum discord. They showed that quantum discord is more robust than entanglement against decoherence. After having analyzed the dynamics of quantum correlations (based on asymmetric and symmetric versions of quantum discord) and classical correlation under noises, Maziero et al. [27] obtained several signiﬁcant results, for instance, decoherence may occur without entanglement between the system and its environment and, in some cases, the initial nonclassical correlation of bipartite system is completely evaporated without transferring to the environments [27]. What needs to be stressed, here, is that asymmetric quantum discord (one-side measures of correlation) and symmetric quantum discord (two-side measures of correlation) refer to two types of quantiﬁers of quantum correlations [27,29], which depend on the measurement process. It has been veriﬁed that symmetric quantum discord can be reduced to asymmetric quantum discord for two-qubit composite states with maximally mixed marginals. More recently, different kinds of bipartite correlations under decoherence were investigated in an all-optical experimental setup [30]. The decoherence-free evolution of quantum and classical correlations under certain Markovian noise and the sudden transition from classical to quantum decoherence regimes are observed in Refs. [30,31]. It is worthwhile emphasizing that most investigations of correlation dynamics are focused on the quantum environments. As

J.-Q. Li, J.-Q. Liang / Physics Letters A 375 (2011) 1496–1503

a result, it is often stated that decoherence of an open quantum system is due to growing entanglement between system and quantum environments. Nevertheless, there are many relevant examples showing that the loss of quantum properties, without invoking a quantum environment, is attributed to the random external ﬁelds. For instance, in the experimental studies with trapped ions, classical ﬂuctuations are believed to be the main source of decoherence. At the same time, in nuclear magnetic resonance (NMR) decoherence studies the ﬂuctuating ﬁelds also play an important role [32]. Moreover, random ﬂuctuations are also of special interest in the context of dynamical suppression [33], quantum gate [34] and geometric quantum phase [35], etc. Motivated by the fundamental importance of random ﬂuctuations, in this Letter, we provide a systematic survey of the correlation dynamics in a two-qubit system which is affected by Gaussian colored (Ornstein–Uhlenbeck) noise that can act both on single-qubit and the joint two-qubit systems. Recently, the Ornstein–Uhlenbeck noise has been widely studied in various cases [36–40]. In the present work, we study the effects of initial parameters of a system and the non-Markovian properties on quantum and classical correlations and discuss the relations between them in some special cases. We show that the system can exhibit a sudden transition from classical to quantum decoherence regimes in local (or independent) noises. The strong non-Markovian properties prolong the constancy regions of quantum discord. However, in the global (or common) noise, there are many new dynamic features. It is found that besides decaying, quantum discord can be greatly ampliﬁed or protected, and so on. Although the sudden transitions emerge in quantum discord, the behavior, that quantum correlation is constant before the transition time and then loses monotonously after it, does not happen. Recently, Ref. [41] studied the dynamics of quantum discord between two noninteracting qubits immersed in a common Ohmic environment and pointed out that quantum discord can be stably ampliﬁed or protected for certain X -type states. Here, we discuss in detail the effects of non-Markovian properties on the dynamics of correlations for the global classical–noise model different from the spin–boson model in Ref. [41], where the memory effect of environment is not involved. Moreover, we consider two identical qubits with different initial parameters and systemically investigate the dynamics of both quantum discord and classical correlation.

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a comparison among entanglement, quantum discord and classical correlation on an equal footing. In this method [44], quantum discord is deﬁned as the distance between ρ A B and its closest classical state X A B , which can be expressed as

Q ρ AB = S X AB − S ρ AB .

(2)

Accordingly, relative entropy of entanglement Rn(ρ A B ) is deﬁned as the distance to the closest separated state σ measured by relative entropy S [45], that is

Rn

ρ A B = min S ρ A B σ = S ρ A B σ , σ ∈D

(3)

where S (ρ A B σ ) = Tr(ρ A B log ρ A B ) − Tr(ρ A B log σ ) is the relative entropy between states ρ A B and σ and D denotes the set of all separable states. The relative entropy Rn(ρ A B ) is a faithful entanglement measure, which is strictly positive if and only if ρ A B is entangled. One can interpret Rn(ρ A B ) as a measure of statistical distinguishability of ρ A B . The more entangled ρ A B is, the more it is distinguishable from σ . In bipartite systems, Rn(ρ A B ) is shown to be a lower bound for the entanglement of formation and an upper bound for the entanglement of distillation. Here, we try to quantify the entanglement in terms of Rn. In the same spirit, the classical correlation is expressed as C (ρ A B ) = S (πX A B ) − S (X A B ), which is the minimal distance between the closest classical-state X A B and its product state πX A B . Since all the distances are measured with relative entropy, we may compare different correlations in a consistent way. It is not diﬃcult to verify that for a class of initial states with maximally mixed marginals (ρ A ( B ) = I A ( B ) /2) described by

ρ

AB

= 1

AB

+

3

A

c i σi ⊗ σ

B i

4,

(4)

i =1

the expression of quantum discord is D (ρ A B ) = Q(ρ A B ) and the classical correlation is C (ρ A B ) = C (ρ A B ) in our model. At the same AB time, Rn(ρ ) with Eq. (4) can be expressed as Rn(ρ A B ) = 1 + λ1 log λ1 + (1 − λ1 ) log(1 − λ1 ) if λ1 0.5, whereas Rn(ρ A B ) = 0 if λ1 ∈ [0, 0.5]. The parameter λ1 is the highest one of the four eigenvalues of density matrix ρ A B . Because of this, we can describe the correlations with D (ρ A B ), C (ρ A B ) and Rn(ρ A B ) in this Letter.

2. Measures of correlations 3. Correlation dynamics with two-qubit Hamiltonian The physical quantities we want to investigate here are quantum and classical correlations between two qubits. We use the quantum discord and entanglement as the measures of quantum correlations. The quantum discord is deﬁned as the difference between two quantum extensions of classically equivalent concepts, which is given by [42]

D

ρ

AB

=I ρ

AB

−C ρ

AB

(1)

,

where I (ρ ) = S (ρ ) + S (ρ ) − S (ρ ) is the so-called quantum mutual information and C (ρ A B ) = sup{Πk } [ S (ρ A ) − S (ρ A B |{Πk })] [43] represents classical correlation where the maximum is taken over the set of projective measurements {Πk }. S (ρ ) = − Tr(ρ log ρ ) is the von Neumann entropy and S (ρ A B |{Πk }) = k pk S (ρk ) denotes the conditional entropy of A, given the knowledge of AB

state B, with

A

ρk =

B

( I ⊗Πk )ρ A B ( I ⊗Πk ) pk

AB

and the probability pk = Tr[( I ⊗

Πk )ρ A B ( I ⊗ Πk )]. It has been shown that D (ρ ) 0 [16] with the equal sign only for classical correlation. In this Letter, all logarithms are of base 2. Recently, Modi et al. [44] provided an uniﬁed view of quantum and classical correlations by employing the relative entropy as a distance measure of correlations, which allows us to make

In the following sections, we will investigate the correlation dynamics of a two-qubit system under non-Markovian dephasing noises described by so-called Ornstein–Uhlenbeck processes. The Hamiltonian of total system can be written as

H (t ) = −

1

2

Ω A (t )σzA + Ω B (t )σzB + Ω A B (t ) σzA + σzB ,

(5)

where Ωi (t ) (i = A , B , A B ) are the independent classical ﬂuctuations of the qubit level spacings, which satisfy the properties [46]

Ωi (t ) = 0,

Γi γ −γ |t −s| Ωi (t )Ωi (s) = , e 2

(6)

where · · · denotes the ensemble time average, and Γi is the dephasing damping rate of the ith qubit. The parameter γ , deﬁning the spectral width of noise, is connected to the reservoir correlation time by the relation τb = γ −1 . Eq. (6) shows that the processes are non-Markovian and can reduce to the well-known Markov case when τb → 0 [40,46–48]. Giving the basis {|1 = |++, |2 = |+−, |3 = |−+, |4 = |−−} and based on Eqs. (5) and (6), the reduced density matrix of

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two-qubit system can be expressed in terms of Kraus operators as

ρs (t ) = ε ρ (0) =

2 3

†

B†

A†

Dk E j E i

ρ (0) E iA E Bj D k ,

(7)

i , j =1 k =1

where we have assumed that the initial state of system ρ (0) is independent of the environment. The interaction terms with the local environments Ω A (t ) and Ω B (t ) are given by the Kraus operators

c 1 , c 2 , c 3 , one can distinguish three different dynamical regimes, viz. Regime 1. |c 3 (0)| |c 1 (0)|, |c 2 (0)|. In this case, the classical correlation C (ρ A B ) in Eq. (11) remains constant in time. However, quantum discord decays monotonically. Noticed that if c 1 (0) = c 2 (0) = −c 3 (0) = r, Eq. (4) reduces to the Werner state ρ I (0) = r |Φ + Φ + | + (1 − r ) I 4 /4 or ρII (0) = r |Ψ + Ψ + | + (1 − r ) I 4 /4 for c 1 (0) = c 3 (0) = −c 2 (0) = r. Based on ρ I (0) or ρII (0) and Eq. (11), classical correlation reads

1+r

Cρ I (ρII ) =

E 2A =

quantum discord in Eq. (12) reads

E 1B E 2B

ω A (t )|−−| ⊗ I B ,

= I A ⊗ |++| + γ B (t )|−−| , = I A ⊗ ω B (t )|−−| ,

(8)

D ρ I (ρII ) =

D 3 = ω3 (t )|44|.

(9)

The parameters appearing in Eqs. (8) and (9) are given by

ω A (t ) = 1 − γ A2 ,

ω B (t ) = 1 − γ B2 ,

ω2 (t ) = −γ A2B 1 − γ A2B , Γi

ω3 (t ) =

ω1 (t ) = 1 − γ A2B ,

1 − γ A2B

1 − γ A4B ,

−γ t

where γi (t ) = e − 2 [t +(e −1)/γ ] (i = A , B , A B ). Next, let us ﬁrst consider the dynamics of initial state in Eq. (4) for one-qubit and two-qubit local dephasing noises. The time evolution of total system under these independent environments is given by

+ Ψ + + λ+ (t )Φ + Φ + ρ A B (t ) = λ+ Ψ (t ) Ψ Φ

− − − − Φ + λ− Ψ , + λ− Φ (t ) Φ Ψ (t ) Ψ

(10)

where

λ± Ψ (t ) = λ± Φ (t ) =

1 ± c 1 (t ) ∓ c 2 (t ) + c 3 (t ) 4 1 ± c 1 (t ) ± c 2 (t ) − c 3 (t ) 4

, ,

√

√

and |Ψ ± = (|1 ± |4)/ 2, |Φ ± = (|2 ± |3)/ 2 are Bell states. Based on Eq. (1), the expressions of classical and quantum correlations can be written as [42] 2

1 + (−1)k X (t ) C ρ AB = log 1 + (−1)k X (t ) ,

D

k =1

ρ AB = 2 +

2

λlk (t ) log λlk (t ) − C (ρ A B ),

4

log[1 − r ],

(13)

1

(1 − 2γ A r + r ) log[1 − 2γ A r + r ] + (1 + 2γ A r + r ) 1+r 2

4

log[1 + r ],

(14)

and the relative entropy of entanglement Rn(ρ ) is

D 1 = γ A B (t ) |11| + |44| + |22| + |33|, D 2 = ω1 (t )|11| + ω2 (t )|44|,

1

2

× log[1 + 2γ A r + r ] −

and the collective interaction is described as

2

log[1 + r ] +

1−r

E 1A = |++| + γ A (t )|−−| ⊗ I B ,

(11) (12)

k,l

where X (t ) = max{|c 1 (t )|, |c 2 (t )|, |c 3 (t )|}, k = Ψ, Φ , and l = ±. We ﬁnd that |c 1 (t )| = |c 1 (0)|γ A , |c 2 (t )| = |c 2 (0)|γ A , |c 3 (t )| = |c 3 (0)| for one-qubit dephasing noise, while |c 1 (t )| = |c 1 (0)|γ A2 , |c 2 (t )| = |c 2 (0)|γ A2 , |c 3 (t )| = |c 3 (0)| for two-qubit dephasing noise (hereafter, Γ A = Γ B = Γ A B = Γ is assumed). Thus, in both cases the similar behaviors are presented for quantum and classical correlations, respectively. The only difference is that the decay of quantum or classical correlations in two-qubit local noise is faster than one-qubit dephasing noise. In other words, two-qubit local noise is even more harmful than one-qubit non-Markovian noise in the application of quantum correlations. Thus, it is enough for us to consider the effects of decoherence on correlations only for one-qubit case in independent environments. Depending on the relations of

Rnρ I (ρII ) = 1 + λ log λ + (1 − λ) log(1 − λ),

(15)

where λ = (1 + r + 2r γ A )/4. We can learn from Eqs. (14) and (15) that both quantum discord and entanglement depend on the purity r and dephasing strength γ A . In Fig. 1 we display the numerical results of Rnρ I (ρII ) and D ρ I (ρII ) as a function of dimensionless time Γ t. From Fig. 1 some useful results can be got: First of all, the non-Markovian property b, purity r can inﬂuence Rnρ I (ρII ) and D ρ I (ρII ) to a great extent. We can see that the strong non-Markovian effect (b 1) and high values of purity r both play a key role in improving the quantum discord and relative entropy of entanglement. With the increase of b (b > 1), the time-evolution of Rnρ I (ρII ) and D ρ I (ρII ) will tend to Markov behavior. It is clearly shown that the decay of Rnρ I (ρII ) and D ρ I (ρII ) can be postponed with high values of purity r. Secondly, the entanglement sudden death (ESD) can happen except for r = 1, however, discord disappears only asymptotically, which is different from the entanglement. Finally, discord is usually larger than entanglement for 0 r 1. In particular, nonvanishing discord emerges for the well-known separable states r 1/3 in which the entanglement is zero. In this sense, discord is a fundamentally different resource and more robust than entanglement. On the other hand, classical correlation Cρ I (ρII ) is equal to 1, 0.42 and 0.082 for r = 1, r = 0.72 and r = 1/3, respectively. Thus, classical correlation, may be smaller or larger than discord. However, it is always larger than entanglement. Regime 2. c 3 (0) = 0. All of the correlations decay in a monotonic way. Regime 3. |c 3 (0)| < |c 1 (0)| and/or |c 2 (0)|. This is the most interesting dynamical region for us, where the sudden transition from classical to quantum decoherence can happen at the critical transition time t c . In order to be speciﬁc, we take c 1 (0) = 1, c 3 (0) = −c 2 (0) = 0.6 into account. Obviously, based on Eq. (11), the classical correlation reads 2

1 + (−1)k γ A (t ) Ct

2

(16)

and 2

1 + 0.6(−1)k Ct >tc ρ A B = log 1 + 0.6(−1)k , k =1

2

(17)

where the critical time t c meets the condition γ A (t c ) = 0.6. Then, one can easy ﬁnd that the discord takes the form of Eq. (17) for D t

J.-Q. Li, J.-Q. Liang / Physics Letters A 375 (2011) 1496–1503

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Fig. 1. Entanglement (Rn) (a)–(b) and quantum discord (D) (c)–(d) as a function of the dimensionless time Γ t in one-qubit local noise with (a) r = 0.9, (b) b = 0.2, and the corresponding values in (c)–(d) are the same as given by (a)–(b), respectively. Here, we set the parameter b = γ /Γ , which reﬂects the strength of non-Markovian noises and controls the approach to the Markov limit.

Fig. 2. Entanglement Rn (dotted line), quantum discord (dashed line) and classical correlation (solid line) as a function of the dimensionless time Γ t with (a) b = 0.2 and (b) b = 5 in one-qubit local noise. The other parameters are: c 1 (0) = 1.0, −c 2 (0) = c 3 (0) = 0.6.

Rn

ρ A B = 1 + λ log λ + 1 − λ log 1 − λ ,

(18)

where λ = 0.4(1 + γ A ). Fig. 2 clearly shows that classical correlation decays monotonically until the critical time t c , and then keeps constant after it. On the contrary, quantum discord remains

constant during the time interval [0, t c ], and then decays monotonically. Moreover, Fig. 2 tells us that strong non-Markovian property b can greatly prolong t c and improve the values of classical and quantum correlations. And, discord is always smaller than classical correlation and larger than entanglement Rn.

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We have analyzed the dynamic evolution of correlations in local noises, however, the dynamic behaviors in global noise are also particularly interesting for us. In the following we investigate the dynamics of two qubits in the common environment. One can easily obtain the four eigenvalues of ρ A B (t ) in the collective noise as

λ± Ψ (t ) = λ± Φ (t ) =

1 ± c 1 (t ) ∓ c 2 (t ) + c 3 (0)

,

4 1 ± c 1 (0) ± c 2 (0) − c 3 (0) 4

,

and the classical and quantum correlations are given by Eqs. (11) and (12), respectively. Here, X (t ) = max{|c 3 (0)|, (|μ(t )| + |ν (t )|)/2} with μ(t ) = c 1 (0) + c 2 (0) and ν (t ) = c 1 (t ) − c 2 (t ) = (c 1 (0) − c 2 (0))γ A4 . It is found that quantum discord can be greatly ampliﬁed or protected for certain initial conditions, which is different obviously from what we have just discussed in local noises. In order to illustrate this point, we adopt the categories as follows Regime 1 . |c 3 (0)| (|μ(0)| + |ν (0)|)/2. In this case, classical correlation C (ρ A B ) is unchanged all the time. Quantum discord presents complicated dynamic behaviors, which depend on c 1 (0), c 2 (0). We ﬁnd that (i) quantum discord is constant if c 1 (0) = c 2 (0). Undoubtedly, ρ I (0) belongs to this case. (ii) Quantum discord disappears monotonously in time for c 1 (0) = −c 2 (0) seen from ρII (0).

Fig. 3. Entanglement (Rnρ I ) (dotted line), quantum discord (D ρ I ) (solid line), and classical correlation (C ) (dashed line) versus r in the global noise.

Therefore, quantum discord D ρ I (t ) displays evidently different behaviors from D ρII (t ), so do Rnρ I (t ) and RnρII (t ). The state ρ I (0) is decoherence-free under collective dephasing and more robust than ρII (0). Quantum discords D ρ I (t ) and D ρII (t ) (or entanglement Rnρ I (t ) and RnρII (t )) can be obtained from Eq. (14) (or Eq. (15)) by replacing γ A with 1 and γ A4 , respectively. Classical correlation C is given by Eq. (13). In Fig. 3, we plot D ρ I (t ), Rnρ I (t ) and C as a function of purity r, from which a clear picture can be got, i.e., quantum discord and classical correlation are always larger than entanglement. At the same time, discord is always greater classical correlation in disagreement with the previous conjecture that classical correlation is always greater than quantum correlations [49]. (iii) Quantum discord will eventually decay to a stable value with |c 1 (0)| = |c 2 (0)| as shown in Fig. 4. We plot the quantum discord and entanglement Rn as a function of the scaled time Γ t with c 1 (0) = c 3 (0) = 0.6, c 2 (0) = −0.2, b = 0.2 (solid line) and b = 5 (dotted line) in Figs. 4(a) and (b), respectively. We can see that discord is more greater than entanglement and non-Markovian property b plays a passive role in the improvement of correlations. In this situation, the classical correlation is C = 0.278. Regime 2 . c 3 (0) = 0. Based on analyzing the behaviors of correlations, we ﬁnd that (i) quantum and classical correlations are immune to decoherence if c 1 (0) = c 2 (0). (ii) Quantum discord and classical correlation tend to zero asymptotically for c 1 (0) = −c 2 (0). (iii) If |c 1 (0)| = |c 2 (0)|, it is diﬃcult for us to ﬁnd general rules of the quantum discord. With various values of c 1 (0) and c 2 (0), quantum discord exhibits different dynamic behaviors. In contrast with quantum discord, classical correlation can reach stable value ultimately. In order to have a vivid illustration, in Fig. 5, we have plotted the quantum discord (dash-dotted line with b = 0.2 and dotted line with b = 5) and classical correlation (solid line with b = 0.2 and dashed line with b = 5) with respect to the different initial parameters c 1 (0) and c 2 (0) in the time evolution. The phenomenon of saturated ampliﬁcation after “smooth creation” of quantum discord is found for parameters c 1 (0) = 0.5, c 2 (0) = 0. It indicates that the common noise plays a role in the generation of quantum discord between two qubits. Moreover, we also can see from Fig. 5 that quantum discord may get an ampliﬁed or relatively small stationary value compared with initial discord after undergoing the monotonic decay and instantaneous vanishing when c 1 (0) = 0.5, c 2 (0) = −0.1 or when c 1 (0) = 0.5, c 2 (0) = −0.2 (c 2 (0) = −0.3). These demonstrate a fact that quantum states ﬁnally become steady states in long-time evolution. The stable values are closely associated with ζ = |c 1 (0) + c 2 (0)||c 1 (0) − c 2 (0)|. The larger ζ is,

Fig. 4. Quantum discord (D) and entanglement (Rn) versus Γ t with c 1 (0) = c 3 (0) = 0.6, c 2 (0) = −0.2 in the global noise. The solid line and dotted line in ﬁgure are for b = 0.2 and b = 5, respectively.

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Fig. 5. Quantum discord (D) and classical correlation (C ) versus Γ t in the global noise. The dash-dotted line and solid line, respectively, represent D and C in the case of b = 0.2. While the dotted line and dashed line correspond to D and C in b = 5, respectively. The other parameters are: (a) c 1 (0) = 0.5, c 2 (0) = 0; (b) c 1 (0) = 0.5, c 2 (0) = −0.1; (c) c 1 (0) = 0.5, c 2 (0) = −0.2; (d) c 1 (0) = 0.5, c 2 (0) = −0.3.

the larger stable value will be obtained (see Fig. 5). In fact, the interesting dynamic evolution in Fig. 5 may be understood with the help of the derivative of quantum discord with respect to time t, i.e.,

∂ D (ρ A B (t )) ∂t

=

∂γA ∂ D (ρ A B (t )) ∂ γ A ∂γA ∂ t [41]. Since ∂ t

< 0, γ A monoton-

ically decreases with respect to time t. Hence, quantum discord ∂ D (ρ A B (t )) ∂γA ∂ D (ρ A B (t )) 0. Moreover, ∂γA

will increase with time if

< 0 or else will decrease

∂ D (ρ A B (t )) if ∂γA

= 0 implies that quantum

>

∂ D (ρ A B (t )) ∂γA ν (t ) 1+ν (t )+c 3 (0) log [ ] γA 1−ν (t )+c 3 (0)

discord is zero or a stable value. Based on Eq. (12), seems to be F (γ A ) + G (γ A ), where F (γ A ) = |ν (t )|

2−|ν (t )|−|μ(t )|

and G (γ A ) = γ log[ 2+|ν (t )|+|μ(t )| ]. Therefore, the dynamics beA haviors of quantum discord in Fig. 5 may be veriﬁed through ∂ D (ρ A B (t )) . Here, the speciﬁc analyses are omitted. On the other ∂t

hand, we observe that non-Markovian property b greatly inﬂuences the evolution of correlations. To our surprise, differently from the local noises, strong non-Markovian properties are not always beneﬁt to quantum discord. On the side, entanglement in Fig. 5 is zero in all cases and classical correlation becomes constant being less than or equal to quantum discord after some timeperiod. Regime 3 . |c 3 (0)| < (|μ(0)| + |ν (0)|)/2. We ﬁnd that the ﬁrst derivative of D (ρ A B (t )) with respect to γ A has the following forms:

∂ D (ρ A B (t )) = ∂γA

F (γ A ) + G (γ A ),

if 0 < t t c ,

F (γ A ),

if t > t c ,

(19)

where F (γ A ) and G (γ A ) are the same as mentioned earlier. t c is determined by |c 3 (0)| = (|μ(t c )| + |ν (t c )|)/2}. With Eqs. (12) and (19), we know that quantum discord is constant for c 1 (0) = c 2 (0) and decays toward zero with a sudden transition for c 1 (0) = −c 2 (0). By changing the parameters c 1 (0), c 2 (0) and c 3 (0), quantum discord may be ampliﬁed or protected after some time-period, which as shown in Fig. 6 where the parameters are: (a) c 1 (0) = −c 2 (0) = 0.5, c 3 (0) = 0.2; (b) c 1 (0) = 1.0, c 3 (0) = −c 2 (0) = 0.2; (c) c 1 (0) = 1.0, c 3 (0) = −c 2 (0) = 0.4; (d) c 1 (0) = 0.7, c 2 (0) = −0.4 and c 3 (0) = 0.2; with solid line for b = 0.2 and dotted line for b = 5. With these parameters, we ﬁnd that classical correlation may be greater or less than the discord, while entanglement is always less than the discord. In addition, the strong nonMarkovian property b may also play a negative role in quantum discord. 4. Conclusion To sum up, we have studied the dynamics of two-qubit quantum and classical correlations in classical Ornstein–Uhlenbeck noise based on the general initial states with maximally mixed marginals. We have shown that both quantum and classical correlations display different behaviors in local and global environments depending on initial conditions and non-Markovian effects. There exist the same dynamic behaviors for discord (or entanglement) when the initial Werner states ρ I (0) and ρII (0) are considered in local noises rather than the collective case, while ρ I (0) is more robust than ρII (0) in dephasing environments. On the other

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Fig. 6. Quantum discord as a function of the dimensionless time Γ t with b = 0.2 (solid line) and b = 5 (dotted line) in the global noise. The other parameters used are: (a) c 1 (0) = −c 2 (0) = 0.5, c 3 (0) = 0.2; (b) c 1 (0) = 1.0, −c 2 (0) = c 3 (0) = 0.2; (c) c 1 (0) = 1.0, −c 2 (0) = c 3 (0) = 0.4; (d) c 1 (0) = 0.7, c 2 (0) = −0.4, c 3 (0) = 0.2.

hand, the system of two qubits can undergo a sudden transition at the critical time t c between classical and quantum decoherence in local environment and the strong non-Markovian effects b can greatly prolong t c and improve the discord. It is worthwhile emphasizing that the dynamics of quantum and classical correlations in two-qubit system under local random telegraph signal noise is investigated recently in [50]. The revivals of correlations and multiple transitions between the classical and quantum decoherence are attributed to the memory effects of non-Markovian environment. However, we noticed that there is no back-action from environment in our local Ornstein–Uhlenbeck noisy model. Here the nonzero correlation time of the bath simply prolongs the survival time of correlations. So the statement that memory feedback of environment may bring into revivals of correlations cannot be a general conclusion. At last, we have investigated the dynamics in global dephasing noise and found that quantum discord can be greatly ampliﬁed or protected in time. To our surprise, in this situation, strong non-Markovian properties b are not always beneﬁt to the quantum discord. These results are different from the local noises. Moreover, it should be pointed out that we have made a comparison among discord, entanglement, and classical correlation and found that discord may be larger or smaller than classical correlation and both of them are larger than entanglement. Acknowledgements This work was supported by the Natural Science Foundation of China under Grants Nos. 10775091, 11074154, 11047167, and

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