Dynamics of entanglement for a two-parameter class of states in a qubit-qutrit system
aa r X i v : . [ qu a n t - ph ] J u l Dynamics of entanglement for a two-parameter class of states in a qubit-qutrit system ∗ Hai-Rui Wei, Bao-Cang Ren, Tao Li, Ming Hua, and Fu-Guo Deng † Department of Physics, Applied Optics Beijing Area Major Laboratory,Beijing Normal University, Beijing 100875, China (Dated: December 7, 2018)We investigate the dynamics of entanglement for a two-parameter class of states in a hybridqubit-qutrit system under the influence of various dissipative channels. Our results show that en-tanglement sudden death (ESD) is a general phenomenon and it usually takes place in a qubit-qutritsystem interacting with various noisy channels, not only the case with dephasing and depolarizingchannels observed by others. ESD can only be avoided for some initially entangled states under someparticular noisy channels. Moreover, the environment affects the entanglement and the coherenceof the system in very different ways.
PACS numbers: 03.65.Yz, 03.67.Mn, 03.65.Ta
I. INTRODUCTION
Quantum entanglement is a vital resource for quantuminformation processing [1]. However, isolating a quantumsystem completely from its environment is plainly an im-possible task and each quantum system will inevitablyinteract with its environment. Therefore, it is impor-tant to investigate the behavior of an entangled quan-tum system under the influence of its environment. Re-cently, Yu and Eberly [2, 3] investigated the dynamics oftwo-qubit entangled states undergoing various modes ofdecoherence. They found that it takes an infinite timeto complete decoherence locally, the global entanglementmay be lost in a finite time, and the decay of a single-qubit coherence can be slower than the decay of a two-qubit entanglement. The abrupt disappearance of en-tanglement in a finite time was named ”entanglementsudden death”(ESD). A geometric interpretation of thephenomenon is given in Ref.[4]. In addition, experimen-tal evidences of ESD have been reported for optical se-tups [5, 6] and atomic ensembles [7]. Clearly, ESD canseriously affect the applications of entangled states in apractical quantum information processing. Recently, dy-namics of entanglement has received increasing attention[8–10].ESD in finite-dimensional systems is not limited onlyto two-qubit systems. It may be occurs in a compositequantum system with a larger dimension and a multi-qubit system as well [11–17]. The dissipative dynamicsfor a specific one-parameter class of states in a qubit-qutrit (2 ⊗
3) system interacting with dephasing and de-polarizing channels was studied by Ann et al. [13] andKhan [14], respectively. Ann et al. [13] conjectured thatESD exists in all bipartite quantum systems. Khan [14]showed that no ESD happens in any density matrix ofa qubit-qutrit system when only the qubit is coupled toits local depolarizing channel but the re-birth of entan- ∗ Published in Commun. Theor. Phys. , 983-990 (2012) † Corresponding author. Email address: [email protected] glement occurs in particular initial states. However, forgeneral qubit-qutrit states and other common noise chan-nels, the dissipative dynamics of a hybrid qubit-qutrit isnot presented.In this paper, we devote to investigate the behaviorof entanglement for a two-parameter class of states in aqubit-qutrit system under the influence of both two inde-pendent (multi-local) and only one (local) various noisechannels, such as dephasing, phase-flip, bit-(trit-) flip,bit-(trit-) phase-flip, and depolarizing channels. Usingnegativity for quantifying entanglement, some analyticalor numerical results are presented. We find that ESD is ageneral phenomenon in a qubit-qutrit system undergoingall these noise channels, not only the case with dephas-ing and depolarizing channels observed by others [13].It is interesting to show that ESD always takes place inany density matrix when each subsystem couples to itsdepolarizing channel or only the qutrit couples to its trit-flip or trit-phase-flip channels. ESD can only be avoidedin some initial states undergoing particular noise chan-nels. For example, no ESD occurs when the system underthe influence of multi-local (local) dephasing, multi-local(local) phase-flip, local bit-flip, and local bit-phase-flipchannels if it is initially in the state shown in Eq.(1) withthe parameter b = 0. Our results show that the noisechannels affect the entanglement and the coherence of ahybrid qubit-qutrit system in very different ways. For lo-cal or multi-local dephasing, phase-flip, and depolarizingnoise channels, a time scale of disentanglement is usu-ally shorter than the decay of the off-diagonal dynamics,and coherence disappears in an infinite-time limit. Formulti-local and local bit-flip and bit-phase-flip channels,disentanglement occurs in an infinite time, but coherencedoes not disappear even though t .This paper is organized as follows. In Sec.II, we mo-tivate the choice of a two-parameter class of states ina qubit-qutrit system, and the physical model are in-troduced. In Sec.III, entanglement dynamics of a two-parameter class of states in a qubit-qutrit system underthe influence of local and multi-local dephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and depolarizingnoise channels are discussed, respectively. Discussion andsummary are shown in Sec.IV. II. INITIAL STATES AND NOISE MODEL
A two-parameter class of states with real parametersin a hybrid qubit-qutrit (2 ⊗
3) quantum system [18] canbe described as ρ bc (0) = a ( | ih | + | ih | ) + b ( | φ + ih φ + | + | φ − ih φ − | + | ψ + ih ψ + | ) + c | ψ − ih ψ − | , (1)where | φ ± i = 1 √ | i ± | i ) , | ψ ± i = 1 √ | i ± | i ) , (2)and a , b , and c are three real parameters, and they sat-isfy the relation 2 a + 3 b + c = 1. | i and | i are thetwo eigenstates of a two-level quantum system (qubit) orthe eigenstates of a three-level quantum system (qutrit)with the other eigenstate | i . The two-parameter classof states ρ bc (0) can be obtained from an arbitrary stateof a 2 ⊗ ρ AB in a 2 ⊗ ⊗ [19, 20] N ( ρ AB ) = k ρ T B k − , (3)which corresponds to the absolute value of the sum ofnegative eigenvalues of ρ T B (the partial transpose ρ T B associated with an arbitrary product orthonormal ba-sis f i ⊗ f j is defined by the matrix elements: ρ T B mµ,nν ≡h f m ⊗ f µ | ρ T B | f n ⊗ f ν i = ρ mν,nµ ), i.e., N ( ρ AB ) = 2 max { , − λ S } , (4)where λ S represents the sum of all negative eigenvaluesof ρ T B . N ( ρ AB ) = 0 for an unentangled state. Therefore,from Eq.(1) one can obtain the range of parameters as3 b < c ≤ − b , i.e., b ∈ [0 , /
6) for the initial entangledstates.In our physical model of noise for a qubit-qutrit system(composed of a two-level subsystem A and a three-levelsubsystem B ), the two subsystems interact with their en-vironments independently. The evolved states of the ini-tial density matrix of such a system when it is influencedby multi-local environments can be given compactly by ρ ABbc ( t ) = X i =1 3 X j =1 F Bj E Ai ρ ABbc (0) E A † i F B † j . (5) Here, the operators E Ai and F Bj are the Kraus operatorswhich are used to describe the noise channels acting onthe qubit A and the qutrit B , respectively. They satisfythe completeness relations E A † i E Ai = I and F B † j F Bj = I for all t . III. DYNAMICS OF ENTANGLEMENT UNDERDECOHERENCE
It is important to consider the possible degradationof any initially prepared entanglement due to decoher-ence. In this section we investigate what happens tothe entanglement in a qubit-qutrit system under com-mon noise channels for qubit (qutrit): dephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and depolarizingchannels. The two specific environment noise situationswill be considered: (i) local and (ii) multi-local. In thecase (i), only one part of a qubit-qutrit system ( S ) in-teracts with its environment. In the case (ii), both thetwo parts of S interact with their local environments,independently. A. Dephasing channels
The set of Kraus operators for a single qubit A and asingle qutrit B that reproduce the effect of a dephasingchannel are given by E A = (cid:18) √ − γ A (cid:19) ⊗ I ,E A = (cid:18) √ γ A (cid:19) ⊗ I , (6) F B = I ⊗ √ − γ B
00 0 √ − γ B ,F B = I ⊗ √ γ B
00 0 0 ,F B = I ⊗ √ γ B . (7)The time-dependent parameters are defined as γ A = 1 − e − t Γ A and γ B = 1 − e − t Γ B . Here γ A , γ B ∈ [0 , A (Γ B ) denotes the decay rate of the subsystem A ( B ).According to Eq.(5), the time-dependent evolved den-sity operator ρ AB ( t ) of a hybrid qubit-qutrit system,which is initially in the entangled state ρ AB (0), is givenby ρ AB ( t ) = b b + c ( b − c ) √ (1 − γ A )(1 − γ B )2 − c − b ( b − c ) √ (1 − γ A )(1 − γ B )2 b + c b
00 0 0 0 0 − c − b . (8)In order to characterize the dynamics of evolution forthe density matrix and consider the entanglement of thissystem quantified by negativity, we should calculate theeigenvalues of the partial transpose of the time-evolveddensity matrix ρ AB ( t ) and determine the potential neg-ative eigenvalues. (1) Multi-local dephasing channel . The eigenvaluewhich can potentially be negative is λ = b − c − b p (1 − γ A )(1 − γ B ) . (9)The entanglement of the qubit-qutrit system under amulti-local dephasing channel is N mul − loc ( ρ AB ) = 2 max { , c − b p (1 − γ A )(1 − γ B ) − b } . (10)It is easy to obtain that all the states which are initiallyentangled (3 b < c ≤ − b ) become separable when (1 − γ A )(1 − γ B ) ≤ (cid:16) bc − b (cid:17) and b = 0. (2) Qubit dephasing channel only . If the qutrit field isturned off (i.e., γ B = 0), that is, only a dephasing noiseacts on qubit A alone, the entanglement of the qubit-qutrit system is N bit ( ρ AB ) = 2 max { , c − b p − γ A − b } . (11)All the states which are initially entangled (3 b < c ≤ − b ) become separable as soon as γ A ≥ − (cid:16) bc − b (cid:17) and b = 0. (3) Qutrit dephasing channel only . By the same ar-gument as that made in the qubit dephasing noise, for adephasing noise acting on qutrit B alone, the entangle-ment is N trit ( ρ AB ) = 2 max { , c − b p − γ B − b } . (12)A single-qutrit dephasing channel will induce ESD in thequbit-qutrit system when γ B ≥ − (cid:16) bc − b (cid:17) and b = 0. Together with these pieces of results, one can see thatESD takes place under local and multi-local dephasingchannels in a general qubit-qutrit system if and only ifthe system is initially in the state shown in Eq.(1) withthe parameter b = 0. Moreover, the smaller c/b , thelonger the death time range. However, if b = 0, no ESDoccurs. B. Phase-flip channels
The Kraus operators describing the phase-flip channelfor a single qubit A are given by E A = r − γ A (cid:18) (cid:19) ⊗ I ,E A = r γ A (cid:18) − (cid:19) ⊗ I , (13)and those for a single qutrit B can be written as F B = I ⊗ r − γ B ,F B = I ⊗ r γ B e − i π/
00 0 e i π/ ,F B = I ⊗ r γ B e i π/
00 0 e − i π/ , (14)where γ A = 1 − e − t Γ A , γ B = 1 − e − t Γ B , and γ A , γ B ∈ [0 , A (Γ B ) represents the decay rate of the subsystem A ( B ).We can obtain the time-evolved density-matrix dynam-ics ρ AB ( t ) of the qubit-qutrit system under a phase-flipchannel, according to Eq.(5). That is, ρ AB ( t ) = b b + c ( b − c )(1 − γ A )(1 − γ B )2 − c − b ( b − c )(1 − γ A )(1 − γ B )2 b + c b
00 0 0 0 0 − c − b . (15)With the same argument as that made in dephasing chan-nels, some results can be presented as follows. (1) Multi-local phase-flip channel . The entanglement of the qubit-qutrit system under a multi-local phase-flipchannel is calculated as N mul − loc ( ρ AB ) = 2 max (cid:26) , ( c − b ) + ( c − b )((1 − γ A )(1 − γ B ) − (cid:27) . That is, all the states which are initially entangled (3 b The Kraus operators describing the bit-flip channel fora single qubit A are given by E A = r − γ A (cid:18) (cid:19) ⊗ I ,E A = r γ A (cid:18) (cid:19) ⊗ I , (18)and those for a single qutrit B can be written as F B = I ⊗ r − γ B ,F B = I ⊗ r γ B ,F B = I ⊗ r γ B , (19)where γ A = 1 − e − t Γ A , γ B = 1 − e − t Γ B , and γ A , γ B ∈ [0 , ρ after the interaction time t under a multi-local bit-(trit-) flip channel are given by ρ ( t ) = ρ ( t ) = 112 (12 b + 3( b − c ) γ A ( γ B − − b ) γ B ) ,ρ ( t ) = ρ ( t ) = 112 (6( b + c ) − b − c ) γ A ( γ B − − b − c ) γ B ) ,ρ ( t ) = ρ ( t ) = 16 (3(1 − b − c ) + (9 b + 3 c − γ B ) ,ρ ( t ) = ρ ( t ) = 112 ( b − c ) γ A (3 − γ B ) ,ρ ( t ) = ρ ( t ) = ρ ( t ) = ρ ( t ) = 112 ( b − c )(2 − γ A ) γ B ,ρ ( t ) = ρ ( t ) = ρ ( t ) = ρ ( t ) = 112 ( b − c ) γ A γ B ,ρ ( t ) = ρ ( t ) = 112 ( b − c )(2 − γ A )(3 − γ B ) , (20)and all the remaining matrix elements are zero. Here ρ = ( ρ ij ) and i, j = 1 , . . . , (1) Qubit bit-flip channel only . We obtain the sameresults as the case with a phase-flip channel. (2) Qutrit trit-flip channel only . The negativity for allthe states which are initially entangled (3 b < c ≤ − b )can be given by N trit ( ρ AB ) = 2 max (cid:26) , b − c − (1 − b + 2 c ) γ B (cid:27) . (21)The states become separable once γ B ≥ c − b − b +2 c . (3) Multi-local bit-(trit-) flip channel . For simplic-ity, we choose the local asympotic bit-(trit-) flip ratesΓ A = Γ B = Γ to discuss the effect of this noise on en-tanglement. Unfortunately, it is difficult to calculate theanalytical eigenvalues of the partial transpose of the time-evolved density matrix. Therefore, we take the numericalcalculation for various initial states as examples to inves-tigate the dynamics of entanglement of the qubit-qutritsystem under a multi-local bit-(trit-) flip channel.As the first example, we consider the initial entangledstates shown in Eq.(1) with the parameter a = 0 whichare equivalent to Werner states [21] in a 2 ⊗ a = b = 0 is just the case for the maximally entangledBell state | ψ − i [22], and the system does not suffer fromESD under a multi-local bit-(trit-) flip channel.As a second example, we consider the initial entangledstates with the parameter b = 0. Their dynamics ofentanglement is shown in Fig.2.Finally, we consider the initial states with a, b, c > b = 0 . Γ N FIG. 1: Dynamics of entanglement for the system undergoingthe multi-local bit- (trit-) flip noise with a = 0. N representsthe negativity of a hybrid qubit-qutrit system. γ is the time-dependent parameter and γ A = γ B = γ = 1 − e − t Γ . Thesolid, dashed, short-dashed, dashed-dotted, and dotted linescorrespond to b = 0, b = 1 / b = 2 / b = 3 / 30, and b = 4 / 30, respectively. b = 0. However, ESD always occurs when the systemundergoes a qutrit-flip channel alone. Γ N FIG. 2: Dynamics of entanglement for the system undergoingthe multi-local bit-(trit-) flip noise with parameter b = 0. Thesolid, dashed, dashed-dotted, and dotted lines correspond to c = 1, c = 3 / c = 1 / 2, and c = 1 / 4, respectively. Γ N FIG. 3: Dynamics of entanglement for the system undergoingthe multi-local bit-(trit-) flip noise with b = 1 / 20. The solid,dashed, dashed-dotted, and dotted lines correspond to c =16 / c = 12 / c = 8 / 20, and c = 4 / 20, respectively. D. Bit-(Trit-) phase-flip channels The Kraus operators describing the bit-phase flip chan-nel for a single qubit A are given by E A = r − γ A (cid:18) (cid:19) ⊗ I ,E A = r γ A (cid:18) − ii (cid:19) ⊗ I , (22)and those for a single qutrit B can be written as F B = I ⊗ r − γ B ,F B = I ⊗ r γ B e i π/ e − i π/ ,F B = I ⊗ r γ B e − i π/ e i π/ ,F B = I ⊗ r γ B e − i π/ 00 0 e i π/ ,F B = I ⊗ r γ B e i π/ 00 0 e − i π/ , (23)where γ A = 1 − e − t Γ A , γ B = 1 − e − t Γ B , and γ A , γ B ∈ [0 , ρ after the interac-tion time t under multi-local bit-(trit-) phase-flip chan-nels are given by ρ ( t ) = ρ ( t ) = b + 14 ( b − c ) γ A ( γ B − 1) + ( 16 − b ) γ B ,ρ ( t ) = ρ ( t ) = 112 (6( b + c ) − b − c ) γ A ( γ B − 1) + (2 − b − c ) γ B ) ,ρ ( t ) = ρ ( t ) = 16 (3(1 − b − c ) + (9 b + 3 c − γ B ) ,ρ ( t ) = ρ ( t ) = − 112 ( b − c ) γ A (3 − γ B ) ,ρ ( t ) = ρ ( t ) = ρ ( t ) = ρ ( t ) = 124 ( b − c )( γ A − γ B ,ρ ( t ) = ρ ( t ) = 112 ( b − c )(2 − γ A )(3 − γ B ) ,ρ ( t ) = ρ ( t ) = ρ ( t ) = ρ ( t ) = 112 ( b − c ) γ A γ B , (24)and all the remaining matrix elements are zero. (1) Local bit-(trit-) phase-flip channel only . We obtainthe same results as the case with a bit-(trit-) flip channel. Γ N FIG. 4: Dynamics of entanglement for the system undergoingthe multi-local bit-(trit-) phase-flip noise with the parameter a = 0. The solid, dashed, short-dashed, dashed-dotted, anddotted lines correspond to b = 0, b = 1 / b = 2 / b =3 / 30, and b = 4 / 30, respectively. Γ N FIG. 5: Dynamics of entanglement for the system undergoingthe multi-local bit-(trit-) phase-flip noise with the parameter b = 0. The solid, dashed, dashed-dotted, and dotted linescorrespond to c = 1, c = 3 / c = 1 / 2, and c = 1 / 4, respec-tively. (2) Multi-local bit-(trit-) phase-flip channel . For sim-plicity, the time dependent parameters are also definedas γ A = γ B = γ . It is difficult to obtain the analyticalresults. The similar work is made as that in the case witha bit-(trit-) flip channel, and the dynamics of entangle-ment of the initial states with the parameter a = 0 isdisplayed in Fig.4. Different from a multi-local bit-(trit-)flip channel, one can see that there exists ESD for themaximally entangled Bell state (solid line). The dynam-ics of entanglement of the initial states shown in Eq.(1)with the parameters b = 0 and a, b, c > Γ N FIG. 6: Dynamics of entanglement for the system undergoingthe multi-local bit-(tri-t) phase-flip noise with the parameter b = 1 / 20. The solid, dashed, dashed-dotted, and dotted linescorrespond to c = 16 / c = 12 / c = 8 / 20, and c = 4 / E. Depolarizing channels A depolarizing channel represents the process in whichthe density matrix is dynamically replaced by the maxi-mally mixed state I/d . Here I is the identity matrix of asingle qudit. The set of Kraus operators that reproducesthe effect of the depolarizing channel for a single qubit A are given by E A = r − γ A I ,E A = r γ A σ ⊗ I ,E A = r γ A σ ⊗ I ,E A = r γ A σ ⊗ I , (25)where σ i ( i = 1 , , 3) are the three Pauli matrices. TheKraus operators describing a single-qutrit depolarizingnoise are given by [ ? ] F B = I ⊗ r − γ B I ,F B = I ⊗ √ γ B Y,F B = I ⊗ √ γ B Z,F B = I ⊗ √ γ B Y ,F B = I ⊗ √ γ B Y Z,F B = I ⊗ √ γ B Y Z, F B = I ⊗ √ γ B Y Z ,F B = I ⊗ √ γ B Y Z ,F B = I ⊗ √ γ B Z , (26)where Y = , Z = e i π/ 00 0 e − i π/ , (27)and γ A = 1 − e − t Γ A , γ B = 1 − e − t Γ B , γ A , γ B ∈ [0 , ρ after the interaction time t under a multi-local depolarizing channel are given by ρ ( t ) = ρ ( t ) = 112 (12 b + 3( b − c )( γ B − γ A + 2(1 − b ) γ B ) ,ρ ( t ) = ρ ( t ) = 112 (6( b + c ) − b − c )( γ B − γ A + (2 − b − c ) γ B ) ,ρ ( t ) = ρ ( t ) = 16 (3(1 − b − c ) + (9 b + 3 c − γ B ) ,ρ ( t ) = ρ ( t ) = 12 ( b − c )(1 − γ A )( − γ B ) , (28)and all the remaining matrix elements are zero. (1) Multi-local depolarizing channel . The negativityfor the composite system with initial entangled states(3 b < c ≤ − b ) is given by N mul − loc ( ρ AB ) = 2 max { , − λ } , (29)where λ = 9( b − c ) γ A ( γ B − 1) + 2 γ B (1 − b + 3 c ) + 18 b − c . (30)It is easy to obtain the result that all the states, whichare initially entangled ones, become separable if and onlyif 9( b − c ) γ A ( γ B − 1) + 2 γ B (1 − b + 3 c ) ≥ c − b ). (2) Qubit depolarizing channel only . The negativityfor all the states which are initially entangled (3 b < c ≤ − b ) is given by N bit ( ρ AB ) = 2 max (cid:26) , c − b − γ A ( c − b )4 (cid:27) . (31)The states become separable for all values of γ A ≥ c − b c − b ) . (3) Qutrit depolarizing channel only . The negativityfor all the states which are initially entangled (3 b < c ≤ − b ) can be written as N trit ( ρ AB ) = 2 max { , c − b − (1 − b + 3 c ) γ B } . (32)The states become separable for all values of γ B ≥ c − b − b +3 c . TABLE I: ESD in 2 ⊗ b = 0 ESD with b = 0 ESD with b = 0phase-flip ESD with b = 0 ESD with b = 0 ESD with b = 0bit-(trit-) flip exist ESD ESD with b = 0 ESDbit-(trit-) phase-flip exist ESD ESD with b = 0 ESDdepolarizing exist ESD ESD ESD IV. DISCUSSION AND SUMMARY Putting all the pieces of our results together, one cansee that ESD is a general phenomenon in a qubit-qutrit system undergoing various independent noise channelsand we show the outcomes explicitly in Table.I. “ESDwith b = 0” denotes ESD takes places in a qubit-qutritsystem if and only if b = 0, that is, if b=0, no ESDoccurs.“exist ESD” denotes ESD may occurs in a qubit-qutrit system, but not necessary. “ESD” denotes ESDalways occurs.Decoherence which is characterized by the decay ofthe off-diagonal elements of the density matrix describingthe system [24], results from the unwanted interactionsof a quantum system with its environment. Accordingto Eqs.(8,15,20,24,28), one can see that the coherenceof a qubit-qutrit system can be destroyed completelywhen the system undergos the multi-local or local de-phasing, phase-flip and depolarizing channels, while thedisappearance of coherence does not occur even though γ A , γ B et al. [13] in 2008. We havegeneralized their study to a more general case, that is, atwo-parameter class of states under the influence of localand multi-local dephasing, phase-flip, bit-(trit-) flip, bit- (trit-) phase-flip, and depolarizing channels. With ana-lytical and numerical analysis, we have shown that ESDis a general phenomenon in a qubit-qutrit system undervarious noise channels, not only the case with dephas-ing and depolarizing ones [13]. It can only be avoided insome initial states undergoing particular noise channels.In summary, our results show that the environment,which causes dephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and depolarizing, affects the entangle-ment and the coherence of a hybrid qubit-qutrit system ina two-parameter class of entangled states in very differentways. ESD is a general phenomenon in a qubit-qutrit sys-tem undergoing various independent noise channels andit can only be avoided in some initial states undergo-ing particular noise channels. Moreover, we can dividethose noise channels into two groups. For multi-local andlocal dephasing, phase-flip, and depolarizing noise chan-nels, a time scale of disentanglement is usually shorterthan the decay of the off-diagonal dynamics, and coher-ence disappears in an infinite-time limit t in which γ A , γ B 1. For multi-local or local bit-(trit-) flip andbit-(trit-) phase-flip channels, disentanglement may oc-cur in the infinite-time limit, but the disappearance ofcoherence does not occur even though γ A , γ B ACKNOWLEDGEMENTS This work is supported by the National Natural Sci-ence Foundation of China under Grant Nos. 10974020and 11174039, NCET-11-0031, and the Fundamental Re-search Funds for the Central Universities. [1] M. A. Nielsen and I. L. Chuang, Quantum Computa-tion and Quantum Information, Cambridge Univ. Press,Cambridge, UK, 2000.[2] T. Yu and J. H. Eberly, Science (2007) 555.[3] T. Yu and J. H. Eberly, Science (2009) 598.[4] M. O. Terra Cunha, New J. Phys. (2007) 237.[5] M. P. Almeida et al., Science (2007) 579.[6] A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P.Walborn, P. H. Souto Ribeiro, and L. Davidovich, Phys.Rev. A (2008) 022322.[7] J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J.Kimble, Phys. Rev. Lett. (2007) 180504.[8] Y. Li, J. Zhou, and H. Guo, Phys. Rev. A (2009)012309.[9] B. K. Zhao, F. G. Deng, F.S. Zhang, and H.Y. Zhou,Phys. Rev. A (2009) 052106.[10] B. K. Zhao and F. G. Deng, Phys. Rev. A (2010)014301.[11] K. Ann and G. Jaeger, Phys. Rev. A (2007) 044101.[12] G. Jaeger and K. Ann, J. Mod. Opt. (2007) 2327. [13] K. Ann and G. Jaeger, Phys. Lett. A (2008) 579.[14] S. Khan, arXiv: quant-ph/1012.1028.[15] Z. X. Man, Y. J. Xia, and N. B. An, Phys. Rev. A (2008) 064301.[16] Z. X. Man, Y. J. Xia, and N. B. An, New J. Phys. (2010) 033020.[17] Z. He, J. Zou, B. Shao, and S. Y. Kong, J. Phys. B: At.Mol. Opt. Phys. (2010) 115503.[18] D. P. Chi and S. Lee, J. Phys. A (2003) 11503.[19] G. Vidal and R. F. Werner, Phys. Rev. A (2002)032314.[20] S. Lee, D. P. Chi, S. D. Oh, and J. Kim, Phys. Rev. A (2003) 062304.[21] R. F. Werner, Phys. Rev. A (1989) 4277.[22] B. Groisman and S. Popescu, A. Winter, Phys. Rev. A (2005) 032317.[23] S. Salimi and M. M. Soltanzadeh, Int. J. Quant. Inf. (2009) 615.[24] R. Omn` e s, Phys. Rev. A56