aa r X i v : . [ qu a n t - ph ] M a y Dynamics of global entanglement under decoherence
Afshin Montakhab ∗ and Ali Asadian Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran.
We investigate the dynamics of global entanglement, the Meyer-Wallach measure, underdecoherence, analytically. We study two important class of multi-partite entangled states, theGreenberger-Horne-Zeilinger and the W state. We obtain exact results for various models ofsystem-environment interactions (decoherence). Our results shows distinctly different scalingbehavior for these initially entangled states indicating a relative robustness of the W state,consistent with previous studies.PACS number(s): 03.67.Mn, 03.65.Ud, 03.65.Yz
I. INTRODUCTION
It is well-known that the notion of quantum entangle-ment is a key concept in quantum mechanics. It is alsoresponsible for “strange” non-local behavior of quantumsystems in marked contrast to classical notions of realityand locality [1,2]. Schr¨odinger refer to it as the “essenceof quantum mechanics” [1]. Besides its fundamental as-pects, entanglement constitutes the central part of newmodes of information technology, quantum computationand quantum communication [3,4,5] and is therefore akey ingredient of many information processing protocols.Recently, a considerable amount of work has been de-voted to characterize, quantify and realize different vari-ety of entangled states [6]. By now, bi-partite entangle-ment is a relatively well-understood phenomena. How-ever, the situation becomes much more complex whenmulti-partite systems are considered [7].On the other hand, entangled states are very fragilewhen they are exposed to environment. The biggest en-emy of entanglement is decoherence which is believed tobe the responsible mechanism for emergence of the clas-sical behavior in quantum systems [8]. Since the mainte-nance and control of entangled states is essential to real-ization of quantum information processing systems, thestudy of deteriorating effect of decoherence in entangledstates would be of considerable importance from theoret-ical as well as experimental point of view [9,10,11].To demonstrate the effects of decoherence on entangledstates, an appropriate entanglement measure which couldbe capable of monitoring the dynamics of entanglementin decoherence processes is needed . However, there areno exact measure of entanglement under general condi-tions for mixed states. Even for bipartite mixed states,apart from the particular case of two-level systems [12],the exact solution is missing. There is an approximategeneralization of concurrence for mixed states which wasproposed by Mintert et al [13] and has bean used to mea-sure entanglement in multi-qubit systems.Global entanglement(GE), defined by Meyer-Wallach ∗ E-mail address: [email protected] (MW) entanglement measure of pure-state [14] which isa monotone[15], is a very useful measure of entanglement.As we will show briefly, GE is a measure of total non-localinformation per particle in a general multi-qubit system.Therefore, GE gives an intuitive meaning to multi-qubitentanglement as well being an experimentally accessiblemeasure [16]. In this paper, we use this measure to mon-itor the entanglement dynamics of two seminal multi-qubit entangled states, the Greenberger-Horne-Zeilinger(GHZ) state and the W state, under different models ofsystem-environment interaction.To demonstrate the GE dynamics of multi-qubit entan-gled state under decoherence, we need to know the gen-eralization of GE to mixed state. Unfortunately, thereis no generalization of the primary definition of the MWmeasure to mixed states, analytically. However, we canmonitor the GE dynamics for two important classes ofmulti-qubit entangled state, the GHZ and W state, byexploiting the relationship between GE and tangles. Toelucidate this point, we adopt informational approachwhich can give an intuitive meaning to GE.
II. GLOBAL ENTANGLEMENT,INFORMATION, AND TANGLES
Finite amount of information can be attributed to N-qubit pure state which is N bit of information accordingto Brukner-Zeilinger operationally invariant informationmeasure [17]. This information can be distributed in lo-cal as well as non-local form, which is associated withentanglement [18].This information has a complimentaryrelation: I total = I local + I non − local . (1)The total information is conserved unless transferredto environment through decoherence. The amount ofinformation in local form is I local = P Ni =1 I i where, I i = 2 T rρ i − I non − local = P Ni =1 − T rρ i ) which can be distributedin different forms of quantum correlations, the tangles,among the system, I non − local = 2 X i
In order to evolve our chosen states under influence ofdecoherence we use the Lindblad form of master equation[22], dρdt = N X i =1 L i ρ. (6)The Lindblad operators, L i , describe the local interac-tion of each qubit with environment independent of otherqubit interaction with the environment. We assume L i is the same form for all qubit, L i = L . For markovianprocess [22] L i ρ = X k γ k J k ρJ † k − { J k J † k , ρ } ] , (7)where operator J k describes the system-environmentmodel of interaction with strength γ k . In this paper weinvestigate dissipative, dephasing, and noise processes,each with a well-defined J k . For the two-level systemsthe operators, J k , are expressed in terms of Pauli matri-ces. The solution of Lindblad form of master equationfor two-level systems are studied in [23].For dissipative environment, J = σ − . In this processthe system interacts with a thermal bath at zero temper-ature. This process could be described as spontaneousemission of a two-state atom coupled with the vacuummodes of the ambient electromagnetic field which leadsthe atom state to the ground state. For dephasing pro-cess, J = σ + σ − . This is a phase-destroying processthat does not have a classical counterpart and is there-fore intrinsically quantum mechanical. It corresponds toa situation where no energy is exchanged with environ-ment, that is, the population of energy eigenstates of thesystem do not change with time. Only the phase informa-tion which includes quantum correlations is lost. For thenoisy environment, J = σ − and J = σ + . Noisy dynam-ics are related to another extreme of thermal bath, i.e.when temperature is extremely high while the system-bath coupling is extremely weak. This process random-izes the state of the system which results in a completelymixed state eventually. The noise process has a partic-ular interest since its effect is basis independent. Thatis, the noisy operation is invariant under unitary oper-ation. All these processes could have different effect onthe multi-qubit entangled state. But the common featureof them is that under the action of each of these environ-ments any initial entangled state asymptotically evolvesto a separable state. IV. RESULTS
Our goal is to obtain the time dependence of the den-sity matrix, ρ ( t ) = e − Lt ρ (0), of the system in order todetermine the time evolution of E gl ( t ) for the initiallyprepared multi-qubit entangled states, i.e. W and GHZ.According to the structure of entanglement in W statewe can deduce the time dependency of the GE from two-qubit entanglement, τ . The two-qubit density matrix, ρ ij (0) = N | ψ + ih ψ + | + ( N − N | ih | , is the same for anypair of qubits, ij . Therefore, for the initial W state inthe dissipative process we have, ρ ij ( t ) = N − pN pN pN pN pN
00 0 0 0 , (8)where p = e − γt is the decoherence parameter. Similarly,for dephasing process, one obtains, ρ ij ( t ) = N − N N pN pN N
00 0 0 0 , (9)and, therefore, for the initial W state, GE has the simpleexact solution, E gl ( t ) = 4( N − N e − γt , (10)for both dissipative as well as dephasing processes. Fornoisy process, the density matrix is ρ ij ( t ) = 2 N − p p p p p
00 0 0 − p + (11)( N − N (1+ p ) − p − p
00 0 0 (1 − p ) , which leads to E gl = N − N [ max { p − (1 − p ) ( N − ( pN − p ) ) , } ] . (12) The dynamics of GE in dephasing, dissipation and noisyenvironment for the initial W state is illustrated in Figs.1and 2. Eq.(10)(Fig.1) shows that the decay rate( α , for E gl ∝ exp( − αt ) for GE is independent of N for the Wstate in dissipative and dephasing environment as foundpreviously using numerical solution for a different mea-sure of entanglement [10]. Note, however, that the rateof change of GE decreases with increasing N. For noisyenvironment, Fig.2, we observe a decay to separable stateafter a finite time t sep which increases linearly with N,also consistent with previous studies [10]. γ t E g l N α W state (dephasing & dissipation)
FIG. 1: E gl vs. γt in an initial W state with dephasing (ordissipative) process for N = 2 , , , N . Note that these are simple graphsof Eq.(10) and are only drawn for comparison with other Figs. γ t E g l N α N / T s ep W state (noise)
FIG. 2: E gl vs. γt in an initial W state with noisy process for N = 2 , , , N for up to N = 14 qubits. For the GHZ state, the density matrix in the dephasingprocess is ρ N ( t ) = 12 ( | i ⊗ N h | ⊗ N + p N | i ⊗ N h | ⊗ N + (13) p N | i ⊗ N h | ⊗ N + | i ⊗ N h | ⊗ N ) , which leads simply to E gl = e − Nγt . The GHZ densitymatrix for the dissipative process is ρ N ( t ) = 12 ( p N | i ⊗ N h | ⊗ N + p N | i ⊗ N h | ⊗ N (14)+ X q ,...,q N =0 λ h Z i | q q ...q N ih q q ...q N | ) , where Z = N X i =1 z i ; z i = 1 − σ z i , and h Z i = h q q ...q N | Z | q q ...q N i , and λ h Z i = [ p h Z i (1 − p ) N −h Z i + 0 h Z i ] . The GHZ density matrix for the noisy process is ρ N ( t ) = 12 ( p N | i ⊗ N h | ⊗ N + p N | i ⊗ N h | ⊗ N (15)+ X q ,...,q N =0 λ h Z i | q q ...q N ih q q ...q N | ) ,λ h Z i = 12 N [(1+ p ) h Z i (1 − p ) N −h Z i +(1 − p ) h Z i (1+ p ) N −h Z i ] . Consequently, our results ( E gl ), for the GHZ state indephasing, dissipative and noisy environment are shownrespectively in Figs. 3, 4, and 5 for various N. As shownin the corresponding insets, the decay rates increases lin-early with system size N for all three processes. Also,the rate of change of E gl increases with system size aswell. All these results are consistent with previous stud-ies using different methods than ours[9,10,11]. For thedynamics of GE in GHZ state, although we have the ex-act solution only for even number of qubit, the behaviorof GE under decoherence for the odd number can be in-ferred from the simplicity and symmetry of our results.For example, our results for E gl in dephasing processholds for any number of qubit in the initial GHZ state. V. CONCLUSION
In conclusion, in this work we have given an intuitiveinformational meaning of MW measure of global entan-glement. Based on the relationship between global en-tanglement and tangles, which constitute the non-local γ t E g l N α GHZ state (dephasing)
FIG. 3: E gl vs. γt in an initial GHZ state with dephasingprocess for N = 2 , , , N up to N = 10 qubits. γ t E g l N α GHZ state (dissipation)
FIG. 4: E gl vs. γt in an initial GHZ state with dissipationprocess for N = 2 , , , N up to N = 10 qubits. form of the information, we identify the exact solution ofits dynamics under different system-environment modelsfor the two qualitatively different multi-qubit entangledstates, the GHZ and W states. In all the cases consid-ered, we obtain an exponential decay of entanglement asa function of time. For the W state, the results showthat the lifetime of the GE is independent of the numberof the qubits in dephasing and dissipative processes andthe lifetime linearly decreases with N in a noisy process.While for the GHZ state, the lifetime of GE decreaseslinearly with N. Our results indicate that the quantumcorrelations in W state are more robust to decoherenceeffects than that of the GHZ state.The authors kindly acknowledge the support of Shiraz γ t E g l N α GHZ state (noise)
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