Scaling laws for the decay of multiqubit entanglement
SScaling laws for the decay of multiqubit entanglement
L. Aolita, R. Chaves, D. Cavalcanti, A. Ac´ın,
2, 3 and L. Davidovich Instituto de F´ısica, Universidade Federal do Rio de Janeiro. Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brasil ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats, Lluis Companys 23, 08010 Barcelona, Spain (Dated: October 24, 2018)We investigate the decay of entanglement of generalized N -particle Greenberger-Horne-Zeilinger (GHZ)states interacting with independent reservoirs. Scaling laws for the decay of entanglement and for its finite-timeextinction (sudden death) are derived for different types of reservoirs. The latter is found to increase with thenumber of particles. However, entanglement becomes arbitrarily small, and therefore useless as a resource,much before it completely disappears, around a time which is inversely proportional to the number of particles.We also show that the decay of multi-particle GHZ states can generate bound entangled states. PACS numbers: 03.67.-a, 03.67.Mn, 03.65.Yz
Introduction.
Entanglement has been identified as a keyresource for many potential practical applications, such asquantum computation, quantum teleportation and quantumcryptography [1]. Being it a resource, it is of fundamentalimportance to study the entanglement properties of quantumstates in realistic situations, where the system unavoidablyloses its coherence due to interactions with the environment.In this context a peculiar dynamical feature of entangled stateshas been experimentally confirmed for the case of two qubits(two-level systems) [2]: even when the constituent parts ofan entangled state decay asymptotically in time, entanglementmay disappear at a finite time [3, 4, 5, 6, 7, 8, 9]. The phe-nomenon of finite-time disentanglement, also known as en-tanglement sudden death (ESD) [2, 7, 8, 9], illustrates the factthat the global behavior of an entangled system, under the ef-fect of local environments, may be markedly different fromthe individual and local behavior of its constituents.Since the speed-up gained when using quantum-mechanicalsystems, instead of classical ones, to process information isonly considerable in the limit of large-scale information pro-cessing, it is fundamental to understand the scaling propertiesof disentanglement for multiparticle systems. Important stepsin this direction were given in Refs. [3, 4, 5]. In particular,it was shown in Ref. [3] that balanced Greenberger-Horne-Zeilinger (GHZ) states, | Ψ (cid:105) = ( | (cid:105) ⊗ N + | (cid:105) ⊗ N ) / √ , subjectto the action of individual depolarization [1], undergo ESD,that the last bipartitions to loose entanglement are the mostbalanced ones, and that the time at which such entanglementdisappears grows with the number N of particles in the sys-tem. Soon afterwards it was shown in Ref. [5] that the firstbipartitions to loose entanglement are the least balanced ones(one particle vs. the others), the time at which this happensdecreasing with N . A natural question arises from these con-siderations: is the ESD time a truly physically-relevant quan-tity to assess the robustness of multi-particle entanglement? In this paper we show that, for an important family ofgenuine-multipartite entangled states, the answer is no . Forseveral kinds of decoherence, we derive analytical expressionsfor the time of disappearance of bipartite entanglement, whichis found to increase with N . However, we show that the time at which bipartite entanglement becomes arbitrarily small de-creases with the number of particles, independently of ESD.This implies that for multi-particle systems, the amount of en-tanglement can become too small for any practical applicationlong before it vanishes. In addition, for some specific cases,we characterize not only the sudden-death time of bipartite en-tanglement but we can also attest full separability of the statesin question. As a byproduct we show that in several cases theaction of the environment can naturally lead to bound entan-gled states [10], in the sense that, for a period of time, it is notpossible to extract pure-state entanglement from the systemthrough local operations and classical communication, eventhough the state is still entangled.The exemplary states we take to analyze the robustness ofmultipartite entanglement are generalized GHZ states: | Ψ (cid:105) ≡ α | (cid:105) ⊗ N + β | (cid:105) ⊗ N , (1)with α and β ∈ C such that | α | + | β | = 1 . Therefore,our results also constitute a generalization of those of Refs.[3, 5]. Although (1) represents just a restricted class of states,the study of its entanglement properties is important in its ownright: these can be seen as simple models of the Schr¨odinger-cat state [11], they are crucial for communication problems[12], and such states have been experimentally produced inatomic and photonic systems of up to N = 6 [13]. Decoherence models.
We consider three paradigmatictypes of noisy channels: depolarization, dephasing, and athermal bath at arbitrary temperature (generalized amplitude-damping channel). We consider N qubits of ground state | (cid:105) and excited state | (cid:105) without mutual interaction, each one indi-vidually coupled to its own noisy environment. The dynamicsof the i-th qubit, ≤ i ≤ N , is governed by a master equationthat gives rise to a completely positive trace-preserving map(or channel) E i describing the evolution as ρ i = E i ρ i , where ρ i and ρ i are, respectively, the initial and evolved reducedstates of the i-th subsystem.The generalized amplitude-damping channel (GAD) isgiven, in the Born-Markov approximation, via its Kraus rep-resentation as [1, 9] E GADi ρ i = E ρ i E † + E ρ i E † + E ρ i E † + E ρ i E † ; (2) a r X i v : . [ qu a n t - ph ] M a r with E ≡ (cid:113) n +12 n +1 ( | (cid:105)(cid:104) | + √ − p | (cid:105)(cid:104) | ) , E ≡ (cid:113) n +12 n +1 p | (cid:105)(cid:104) | , E ≡ (cid:113) n n +1 ( √ − p | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) and E ≡ (cid:113) n n +1 p | (cid:105)(cid:104) | being its Kraus operators. Here n isthe mean number of excitations in the bath, p ≡ p ( t ) ≡ − e − γ (2 n +1) t is the probability of the qubit exchanging aquantum with the bath at time t , and γ is the zero-temperaturedissipation rate. Channel (2) is a generalization to finite tem-perature of the purely dissipative amplitude damping chan-nel (AD), which is obtainen from (2) in the zero- temperaturelimit n = 0 . On the other hand, the purely diffusive case isobtained from (2) in the composite limit n → ∞ , γ → , and nγ = Γ , where Γ is the diffusion constant.The depolarizing channel (D) describes the situation inwhich the i-th qubit remains untouched with probability − p ,or is depolarized - meaning that its state is taken to the maxi-mally mixed state (white noise) - with probability p . It can beexpressed as E Di ρ i = (1 − p ) ρ i + ( p ) / , (3)where is the identity operator.Finally, the phase damping (or dephasing) channel (PD)represents the situation in which there is loss of quantuminformation with probability p , but without any energy ex-change. It is defined as E P Di ρ i = (1 − p ) ρ i + p (cid:0) | (cid:105)(cid:104) | ρ i | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ρ i | (cid:105)(cid:104) | (cid:1) . (4)The parameter p in channels (2), (3) and (4) is a convenientparametrization of time: p = 0 refers to the initial time 0 and p = 1 refers to the asymptotic t → ∞ limit.The density matrix corresponding to state (1), ρ ≡ | Ψ (cid:105)(cid:104) Ψ | ≡ | α | ( | (cid:105)(cid:104) | ) ⊗ N + | β | ( | (cid:105)(cid:104) | ) ⊗ N + αβ ∗ ( | (cid:105)(cid:104) | ) ⊗ N + α ∗ β ( | (cid:105)(cid:104) | ) ⊗ N , evolves in time into amixed state ρ given simply by the composition of all N individual maps: ρ ≡ E E ... E N ρ , where, in what follows, E i will either be given by Eqs. (2), (3) or (4). Entanglement sudden death . In order to pick up the en-tanglement features of the studied states we will use the neg-ativity as a quantifier of entanglement [14], defined as theabsolute value of the sum of the negative eigenvalues of thepartially transposed density matrix. In general, the negativ-ity fails to quantify entanglement of some entangled states(those ones with positive partial transposition) in dimensionshigher than six [15]. However, for the states considered here,their partial transposes have at most one negative eigenvalue,and the task of calculating the negativity reduces to a four-dimensional problem. So, in the considered cases, the nega-tivity brings all the relevant information about the separabilityin bipartitions of the states, i.e., null negativity means separa-bility in the corresponding partition.Application of channel (2) to every qubit multiplies the off-diagonal elements of ρ by the factor (1 − p ) N/ , whereasapplication of channels (3) or (4), by the factor (1 − p ) N . Thediagonal terms ( | (cid:105)(cid:104) | ) ⊗ N and ( | (cid:105)(cid:104) | ) ⊗ N in turn give rise to new diagonal terms of the form ( | (cid:105)(cid:104) | ) ⊗ N − k ⊗ ( | (cid:105)(cid:104) | ) ⊗ k ,for ≤ k < N , and all permutations thereof, with coefficients λ k given below. In what follows we present the main resultsconcerning the entanglement behavior of these states. Generalized amplitude-damping channel:
Consider a bi-partition k : N − k of the quantum state. For channel (2),the coefficients λ GADk are given by λ GADk ≡ | α | x N − k y k + | β | w N − k z k , with ≤ x ≡ − pn n +1 + 1 , y ≡ pn n +1 , w ≡ p ( n +1)2 n +1 and z ≡ − p ( n +1)2 n +1 + 1 ≤ . From them, the mini-mal eigenvalue of the states’ partial transposition, Λ GADk ( p ) ,is immediately obtained for the generalized amplitude damp-ing channel [16]: Λ GADk ( p ) ≡ δ k − (cid:113) δ k − ∆ k . (5)Here δ k = 1 / λ GADk ( p ) + λ GADN − k ( p )] and ∆ k = λ GADk ( p ) λ GADN − k ( p ) − | αβ | (1 − p ) N . From (5) one can seethat | Λ GAD ( p ) | ≤ | Λ GAD ( p ) | ≤ ... ≤ | Λ GAD N ( p ) | , for N even, and | Λ GAD ( p ) | ≤ | Λ GAD ( p ) | ≤ ... ≤ | Λ GAD N − ( p ) | , for N odd.The condition for disappearance of bipartite entanglement, Λ GADk ( p ) = 0 , is a polynomial equation of degree N . Inthe purely dissipative case n = 0 , a simple analytical solutionyields the corresponding critical probability for the amplitude-decay channel, p ADc (with β (cid:54) = 0 ): p ADc ( k ) = min { , | α/β | /N } . (6)For | α | < | β | probability (6) is always smaller than 1, mean-ing that bipartite entanglement disappears before the steadystate is asymptotically reached. Thus, contition (6) is the di-rect generalization to the multiqubit case of the ESD conditionof Refs. [2, 7] for two qubits subject to amplitude damping. Aremarkable feature about contition (6) is that it displays no de-pendence on the number of qubits k of the sub-partition. Thatis, the negativities corresponding to bipartitions composed ofdifferent numbers of qubits all vanish at the same time, eventhough they follow different evolutions. In the appendix weprove that at this point the state is not only separable accord-ing to all of its bipartitions but it is indeed fully separable, i.e.,it can be written as a convex combination of product states.For arbitrary temperature, it is enough to consider the case k = N/ , as the entanglement corresponding to the most bal-anced bipartitions is the last one to disappear (we take N evenfrom now on just for simplicity). For arbitrary temperature,the condition Λ GADN/ ( p ) = 0 reduces to a polynomial equationof degree N , which for the purely diffusive case yields: p Diffc ( N/
2) = 1 + 2 | αβ | /N − (cid:113) | αβ | /N . (7) Depolarizing channel:
For channel (3), the coefficients λ Dk of ρ are given by λ Dk ≡ | α | (1 − p ) N − k ( p ) k + | β | (1 − p ) k ( p ) N − k . One obtains again Λ Dk ( p ) ≡ δ k − (cid:112) δ k − ∆ k ,with δ k = 1 / λ Dk + λ DN − k ] and ∆ k = λ Dk λ DN − k − | αβ | (1 − p ) N . Also here it is easy to show that the negativity asso-ciated to the most balanced bipartition is always higher thanthe others, while the one corresponding to the least balancedpartition is the smallest one. The critical probability for thedisappearance of entanglement in the N/ N/ partition isgiven by: p Dc ( N/
2) = 1 − (1 + 4 | αβ | /N ) − / . (8) Phase damping channel:
Finally, for the phase dampingchannel, whereas the off-diagonal terms of the density matrixevolve as mentioned before, all the diagonal ones remain thesame, with λ P Dk ≡ ≡ λ P DN − k for ≤ k < N . In this case, Λ P Dk ( p ) ≡ −| αβ | (1 − p ) N . This expression is independentof k , and therefore of the bipartition, and for any α, β (cid:54) = 0 it vanishes only for p = 1 , i. e., only in the asymptotictime limit, when the state is completely separable: general-ized GHZ states of the form (1), subject to individual dephas-ing, never experience ESD. The environment as a creator of bound entanglement.
Some effort has been recently done in order to understandwhether bound entangled (i.e. undistillable) states naturallyarise from natural physical processes [17]. In this context, ithas been found that different many-body models present ther-mal bound entangled states [17]. Here we show, in a concep-tually different approach, that bound entanglement can alsoappear in dynamical processes, namely decoherence.For all channels here considered, the property | Λ ( p ) | ≤| Λ ( p ) | ≤ ... ≤ | Λ N ( p ) | holds. Therefore, when | Λ ( p ) | =0 , there may still be entanglement in the global state for sometime afterwards, as detected by other partitions. When thishappens, the state, even though entangled, is separable accord-ing to every N − partition, and then no entanglement canbe distilled by (single-particle) local operations.An example of this is shown in Fig. 1, where the negativityfor partitions N − and N/ N/ is plotted versus p ,for N = 4 and α = 1 / √ β , for channel D. After the negativity vanishes, the negativity remains positive until p = p Dc (2) given by Eq. (8). Between these two values of p , the state is bound entangled since it is not separable but noentanglement can be extracted from it locally. Therefore, theenvironment itself is a natural generator of bound entangle-ment. Of course, this is not the case for channels AD and PD,since for the former the state is fully separable at p ADc ( k ) (seeEq. (6) and Appendix) while the latter never induces ESD. Does the time of ESD really matter for large N?
Inspec-tion of critical probabilities (6), (7) and (8) shows that in allthree cases p c grows with N . In fact, in the limit N → ∞ wehave, for | αβ | (cid:54) = 0 , p ADc ( k ) → , p Diffc ( N/ → − √ ≈ . and p Dc ( N/ → − √ ≈ . . This might be in-terpreted as the state’s entanglement becoming more robustwhen the system’s size increases. However, what really mat-ters is not that the initial entanglement does not disappear butthat a significant fraction of it remains, either to be directlyused, or to be distilled without an excessively large overheadin resources. The idea is clearly illustrated in Fig. 2, where Balanced 2:2 qubit partitions
Depolarization on 4 Qubits
Unbalanced 1:3 qubit partitions
Out[398]= p N e ga ti v iti e s Figure 1: Negativity as a function of p for a balanced, α = 1 / √ β , four-qubit GHZ state and independent depolarizing channels. Asimilar behavior is observed with channel GAD with n (cid:54) = 0 , but theeffect is not so marked (the smaller n , the weaker the effect). the negativity corresponding to the most balanced partitionsis plotted versus p for different values of N . Even though theESD time increases with N , the time at which entanglementbecomes arbitrarily small decreases with it. The channel usedin Fig. 2 is the depolarizing channel, nevertheless the behavioris absolutely general, as discussed in the following.For an arbitrarily small real (cid:15) > , and all states for which | αβ | (cid:54) = 0 , the critical probability p (cid:15) at which Λ N/ ( p (cid:15) ) = (cid:15) Λ N/ (0) , becomes inversely proportional to N in the limit oflarge N . For channel (2), this is shown by letting k = N/ in (5), which simplifies to Λ GADN/ ( p ) = −| αβ | (1 − p ) N/ + | α | x N/ y N/ + | β | w N/ z N/ . For any mean bath exci-
40 qubits
Out[401]=
Out[400]= µ - µ - µ - µ - µ - Depolarization, most balanced partitions N e ga ti v iti e s p Figure 2: Negativity versus p for N = 4 , and , for channelD and for the most balanced partitions. In this graphic α = 1 / and β = √ / , but the same behavior is displayed for all other parame-ters and maps. The inset shows a magnification of the region in which | Λ D ( p ) | vanishes. Even though | Λ D ( p ) | and | Λ D ( p ) | cross the lat-ter and vanish much later, they become orders of magnitude smallerthan their initial value long before reaching the crossing point. tation n , x N/ and z N/ are at most of the same order ofmagnitude as (1 − p ) N/ , whereas y N/ and w N/ are muchsmaller than one. Therefore, for all states such that | αβ | (cid:54) = 0 we can neglect the last two terms and approximate (5), at k = N/ , as Λ GADN/ ( p ) = −| αβ | (1 − p ) N/ . We set now Λ GADN/ ( p (cid:15) ) = (cid:15) Λ GADN/ (0) ⇒ (cid:15) = (1 − p (cid:15) ) N/ ⇒ log( (cid:15) ) = N log(1 − p (cid:15) ) . Since p (cid:15) (cid:28) p GADc ( N/ ≤ , we can approx-imate the logarithm on the right-hand side of the last equal-ity by its Taylor expansion up to first order in p (cid:15) and write log( (cid:15) ) = − N p (cid:15) , implying that p GAD(cid:15) ≈ − (2 /N ) log( (cid:15) ) . Similar reasonings applied to channels (3) and (4) lead to p D,P D(cid:15) ( t ) ≈ − (1 /N ) log( (cid:15) ) . These expressions assess the ro-bustness of the state’s entanglement better than the ESD time.Much before ESD, negativity becomes arbitrarily small. Thesame behavior is observed for all studied channels, and all co-efficients α , β (cid:54) = 0 , despite the fact that for some cases, likefor instance for channel (4), no ESD is observed. The pres-ence of log (cid:15) in the above expression shows that our result isquite insensitive to the actual value of (cid:15) (cid:28) . Conclusions . We probed the robustness of the entangle-ment of generalized GHZ states of arbitrary number of par-ticles, N , subject to independent environments. The statespossess in general longer ESD time, the bigger N , but thetime at which such entanglement becomes arbitrarily small isinversely proportional to N . The latter time characterizes bet-ter the robustness of the state’s entanglement than the time atwhich ESD itself occurs. In several cases the action of theenvironment can naturally lead to bound entangled states. Anopen question still remains on how other genuinely multipar-tite entangled states, such as graph states, behave. W statesare expected to be more robust, since they have always onlyone excitation, regardless of N [18]. For example, it is possi-ble to show that, for W states, channel AD induces no ESD;however, the negativity of the least balanced partitions decayswith / √ N [19]. This is another instance in which the ESDtime is irrelevant to assess the robustness of multi-particle en-tanglement. Our results suggest that maintaining a significantamount of multiqubit entanglement in macroscopic systemsmight be an even harder task than believed so far.We thank F. Mintert and A. Salles for helpful comments andFAPERJ, CAPES, CNPQ, Brazilian Millenium Institute forQuantum Information, EU QAP project, Spanish MEC underFIS2004-05639, and Consolider-Ingenio QOIT projects for fi-nancial support. Appendix.
Here we prove that the amplitude dampingchannel leads the state (1) to a fully separable state when allof its bipartite entanglements vanish.The evolved state can be written as ρ = | α | ( | (cid:105)(cid:104) | ) ⊗ N + ρ s , where ρ s is an unnormalized state. The goal is to showthat ρ s is fully separable. This is done by showing that ρ s is obtained, with a certain probability, from a fully separablestate σ through a local positive-operator-valued measurement(POVM) [1]. Because only local operations are applied, weconclude that ρ s , and thus ρ , must be fully separable. The (unnormalized) state σ is defined as σ = 2 − N | β | { | β/α | [ αβ ( | (cid:105)(cid:104) | ) ⊗ N + α ∗ β ∗ ( | (cid:105)(cid:104) | ) ⊗ N ] } , being the N × N identity matrix. State σ is GHZ-diagonal (see definition inRef. [20]) and all of its negativities are null, then σ isfully separable [20]. Consider, for each qubit i , the localPOVM { A ( i ) m } m =1 with elements A ( i )1 = δ ( (cid:112) p AD c ( k ) | (cid:105)(cid:104) | + (cid:112) − p AD c ( k ) | (cid:105)(cid:104) | ) , where δ is such that A ( i ) † A ( i )1 ≤ , and A ( i ) † A ( i )2 = − A ( i ) † A ( i )1 . Applying this POVM to everyqubit of state σ yields ρ s when the measurement outcome is m = 1 (corresponding to A ) for every qubit. (cid:3) [1] M. A. Nielsen, I.L. Chuang, Quantum Computation and Quan-tum Information (Cambridge, Cambridge, 2000).[2] M. P. Almeida et al. , Science , 579 (2007).[3] C. Simon and J. Kempe, Phys. Rev. A , 052327 (2002).[4] A. R. R. Carvalho, F. Mintert, A. Buchleitner, Phys. Rev. Lett , 230501 (2004).[5] W. D¨ur and H.-J Briegel, Phys. Rev. Lett. , 180403 (2004);M. Hein, W. D¨ur, H.-J. Briegel, Phys. Rev. A , 032350(2005).[6] L. Di´osi, Irreversible Quantum Dynamics , edited by F. Benattiad R. Floreanini (Springer, Berlin, 2003); P. J. Dood and P.J. Halliwell, Phys. Rev. A , 052105 (2004); T. Yu and J.H. Eberly, Phys. Rev. Lett. , 140404 (2004); M. F. Santos,P. Milman, L. Davidovich, and N. Zagury, Phys. Rev. A ,040305(R) (2006); M. O. Terra Cunha, New J. Phys , 273(2007).[7] T. Yu and J. H. Eberly, Phys. Rev. Lett. , 140403 (2006).[8] A. Al-Qasimi, D. F. V. James, arXiv: 0707.2611.[9] T. Yu and J. H. Eberly, arXiv: 0707.3215.[10] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. , 5239 (1998).[11] E. Schr¨odinger, Die Naturwissenschaften 23, 807 (1935).[12] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A , 822(1998); M. Hillery, V. Buˇzek, A. Berthiaume, Phys. Rev. A , 173 (2005).[13] D. Leibfried et al. , Nature , 639 (2005); Chao-Yang Lu etal. , Nat. Phys. , 91 (2007).[14] G. Vidal and R.F. Werner, Phys. Rev. A , 032314 (2002).[15] A. Peres, Phys. Rev. Lett. , 1413 (1996); M. Horodecki, P.Horodecki, and R. Horodecki, Phys. Rev. Lett. , 5239 (1998).[16] Since the analized states are permutationally invariant, Λ GADk will correspond to the minimum eigenvalue of the matrix’s par-tial transposition according to all possible k : N − k partitions.This is also true for the other channels.[17] D. Cavalcanti, A. Ferraro, A. Garc´ıa-Saez, and A. Ac´ın,arXiv:0705.3762; G. T´oth, C. Knapp, O. G¨uhne, and H. J.Briegel, arXiv:quant-ph/0702219; D. Patan`e, R. Fazio, and L.Amico, New J. Phys. , 322 (2007).[18] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A , 062314(2000).[19] R. Chaves et. al. , to be published.[20] W. D¨ur, J. I. Cirac, and R. Tarrach , Phys. Rev. Lett.83