Buchleitner, A.

Melo, F. (Fernando) de

Tiersch, M.

English

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Universality in open system entanglement dynamicsMarkus Tiersch,1, 2 Fernando de Melo,1, 3 and Andreas Buchleitner11Physikalisches Institut der Albert–Ludwigs–Universita¨t, Freiburg, Germany2Institute for Quantum Optics and Quantum Information, Innsbruck, Austria3Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, BelgiumWe show that the entanglement evolution of an open quantum system is the same for the vastmajority of initial pure states, in the limit of large Hilbert space dimensions.PACS numbers: 03.67.-a, 03.67.Mn, 03.65.Yz.The ability to characterize and control quantum sys-tems with an ever increasing number of constituents isan ever more desired and necessary skill in many fieldsof modern physics [1], materials science [2], and even bi-ology [3]. From a strictly deterministic point of view,however, the rapidly growing generic complexity of suchlarge systems apparently renders this task ineffable. Yet,similarly to thermodynamics, a statistical description of-ten allows for the extraction of robust, generic featureswhich emerge in the limit of large system size, and implyquantitative predictions.Entanglement, an unmistakeable quantum signature,is a prime example of the above, apparent illusiveness ofan exhaustive characterization as the system size is scaledup: The structure of many-particle entanglement turnsmore and more intricate with the exponentially increas-ing number of possible correlations between different sub-groups of particles. Thus, a complete characterization ofa large, composite quantum state requires an experimen-tal overhead that increases exponentially with the num-ber of system constituents. Even worse, entanglementtends to get ever more fragile when enlarging the systemsize: the more degrees of freedom, the more difficult it be-comes to shield quantum coherences, which are necessaryfor entanglement, against the detrimental influence of anoisy environment. In such situations, the strong quan-tum correlations due to entanglement additionally needto be distinguished from classical correlations inducedby the ambient noise. This is in general accomplished byhigh dimensional optimization procedures on the spaceof all quantum states [4, 5], leaving little hope for quan-titative predictions on entanglement evolution in large,and noisy systems. On the other hand, the signaturesof entanglement in such systems are of high fundamen-tal and, potentially, practical interest, e.g. when it comesto harnessing the computational power of quantum algo-rithms [6] or assessing the role of quantum correlationsin intrinsically noisy biological systems [3]. It is thereforea key issue and the subject of the present Letter to esti-mate the characteristic time scales in which entanglementis present in such adverse situations.Here we consider the fate of entangled states of largequantum systems, which require a high dimensionalHilbert space, in contact with an incoherent environment.We show that a statistical analysis over generic initialstates unveils universal – state independent – open sys-tem entanglement evolution in the limit of large Hilbertspace dimensions. In this thermodynamic limit, an effi-cient characterization of entanglement dynamics is thusagain possible.More specifically, let us start with a composite quan-tum system in a pure state, characterized by a vector |χ〉in a Hilbert space H = HA ⊗HB · · · ⊗ HN with dimen-sion d = dAdB · · · dN . The system then undergoes somedynamics by virtue of its Hamiltonian, and also couplesto uncontrollable degrees of freedom, which define its en-vironment (denoted by subscript E). When focusing onthe system alone, its state ρ evolves as described by atime-dependent map Λt:ρ(0) 7→ ρ(t) = TrE[U(t)ρSE(0)U(t)†] =: Λt[ρ(0)], (1)where U(t) is the unitary time evolution operator ob-tained as solution of the Schro¨dinger equation for sys-tem and environment (subscript SE). Such maps Λtare only independent of the system’s initial state ρ(0)if system and environment are initially uncorrelated orat most classically correlated [7]. This is the case formany implementations of quantum information tasks [8],where initial system state are almost pure, and there-fore uncorrelated with the environment in very good ap-proximation. In what follows, we thus assume the ini-tial state of system and environment to be of the formρSE(0) = |χ〉〈χ|⊗ρE(0). Since, beyond this latter factor-ization, our ansatz makes no assumption on the specificform of the Hamiltonian and thus of U(t), and neitherof ρE(0), our subsequent results are applicable for arbi-trary open system dynamics of the given initial states,including non-Markovian effects that may arise in thecourse of the evolution. The following derivations aresolely grounded in few geometrical properties of the set ofstates, and properties of entanglement measures thereon.The next steps and properties needed to prepare our mainresult are visualized in Fig. 1.As a measure for the relative effect of the open sys-tem dynamics Λt onto two states ρ and ω, we choose themetric distanceDTr(ρ, ω) = ‖ρ− ω‖Tr := Tr|ρ− ω|, (2)arXiv:0810.2506v2  [quant-ph]  8 Jun 20102separableentangledρΛt(ρ) Λt(ω)ωED(ρ)ρDTrFIG. 1: State space is a convex set containing all quantumstates of a given dimension. Open system dynamics due to amap Λt cannot increase the distance between any two states(as measured by the trace distance). Entanglement measures,as ED, monotonically increase with the distance to the set ofseparable states (in gray).induced by the trace norm ‖·‖Tr on state space. Thedistance of two states measured by DTr decreases mono-tonically upon application of any such Λt [9],DTr[Λt(ρ),Λt(ω)] ≤ ηΛt DTr(ρ, ω) with ηΛt ≤ 1. (3)Note that, for closed dynamics, i.e., without coupling tothe environment, equality holds with ηΛt = 1, as a con-sequence of the unitarity of quantum mechanics. Opendynamics, however, introduces additional mixing of thesystem state, such that the state space is effectively con-tracted. This dynamical effect is captured by the changeof ηΛt . For example, when coupled to a thermal bath, allinitial states converge to thermal equilibrium, and ηΛtapproaches zero asymptotically in time.Turning to entanglement, we do not focus on a specificentanglement quantifier, but rather only require that itslowly varies on the set of states. More specifically, wedemand a “strong” form of continuity, namely Lipschitzcontinuity [10],∣∣E(ρ)− E(ω)∣∣ ≤ ηE DTr(ρ, ω) , (4)where ηE is called the Lipschitz constant. Examples ofsuch entanglement measures E(ρ) are the ones definedas the minimum distance of the state ρ to the set Sof separable states: ED(ρ) = minσ∈S D(ρ, σ) [5]. Alldistances D which abide to the dynamic monotonicitycondition (3) [11], fulfill the Lipschitz requirement withconstant ηED = 1, by virtue of the triangle inequality.Another example is negativity N , a computable entan-glement monotone [12], but with a different constant ηN(see the appendix).Under conditions (3) and (4), we can asses how muchtwo initially pure states differ in their entanglement afterexposure to the same incoherent dynamics. For statesρ(t) = Λt(|χ〉〈χ|) and ω(t) = Λt(|ψ〉〈ψ|) we obtain∣∣E(ρ)− E(ω)∣∣ ≤ ηE DTr[Λt(|χ〉〈χ|),Λt(|ψ〉〈ψ|)]≤ ηE ηΛt DTr[|χ〉〈χ|, |ψ〉〈ψ|]≤ 2ηE ηΛt∥∥|χ〉 − |ψ〉∥∥. (5)That is, the entanglement quantifier E inherits Lipschitzcontinuity, with constant 2ηE ηΛt , even with respect tothe Euclidean distance between the two initial state vec-tors in H. The difference in entanglement of the twomixed final states is essentially bounded by the distancebetween the initial states. With the knowledge of theentanglement of a single probe state, say of ρ(t), for aninitial state |χ〉, we are able to predict the entanglementevolution of any other pure initial state |ψ〉 within an er-ror margin given by (5) – this without the need to evolvethe state, and to compute the resulting mixed state’s en-tanglement.Rather than studying specific instances of initial states,we aim at a statement about generic pure states in a sta-tistical manner. To avoid any bias, we use random stateswhich uniformly cover the space of pure states. Statesthat exhibit the properties of random states naturallyappear in registers of quantum computers after long se-quences of gates [13], in quantum systems with a chaoticclassical counterpart [14], and also subsets of pure states,e.g. graph states which emerge in spin gases [15], showthe properties of a uniform distribution [16]. In addition,for many interesting scenarios of biology and chemistrysuch as transport phenomena in proteins, the initial stateand/or the Hamiltonian are not exactly known, e.g. [17].In these cases a uniform distribution of initial states inthe potential subspace is the best prior.Let us first note that the set of state vectors in His isomorphic to a unit sphere S2d−2 in R2d−1, wherethe real and imaginary part of the expansion coefficientsinto any basis constitute the coordinates, constrained bynormalization and ignoring the global phase by choosingthe first component real. With the above preparation,we can now infer our central result by employing thewell-known Levi’s lemma [18–20]: The probability for adeviation larger than  of E[ρ(t)] := E[Λt(|χ〉〈χ|)] fromits mean 〈E〉(t) := ∫ dψ E[Λt(|ψ〉〈ψ|)] over all initialstates, given the uniform initial distribution on S2d−2,exhibits an exponential suppression: (i) in the deviation , and moreover, (ii) in the system dimension d:Pr(∣∣E[ρ(t)]− 〈E〉(t)∣∣ > ) ≤ 4 exp(−C 2d− 14η2Eη2Λt2).(6)The constant C can be chosen (24pi2)−1 [21]. This holdsfor all entanglement quantifiers E that fulfill (4), evenwhen quantifying multipartite entanglement.As a concrete application of this result, we considerthe normalized negativity N/Nmax of a bipartite systemon HA⊗HB . Its Lipschitz constant (see appendix) readsη(N/Nmax) = dA/(dA − 1), and the following inequalityfor a deviation from the mean entanglement holds:30.0 0.2 0.4 0.6 0.8 1.002468Negativity H1 vs. N-1Lprobabilitydensityp=0p=0.1p=0.3p=0.5 N=30.0 0.2 0.4 0.6 0.8 1.0020406080100120Negativity H1 vs. N-1Lp=0p=0.1p=0.3p=0.5N=80.0 0.2 0.4 0.6 0.8 1.005101520Negativity H1 vs. N-1Lprobabilitydensity p=0p=0.1p=0.3p=0.5N=5 æææææææàààààààììììììì2 3 4 5 6 7 80.0050.0100.0200.0500.1000.200Nstandarddeviationp=0p=0.3p=0.5FIG. 2: (Color online) Distributions of negativity. The data for negativity is obtained for the least balanced partition (one vs.N − 1 qubits) of N = 3, 5, 8 qubits, at different snapshots of the evolution. We sample over 10 000 uniformly distributed pureinitial states of N qubits which evolve under local coupling to a decoherence reservoir. The system–environment interactiontime t is parametrized by p = (1− e−Γt), with Γ the local decoherence rate. Clearly, the bigger N , the more concentrates thenegativity distribution around its mean value, for all times, as exemplified in the logarithmically plotted standard deviation forsystem sizes N = 2, . . . , 8 and three time steps of the evolution p = 0, 0.3, 0.5 (bottom right). Lines are linear fits to the data.Pr(∣∣∣∣∣N[ρ(t)]− 〈N〉(t)Nmax∣∣∣∣∣ > )≤ 4 exp(−C (2dAdB − 1)(dA − 1)24d2A η2Λt2). (7)This expresses the concentration effect visualized inFig. 2 for ensembles of N = 2, . . . , 8 qubits (two-levelsystems), each locally coupled to an environment that de-stroys only the coherences. Negativity is here evaluatedwith respect to the least balanced bipartition of the qubitregister, i.e., it quantifies the entanglement of one qubit(dA = 2) with the remaining N − 1 qubits (dB = 2N−1),after application of the decoherence dynamics. We uni-formly sample 10 000 pure initial states, and parametrizethe dynamics by the probability p for complete decoher-ence of a single qubit. Within the Markov approximationp = 1 − e−Γt, where Γ is the decoherence rate [22]. Thelog-plot of the variances for increasing system size, shownat the bottom right of Fig. 2, establishes the exponentialconcentration around the mean. Thereby, our numeri-cal results underpin the mixed state entanglement con-centration predicted by (6) and (7). Furthermore, theconcentration observed by the numerical experiment iseven stronger than estimated – the measured variancesare approximately one order of magnitude smaller thanthe variance which we would infer from (7), with ourabove estimate of C and the Lipschitz constants.The above results determine the open system evolu-tion of entanglement with an error exponentially smallin the dimension of the underlying Hilbert space (pro-vided ηE does not increase faster than√d). Therefore,in high dimensions, it suffices to monitor the entangle-ment evolution of a single, generic pure state, in orderto predict the fate of any other typical state subject tothe same dynamics. Qualitatively different entanglementdynamics for different initial states [23] will occur onlyas singular effects in sufficiently large quantum systems.Inasmuch as knowledge of a single state’s entangle-ment evolution fully determines (in the present, asymp-totic sense) the result for arbitrary states, our result isreminiscent of previously derived entanglement evolutionequations [24, 25], where the final entanglement of anarbitrary initial pure state was shown to be fully char-acterized by the entanglement evolution of a maximallyentangled state.4The concentration of open system entanglement evo-lution, as spelled out by (6) and (7), has yet anotherbearing for the optimization problem routinely encoun-tered [4, 5] when evaluating the entanglement of arbitrarymixed states. For mixed states generated through arbi-trary physical dynamics Λt from a uniform distributionof pure states, mixed state entanglement concentrationsuggests a reduction of the optimization space: A singlerepresentative of the obtained sample generated under aspecific physical evolution, selected with convenient prop-erties (symmetries in terms of its pure state decomposi-tion), will suffice to impose exponentially narrow con-straints on the optimization for all other states.Acknowledgements. We enjoyed discussions with MarkFannes, Alexander Holevo, and particularly thank Rein-hard Werner for discussions on the Lipschitz constant ofnegativity. We also profited from exchanges with Car-los Viviescas, Alejo Sales, and Daniel Cavalcanti, andexpress our gratitude for the hospitality of Berge En-glert and the NUS, where part of the work was done, aswell as Luiz Davidovich’s group at the Universidade Fed-eral do Rio de Janeiro, where this work was finished.DAAD/CAPES is acknowledged for financial supportthrough the PROBRAL program. M.T. was partiallysupported by the IMS. F.M. acknowledges support bythe Alexander von Humboldt Foundation, and BelgiumInteruniversity Attraction Poles Programme P6/02.Appendix. In order to calculate the Lipschitz constantfor the negativity of a bipartite system, we use its defini-tion through the trace norm of the partially transposedstate [12], N (ρ) := (‖(I⊗ T )(ρ)‖Tr − 1)/2, where T rep-resents the transposition acting just on HB . 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Universality in open system entanglement dynamics

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