We consider pure quantum states of $N\gg 1$ spins or qubits and study the
average entanglement that can be \emph{localized} between two separated spins
by performing local measurements on the other individual spins. We show that
all classical correlation functions provide lower bounds to this
\emph{localizable entanglement}, which follows from the observation that
classical correlations can always be increased by doing appropriate local
measurements on the other qubits. We analyze the localizable entanglement in
familiar spin systems and illustrate the results on the hand of the Ising spin
model, in which we observe characteristic features for a quantum phase
transition such as a diverging entanglement length.Comment: 4 page

D. D. Awschalom

P. Pfeuty

R. A. Horn

T. Laustsen

English

arXiv

Verstraete, F.

Popp, M.

Cirac, J.

MPG.PuRe

Entanglement versus correlations in spin systems

We consider pure quantum states of N>1 spins or qubits and study the average entanglement that can be localized between two separated spins by performing local measurements on the other individual spins. We show that all classical correlation functions provide lower bounds to this localizable entanglement, which follows from the observation that classical correlations can always be increased by doing appropriate local measurements on the other qubits. We analyze the localizable entanglement in familiar spin systems and illustrate the results on the hand of the Ising spin model, in which we observe characteristic features for a quantum phase transition such as a diverging entanglement length

Verstraete, Frank

Popp, M

Cirac, JI

Ghent University Academic Bibliography

arXiv:quant-ph/0307009v1  1 Jul 2003Entanglement versus Correlations in Spin SystemsF. Verstraete, M. Popp and J.I. CiracMax-Planck-Institut für Quantenoptik, 85748 Garching, Germany(Dated: October 25, 2018)We consider pure quantum states of N ≫ 1 spins or qubits and study the average entanglementthat can be localized between two separated spins by performing local measurements on the otherindividual spins. We show that all classical correlation functions provide lower bounds to thislocalizable entanglement, which follows from the observation that classical correlations can always beincreased by doing appropriate local measurements on the other qubits. We analyze the localizableentanglement in familiar spin systems and illustrate the results on the hand of the Ising spin model,in which we observe characteristic features for a quantum phase transition such as a divergingentanglement length.PACS numbers: 03.67.Mn, 03.67.-a, 73.43.Nq, 05.50.+qThe mathematical description of multiparticle quan-tum systems plays an important role in several branchesof physics. The main difficulty stems from the fact thatthe number of parameters needed to describe a quantumstate grows exponentially with the number of particles.However, sometimes it is possible to capture the mostrelevant physical properties by describing these systemsin terms of very few parameters. This is the case, forexample, in quantum statistics, where two–particle cor-relations play a fundamental role. They allow us to un-derstand several complex physical phenomena, like phasetransitions. Furthermore, they give rise to concepts likecorrelation length, which quantifies a very intuitive prop-erty of these systems.Multiparticle systems are also of central interest inthe field of quantum information and, in particular, thequantification of the entanglement contained in quantumstates. The reason is that entanglement is the physicalresource to perform some of the most important quantuminformation tasks, like quantum information transfer (cfr.teleportation) or quantum computation.Given the common interest of quantum statistical me-chanics and quantum information in multiparticle sys-tems it is natural to try to describe the physical phenom-ena, like quantum phase transitions, appearing in (e.g.)spin systems from the point of view of entanglement. Themain restriction one encounters is the fact that there ex-ist very few measures of multiparticle entanglement witha clear physical meaning. In any case, since entangle-ment measures correlations (for pure states), one wouldexpect that reasonable entanglement measures be inti-mately connected to the correlation functions widely usedin the context of quantum statistical mechanics. This isnot the case, however, if one studies the behavior of theentanglement of formation between two separate spinsafter tracing out the rest [1, 2, 3, 4, 5]. Although this ap-proach exhibits a very pronounced (universal) behaviorof this quantity at the transition point, it rapidly van-ishes as the distance between the spins goes beyond 3 or4, and thus it is not related to the correlations possessedby the state. Note also that the approach by Vidal et al.[6], in which they study the scaling of entanglement ofa block of spins with the other ones, does not quantifythe entanglement as a resource (that could be used, forexample, for teleportation).In this Letter we introduce a new concept which we callLocalizable Entanglement (LE). On the one hand, thisquantity has a very well defined physical meaning whichtreats entanglement as a truly physical resource. On theother, it establishes a very close connection between en-tanglement and correlation functions, as one would nat-urally expect. The LE of two particles is the maximalamount of entanglement that can be localized in thesetwo particles, on average, by doing local measurementsin the rest of the particles. The LE naturally leads tothe definition of entanglement length, which measures thetypical length scale at which the LE decays. The LEhas an operational meaning which applies to situationsin which out of some multiparticle entangled state onewould like to concentrate as much entanglement as possi-ble in two particular particles. This occurs, for example,in the context of quantum repeaters [7] or in the con-text of quantum transport with spin systems (spintronics[8]). The determination of the LE is a formidable tasksince it involves an optimization over all possible localmeasurement strategies, and thus cannot be determinedin general. We have nevertheless managed to determinetight upper and lower bounds. The first ones stem fromconsidering joint measurements on the rest of the spins,which is closely related to the concept of entanglementof assistance [9]. The second and more interesting onescan be derived by proving that there always exist localmeasurement which do not decrease the existing correla-tions between the two spins, something that despite itsgenerality, up to our knowledge has not been consideredin the context of quantum information.The usefulness of these findings will be illustrated onthe hand of the entanglement present in the ground statesof standard spin Hamiltonians. Let us however first con-2sider a simple example involving the N -qubit GHZ-state:|GHZ〉 = 1√N(|00 · · ·0〉+ |11 · · ·1〉) .In this case, the LE is maximal as it is possible to createa Bell-state between two arbitrary qubits by measuringthe other qubits in the |+〉, |−〉 basis (here |±〉 = (|0〉 ±|1〉)/√2). The existence of these quantum correlationscould also have been revealed by studying the classicalcorrelation functionsQijαβ = 〈ψ|σiα ⊗ σjβ |ψ〉 − 〈ψ|σiα|ψ〉〈ψ|σjβ |ψ〉,where i, j denote the positions of the spins under interestand α, β label the Pauli matrices. The correlation in theα = β = z-direction is the maximal possible one. It willindeed be shown that correlation functions yield lowerbounds to the LE. However, in general the presence or ab-sence of classical correlations only gives a coarse-grainedpicture of the entanglement that ought to be created be-tween two distant spins. It is easy to find examples ofhighly entangled quantum states exhibiting no classicalcorrelations whatsoever between any pair of spins. Asan example, consider the so-called cluster states [10], ob-tained by the unitary evolution of an initially separablestate under the action of the Ising Hamiltonian:|ψ〉 = 12N/2(⊗N−1i=1 (|0〉iσ(i+1)z + |1〉i))(|0〉N + |1〉N )When N ≥ 5, all reduced 2-qubit density operators areproportional to the identity and hence no correlationsexist between any two spins. However, suitable localmeasurements on any N − 2 qubits can always createa Bell-state between the two remaining ones [10], henceindicating maximal LE.Let us next give a formal definition of the LE. Considera pure state |ψ〉 of N spins. Then the localizable entan-glement Eij(ψ) is variationally defined as the maximalamount of entanglement that can be created (i.e. local-ized), on average, between the spins i and j by performinglocal measurements on the other spins. More specifically,every measurement basis specifies a pure state ensem-ble E = {ps, |φs〉} consisting of at least 2(N−2) elementscounted by the index s. In this notation ps denotes theprobability to obtain the two-spin state |φs〉 after per-forming the measurement |s〉 on the assisting spins ofour N partite spin system. The LE is then defined asEij = maxE∑sps E(|φs〉),where E(|φs〉) denotes the entanglement of |φs〉. Aswe deal with pure states of two qubits, all entangle-ment measures are essentially equivalent, and in orderto make the connection with correlation functions, wewill measure the entanglement on the hand of the con-currence [11]: indeed, it can readily be checked thatthe maximal correlation function [18] for a pure stateof two qubits coincides with the concurrence, given byC(|ψ〉 = a|00〉+ b|01〉+ c|10〉+ d|11〉) = 2|ad− bc|. Notethat the behavior of the LE as defined in terms of theentropy of entanglement would be very similar [19].Due to the variational definition, the LE is very diffi-cult to calculate in general. Moreover, in typically largespin systems one does not have an explicit parametriza-tion of the state under interest, but just informationabout the classical 1- and 2-particle correlation functions(which parameterize completely the 2-qubit reduced den-sity operator). It would therefore be interesting to derivetight upper and lower bounds to the LE solely based onthis information. The upper bound can readily be ob-tained using the concept of entanglement of assistance[9], which would correspond to the LE if global or jointmeasurements were allowed on the other spins. It wasshown in [14] that the entanglement of assistance can becalculated as follows: given the reduced density operatorρij and a square root X , ρij = XX†, then the entan-glement of assistance Eij(ψ) as measured by the concur-rence is equal to the trace norm Tr|XT (σy ⊗ σy)X |.The lower bound is more subtle, and will be shownto follow from the following interesting theorem: given a(pure or mixed) state of N qubits with classical correla-tion Qijαβ between the spins i and j and directions α, β,then there always exist directions in which one can mea-sure the other spins such that this correlation do not de-crease, on average. This automatically implies that therealways exist local measurements that increase (or keep)the classical correlations. Surprisingly, this very generaltheorem seems not to have been noticed before, and isinteresting on its own. It could be very useful in thecontext of cryptography (e.g. multipartite distributionof common randomness [15]).Let us next proof this theorem. Note that it is suf-ficient to consider mixed states of three qubits. Let usparameterize a mixed 3-qubit density operator by four4× 4 blocksρ =[ρ1 σσ† ρ2].Without loss of generality, let us consider the Q12zz cor-relations. The original correlations are completely de-termined by the diagonal elements of the density op-erator ρ1 + ρ2. A von Neumann measurement in the|±〉 := cos(θ/2)|0〉 ± sin(θ/2) exp(±iφ)|1〉 basis on thethird qubit results in the hermitian unnormalized 2-qubitoperatorsX± =ρ1 + ρ22± cos(θ)ρ1 − ρ22± sin(θ)(cos(φ)σ + σ†2+ sin(φ)i(σ − σ†)2).Defining p± = Tr(X±), we have to prove that there al-ways exist parameters θ, φ such that the correlation, on3average, does not decrease:p+|Qzz(X+/p+)|+ p−|Qzz(X−/p−)| ≥ |Qzz(X+ +X−)|.Without loss of generality we can assume that α =Qzz(X+ +X−) is positive. Removing the absolute valuesigns and parameterizingx̄ := [cos(θ); sin(θ) cos(φ); sin(θ) sin(φ)],some straightforward algebra yields the sufficient inequal-ityx̄T[α(c̄− β̄α)(c̄− β̄α)T+Q− β̄β̄Tα]x ≥ 0 (1)where c̄ is such that p± = (1 ± c̄T x̄)/2, and β̄, Q aredefined as 3× 1 and 3× 3 blocks of the matrixS = RT (σy ⊗ σy)R =[α β̄Tβ̄ Q].Here R is the real 4× 4 matrix whose columns consist ofthe diagonal elements of the matrices (ρ1+ρ2), (ρ1−ρ2),(σ + σ†), i(σ − σ†).Due to Sylvester’s law of inertia [12], we know thatS has two positive and two negative eigenvalues. NowQ− β̄β̄T /α is the Schur complement of α, and hence cor-responds to a principal 3 × 3 block of the matrix S−1.Due to the interlacing properties of eigenvalues of prin-cipal blocks [12], it follows that it has either two positiveand one negative eigenvalue or two negative and one pos-itive one. And this of course ensures that there alwaysexists an x̄ such that the inequality (1) is fulfilled, com-pleting the proof.Note that the proof is constructive and allows to deter-mine a measurement strategy that would at least achievethe bound reported. Note also that exactly the sameproof applies when correlations between larger blocks ofspins would be considered [13].Let us now show how this theorem yields a lower boundto the LE. Given an initial pure state of N qubits, weknow that the measurement of the first, second, ... N −3’th qubit can be chosen such that on average the finalcorrelations do not decrease. But we end up with a purestate of two qubits, for which the concurrence is equalto the maximal correlation. A lower bound to the LEis therefore given by the maximal correlation function,and following the previous theorem, there is always aconstructive way of determining a measurement strategythat achieves this lower bound. Surprisingly, we will seethat this lower bound seems to be the exact value for theLE in the case of many systems of interest.The previous findings can readily be applied to thestudy of entanglement in translational invariant groundand excited states of spins arranged in a regular lattice.In quantum statistics and more specifically in the studyof quantum phase transitions, the correlation length isthe canonical parameter of interest. The concept of LEreadily lends itself to define the related entanglementlength ξE as the typical length scale at which it is possibleto create Bell states by doing local measurements on theother spins. More specifically, the entanglement lengthis finite if and only if the the LE Ei,i+n ≃ exp(−n/ξE)decays exponentially in n, and the entanglement lengthξE is defined as the constant in the exponent in the limitof an infinite system (see also Aharonov [16]):ξ−1E = limn→∞(− logEi,i+nn)Note that a diverging correlation length automaticallyimplies a diverging entanglement length and hence long-range quantum correlations, although the converse is notnecessarily true[20].More specifically, we have studied ground states oftwo spin interaction Hamiltonians with parity symmetry⊗Ni=1σiz of the form:H = −∑i,j∑α=x,y,zγijα σiασjα −∑iγiσz ,with coupling coefficients γijx ≥ γijy ≥ 0 (note that thismodel also includes spin systems in higher dimensionallattices). Extensive numerical calculations on systems ofup to 20 qubits showed that our lower bound is alwaysvery close to the LE, and typically is even equal to theexact value of the LE [13]: this is surprising and high-lights the power of the given lower bound. Note alsothat whenever the parity symmetry is present, the upperand lower bounds can easily be calculated in terms of thecorrelation functions:max(Qijxx, Qijyy, Qijzz)≤ Eij ≤√sij+ +√sij−2sij± =(1± 〈σizσjz〉)2 −(〈σiz〉 ± 〈σjz〉)2As an illustration, we consider the ground state of theIsing Hamiltonian (γijα = λδα,xδj,i+1; γi = 1), which hasbeen solved exactly [17] and exhibits a quantum phasetransition at λ = 1. In this case, the maximal classicalcorrelation is always given by Qxx, and we conjecturethat it is equal to the LE (for more details, we refer to[13]). For λ < 1, the LE is small as the ground state isalmost separable, and the entanglement length is finite(see Fig. 1) as the LE decreases exponentially with thespin distance. At the quantum critical point λ = 1, thebehavior of the LE changes drastically, as it decreasespolynomially Ei,i+n ∼ n−1/4, thus leading to a divergingentanglement length ξE . In the case λ > 1 we also getξE = ∞, as the LE saturates to a finite value given byM2x = 1/4(1− λ−2)1/4. Indeed, the ground state is thenclose to the GHZ-state. This behavior is illustrated inFig. 1.40 1 2 300.20.40.60.81λ0 1 2 300.20.40.60.81λFIG. 1: Localizable Entanglement Eij and Correlation func-tion Qijxx (solid) and upper bound (dashed) as a function ofthe coupling parameter λ for the infinite Ising chain with spindistance n = 1 (left) and n = 10 (right).0 0.5 1 1.5 2 2.500.10.20.30.40.5λFIG. 2: Correlation function Qi,i+4xx (solid), Localizable En-tanglement Ei,i+4 (diamonds), and its upper bound (dashed)as a function of the coupling parameter λ for the ground stateof the finite Ising chain (N = 14) in the case of broken par-ity symmetry by a small perturbating magnetic field in thex-direction.In a more realistic setup however, the parity symmetryof the Ising Hamiltonian will be broken by a perturba-tion and the ground state for large coupling will also beseparable. Indeed, the energy gap between the lowest en-ergy states with different parity decays exponentially inthe number of qubits in the region λ > 1, and henceforththe ground state becomes a superposition of these twostates. The calculation of its LE and the correspondinglower and upper bounds are depicted in Fig. 2.These results give a clear illustration of the intimateconnection between classical correlations and entangle-ment in the case of ground states of translational invari-ant Hamiltonians. The plethora of results concerningclassical correlation functions, such as diverging correla-tion length at quantum phase transitions, can now beinterpreted from the perspective of quantum informationtheory; one could argue that the status of classical cor-relations has been lifted to the one of (useful) quantumcorrelations.In conclusion, we have introduced the notion of local-izable entanglement. It has a nice operational meaningas it quantifies the amount of useful entanglement thatcan be created between two spins by doing local mea-surements on all other spins. We proved that classicalcorrelation functions always provide lower bounds to theLE, showing that the presence of classical correlationsis sufficient to be able to create Bell-like quantum cor-relations. As a side-product, we proved that classicalcorrelations in multipartite mixed quantum states canalways increase by doing appropriate measurements onthe other qubits. Finally, we demonstrated the useful-ness of these concepts in the context of spin systems ona lattice, provided a natural definition of entanglementlength, and showed that it diverges at a quantum phasetransition. Further generalizations and applications willbe presented elsewhere [13].We acknowledge interesting discussions with J. GarciaRipoll. This work was supported in part by the E.C.(projects RESQ and QUPRODIS) and the Kompeten-znetzwerk ”Quanteninformationsverarbeitung” der Bay-erischen Staatsregierung.[1] A. Osterloh et al., Nature 416, 608 (2002).[2] T.J. Osborne and M.A. Nielsen, Phys. Rev. A 66, 032110(2002)[3] K.M. O’Connor and W.K. Wootters, Phys. Rev. A 63,052302 (2001)[4] P. Zanardi and X. Wang, J. Phys. A 35, 7947 (2002)[5] M.C. Arnesen, S. Bose and V. Vedral, Phys. Rev. Lett.87, 017901 (2001).[6] G. Vidal et al., Phys.Rev.Lett. 90, 227902 (2003).[7] H.J. Briegel et al, Phys. Rev. Lett. 81, 5932 (1998).[8] D.D. Awschalom, D. Loss, and N. Samarth, Semicon-ductor Spintronics and Quantum Computation, Springer-Verlag, Berlin, 2002.[9] D. P. DiVincenzo et al. quant-ph/9803033.[10] H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86:910 (2001).[11] W.K. Wootters, Phys. Rev. Lett. 80:2245 (1998).[12] R.A. Horn and C.R. Johnson, Matrix Analysis, Cam-bridge University Press, 1985.[13] M. Popp, F. Verstraete and J.I. Cirac, in preparation.[14] T. Laustsen, F. Verstraete, and S. J. van Enk, Quant.Inf. and Comp. 3, 64 (2003).[15] I. Devetak and A. Winter, quant-ph/0304196.[16] D. Aharonov, Phys. Rev. A 62, 062311 (2000).[17] P. Pfeuty, Ann. Phys. 57, 79 (1970).[18] The basis with maximal possible correlation can in gen-eral be found by calculating the singular value decompo-sition of the 3 × 3 matrix Qαβ, and the largest singularvalue corresponds to the maximal possible correlation.[19] For a given concurrence C, the entropy of entanglementis given by f(C) = H((1 +√1− C2)/2)with H(x) theShannon entropy. Due to the convexity of f(C), thefollowing bounds hold: f(∑spsCs)≤∑spsf(Cs) ≤∑spsCs.[20] This leaves open the possibility of Hamiltonians exhibit-ing a phase transition as indicated by a diverging entan-glement length at the critical point but with a correlationlength remaining finite.

Entanglement versus Correlations in Spin Systems

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