Nature and measure of entanglement in quantum phase ...viola/publications/PRA_70_042311.pdf ·...

Nature and measure of entanglement in quantum phase transitions

Rolando Somma,1,2,* Gerardo Ortiz,1 Howard Barnum,1 Emanuel Knill,1,† and Lorenza Viola1,‡

1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2Centro Atómico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina

(Received 12 March 2004; published 18 October 2004)

Characterizing and quantifying quantum correlations in states of many-particle systems is at the core of afull understanding of phase transitions in matter. In this work, we continue our investigation of the notion ofgeneralized entanglement[Barnum et al., Phys. Rev. A68, 032308(2003)] by focusing on a simple Lie-algebraic measure of purity of a quantum state relative to an observable set. For the algebra of local observ-ables on multi-qubit systems, the resulting local purity measure is equivalent to a recently introduced globalentanglement measure[Meyer and Wallach, J. Math. Phys.43, 4273(2002)]. In the condensed-matter setting,the notion of Lie-algebraic purity is exploited to identify and characterize the quantum phase transitions presentin two exactly solvable models, namely the Lipkin-Meshkov-Glick model and the spin-1

2 anisotropicXY modelin a transverse magnetic field. For the latter, we argue that a natural fermionic observable set arising after theJordan-Wigner transformation better characterizes the transition than alternative measures based on qubits.This illustrates the usefulness of going beyond the standard subsystem-based framework while providing aglobal disorder parameter for this model. Our results show how generalized entanglement leads to useful toolsfor distinguishing between the ordered and disordered phases in the case of broken symmetry quantum phasetransitions. Additional implications and possible extensions of concepts to other systems of interest incondensed-matter physics are also discussed.

DOI: 10.1103/PhysRevA.70.042311 PACS number(s): 03.67.Mn, 03.65.Ud, 05.70.Jk, 05.30.2d

I. INTRODUCTION

Quantum phase transitions(QPTs) are qualitative changesoccurring in the properties of the ground state of a many-body system due to modifications either in the interactionsamong its constituents or in their interactions with an exter-nal probe[1], while the system remains at zero temperature.Typically, such changes are induced as a parameterg in thesystem HamiltonianHsgd is varied across a point at whichthe transition is made from one quantum phase to a differentone. Often some correlation length diverges at this point, inwhich case the latter is called aquantum critical point.Be-cause thermal fluctuations are inhibited, QPTs are purelydriven by quantum fluctuations: fluctuations or correlationsin the value of some observable or observables that occur ina pure state. Thus, these are purely quantum phenomena: aclassical system in a pure state cannot exhibit correlations.Prominent examples of QPTs are the quantum paramagnet toferromagnet transition occurring in Ising spin systems underan external transverse magnetic field[2–4], the supercon-ductor to insulator transition in high-Tc superconducting sys-tems, and the superfluid to Mott insulator transition origi-nally predicted for liquid helium and recently observed inultracold atomic gases[5].

Since entanglement is a property inherent to quantumstates and intimately related to quantum correlations[6], one

would expect that, in some appropriately defined sense, theentanglement present in the ground state undergoes a sub-stantial change across a point where a QPT occurs. Recently,several authors attempted to better understand QPTs bystudying the behavior of different measures of entanglementin the ground state of exactly solvable models(see[7–12] forrepresentative contributions). Such investigations primarilyfocused on characterizing entanglement using information-theoretic concepts, such as the entropy of entanglement[13]or the concurrence[14], developed forbipartite systems. Inparticular, a detailed analysis of the two-spin concurrencehas been carried out for theXY model in a transverse mag-netic field [7,8], whereas the entanglement between a blockof nearby spins and the rest of the chain has been consideredin [10]. While a variety of suggestive results emerge fromsuch studies, in general a full characterization of the quan-tum correlations near and at a quantum critical point will notbe possible solely in terms of bipartite entanglement. Identi-fying the entanglement measure or measures that best cap-ture the relevant properties close to criticality, including thecritical exponents and universality class of the transition, re-mains an open problem in quantum information andcondensed-matter theory.

In Refs.[11,15], we introducedgeneralized entanglement(GE) as a notion extending the essential properties of en-tanglement beyond the conventional subsystem-based frame-work. This notion is general in the sense that it is definablerelative toany distinguished subset of observables,withoutexplicit reference to subsystems, which makes it directly ap-plicable to any algebraic language used to describe the sys-tem (fermions, bosons, spins, etc.) [16–18]. Founding thenotion on a distinguished set of observables makes it espe-cially well suited to studying QPTs, as our definition makesthe existence of GE equivalent to the existence of nonzero

*Corresponding author. Email address: [email protected]†Present address: National Institute of Standards and Technology,

Boulder, CO 80305, USA.‡Present address: Department of Physics and Astronomy, Dart-

mouth College, Hanover, NH 03755, USA.

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correlations or fluctuations in those observables. The basicidea is that any quantum state gives rise to a reduced state onthe distinguished subset of observables[19]. These reducedstates form a convex set; as with standard quantum states,there are pure(extremal) and mixed(nonextremal) ones[20].We define ageneralized entangledpure state, relative to asubspace of observables, to be one whose reduced state onthat subspace ismixed.Although we will have little occasionin the present context to apply it to states that are mixedrelative to the full set of observables, we extend this notionto include mixed states by defining a generalized entangledmixed state to be one that cannot be written as a convexcombination of generalized-unentangled pure states.

The special case in which the observable set is a Lie al-gebra is often important in physics. In a broad class of suchalgebras described below, the algebra is not only a subspaceof operators, but is such that we can define a natural Hermit-ian projection onto that subspace. Then a simple(global)measure of GE for quantum states is provided by what wecall thepurity relative to the algebra.This is defined as thesquared length of the projection of the Hermitian operator(density matrix) representing the state onto the algebra. Asargued in[11], if the correct algebra is chosen, the puritycontains information about the relevant quantum correlationsthat uniquely identify and characterize QPTs of the system.

In this paper, we deepen and expand the analysis initiatedin [11] by focusing on the detection of QPTs due to a brokensymmetry as revealed by the behavior of an appropriate rela-tive purity of the ground state. In Sec. II, we recall the rel-evant mathematical setting and the definition of the relativepurity as a function of the expectation values of the distin-guished observables. In Sec. III, we discuss several exampleswhere the relative purity is seen to provide a natural measureof entanglement. In Sec. IV, we illustrate some physical cri-teria that are relevant in choosing the appropriate observablesubalgebra and using GE as an indicator of QPTs. In Secs. Vand VI we explicitly characterize the QPTs present in theso-called Lipkin-Meshkov-Glick(LMG) model [21,22] andin the one-dimensional spin-1

2 anisotropicXY model in atransverse magnetic field, respectively. This is done bystudying the properties of the purity relative to different al-gebras of observables in the ground state of both models. Wefind the relevant critical exponents for these models, and inthe case of the anisotropicXY model in a transverse mag-netic field, obtain a new “global” disorder parameter, thevariance of the number of spinless fermionic excitations in aJordan-Wigner-transformed representation of the system. Fi-nally, we provide in separate Appendixes the details under-lying various statements made in the main body of the paper.These include the relationship between standard separabilityand GE (Appendix I), the GE properties of two specialclasses of spin states, the cluster and valence bond solidstates(Appendix II), the proof of the relationship betweenthe local purity and the Meyer-Wallach entanglement mea-sure (Appendix III), and the semiclassical properties of theLMG model in the thermodynamic limit(Appendix IV).

II. GENERALIZED ENTANGLEMENT AND RELATIVEPURITY

In the GE approach, entanglement is considered as anobserver-dependentproperty of a quantum state, which is

determined by the physically relevant point of view throughthe expectation values of a distinguished subset of observ-ables. Whenever a preferred decomposition into subsystemsis specified in terms of an appropriate(physical or encoded[22–25]) tensor product structure, GE becomes identical tostandard entanglement provided that distinguished observ-ables corresponding to alllocal actions on the individualsubsystems are chosen: in particular, forH= ^ iHi withdimsHid=di, standard entanglement of states inH is recov-ered as GE relative tohloc= % isusdid [11,15] (see also Appen-dix I). In fact, the subsystems relative to which standardentanglement is defined(whether directly identifiable withphysical degrees of freedom or related to “encoded” or “vir-tual” ones) are always understandable in terms of appropriate(associative) algebras of local observables. This has beenobserved before, e.g., in[24,25] (see also[26] for a recentanalysis). However, it is important to realize that the GEnotion genuinely extends the standard entanglement defini-tion, and does not coincide with or reduce to it in general. Onone hand, this may be appreciated by noticing that even forsituations where a subsystem partition is naturally present,states which are manifestlyseparablerelative to such a par-tition may possess GE relative to an algebra different fromhloc (see the two-spin-1 example of Sec. III). On the otherhand, as also emphasized in[11], GE is operationally mean-ingful in situations whereno physically accessible decompo-sition into subsystems exists, thus making conventional en-tanglement not directly definable.

A. Relative purity for faithfully represented Lie algebras

As mentioned in the Introduction, we will focus on thecase where the distinguished observables form a Lie algebrah of linear operators, acting on a finite-dimensional statespaceH for the system of interest,S. (Note that we will notusually distinguish between the abstract Lie algebra isomor-phic to h and the concrete Lie algebrah of operators thatfaithfully represents it onH.) We will assumeh to be a realLie algebra consisting of Hermitian operators, with thebracket of two linear operatorsX andY being given by

fX,Yg = isXY− YXd. s1d

In this way, operators inh can be directly associated withphysical observables. For the same reason, we will also use aslightly nonstandard(but familiar to physicists) notion of theLie group generated byh, involving the mapX°eiX insteadof the mathematicians’X°eX, for XPh. No assumption thatthe Lie algebra acts irreducibly onH (i.e., that it admits nonontrivial invariant subspaces) will be made, but importantconsequences of making such an assumption will be dis-cussed. We will also assume the Lie algebra to be closedunder Hermitian conjugation. This implies that it is areduc-tive algebra(not to be confused with reducibility of the rep-resentation). In our context, a reductive Lie algebra is bestthought of as the product(direct sum, as a vector space) of afinite number ofsimpleLie algebras, and a finite number ofcopies of a one-dimensional Abelian Lie algebra. AsimpleLie algebra is a non-Abelian one possessing no nontrivialideals, where an ideal is a subalgebra invariant under com-

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mutation with anything in the algebra; the relevant propertyof ideals here is that they can be quotiented out of the alge-bra, allowing it to be written as a nontrivial product of idealand quotient; thus simple Lie algebras are non-Abelian onesthat cannot be decomposed into factors, so the factorizationused in defining reductive Lie algebras above is maximal.The product(direct vector-space sum) of a finite number ofsimple Lie algebras is calledsemisimple,and thus a reduc-tive algebra is the product of a semisimple and an Abelianpart. The reader is referred to[27–30] for relevant back-ground on Lie algebras and their representation theory. Asthis subsection unfolds, we will summarize much of this rep-resentation theory in a way suited to our needs, and thereader should concentrate on understanding the content ofthe statements, and not vex him or herself unduly about un-derstanding why they are true.

We will consider pure quantum states ofS, uclPH, aswell as mixed quantum states ofS, described by densitymatricesr acting onH. Sinceh was assumed closed underHermitian conjugation, the projection of a quantum stateronto h with respect to the trace inner product is uniquelydefined. LetPh denote the projection map,r°Phsrd. If r isa pure state,r= uclkcu, the purity of ucl relative to h (orh-purity) is defined as the squared length of the projectionaccording to the trace inner product norm[15]; that is,

Phsucld = TrhfPhsuclkcudg2j. s2d

The h-purity may be explicitly evaluated upon selecting anoperator basisB=hA1,… ,ALj for h. By assuming theAa tobe Hermitian,

Aa = Aa† , s3d

and orthogonal,

TrsAaAbd = da,b, s4d

Eq. (2) may be rewritten as

Phsucld = TrF oa,b=1

L

TrsAardTrsAbrdAaAbG= o

a=1

L

kAal2, s5d

wherekAal denotes the expectation value of the observableAa in the pure stateucl.

An important property following is that theh-purity isinvariant under group transformations: if a new basis forh is

introduced by lettingAa=D†AaD, with D=expsiob=1L tbAbd,

D†D=1, andtb real numbers, then one finds

Phsucld = oa=1

L

kAal2 = oa=1

L

kAal2 = Phsucld. s6d

Sometimes it is useful to introduce a common normalizationfactorK in order to set the maximum value of the purity to 1,in which case Eq.(5) becomes

Phsucld = Koa=1

L

kAal2. s7d

As mentioned earlier, a pure quantum stateucl is definedto begeneralized entangled(generalized unentangled) rela-tive to h if it induces a mixed(pure) state on that set ofobservables. Whenh is a complex semisimple Lie algebraacting irreducibly on H, it was shown in[15] [Theorem 14,part (4)] that ucl is generalized unentangled with respect tohif and only if it has maximumh-purity, and generalized en-tangled otherwise. Under the same assumptions, the above-mentioned Theorem[part (3)] also leads to the identificationof the generalized unentangled pure states as thegeneralizedcoherent states(GCSs) associated withh [31–33]. In otherwords, all generalized unentangled states are in the(unique)orbit of a minimum weight state ofh (taken as a referencestate) under the action of the Lie group. Remarkably, GCSsare known to possessminimum invariant uncertainty,sDFd2sucld=oafkAa

2l−kAal2g [34,35], so that, similar to thefamiliar harmonic-oscillator ones, they may be regarded insome sense as closest to “classical” states.

Our characterization theorem for generalized unentangledstates on irreducible representations used some standard factsfrom the theory of semisimple Lie algebras and their repre-sentations that will also be useful in the discussion of reduc-ible representations in the next subsection. These are the ex-istence of Cartan(in the semisimple context, maximalAbelian) subalgebras, their conjugacy under the action of theLie group associated with the algebra, and the fact that anyfinite-dimensional representation, given a choice of Cartansubalgebra(CSA), decomposes into mutually orthogonal“weight spaces,” which are simultaneously eigenspaces of allCSA elements. The map from CSA elements to their eigen-values on a given weight space is a linear functional on theCSA called the “weight” of that weight space. The theoremalso uses the observation that the projection of the state intothe Lie algebra is necessarily a Hermitian element of thatalgebra, hence semisimple(diagonalizable), hence belongingto some CSA, which we call itssupporting CSA.Frequently,semisimple Lie algebras are presented by giving aCartan-Weylbasis, consisting of a set of commuting, jointly diago-nalizable operators that generate a CSA of the algbera, and aset of so-called “Weyl operators” that are nondiagonalizable,and act to take a state in one weight space to a state inanother(or else annihilate it): in physical examples these areoften called “raising and lowering operators.” Normalizedstates correspond to normalized linear functionals on the Liealgebra; when a Cartan-Weyl basis for the algebra is chosensuch that the CSA distinguished by the basis is the support-ing CSA for a given state, the state is zero except on the CSApart of the basis. On the CSA, the state is some convexcombination of the weights, that is, an element of theweightpolytope(which is defined as the convex hull of the weights).So it turns out that extremal states on the Lie algebra corre-spond to extremal points of the weight polytope. This appliesregardless of whether the representation is irreducible or not.For irreducible representations(irreps), the extremal pointsof the weight polytope are also highest-weight states of theirrep. Reducible representations are discussed in the next

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subsection(along with some comments on reductive alge-bras).

In preparation for that, we introduce another aspect ofstandard Lie theory: the Weyl group. Besides being able totake any CSA to any other CSA, the Lie group also acts onthe weight polytope for a given CSA, by reflections in a setof hyperplanes through the origin. The group these generateis called the Weyl group. Considered together, the hyper-planes divide the weight space into a set of convex cones,sometimes calledWeyl chambers,whose points are at theorigin, and whose union with the hyperplanes is the entirespace. Any such cone can be mapped to any other via theWeyl group action, and the weight polytope of the represen-tation is the convex hull of the Weyl group orbits of theweights in the closure of any single Weyl chamber.

B. Irreducibly versus reducibly represented Lie algebras

It is important to realize that the relationships just men-tioned between maximal purity, generalized coherence, andgeneralized unentanglement established for a pure state rela-tive to an irreducibly represented algebrah do not automati-cally extend to the case whereh acts reducibly onH. Wewill discuss semisimple algebras first and then, because thealgebra we use to analyze the LMG model is Abelian, thecase of reductive algebras.

If h is semisimple, a generic finite-dimensional represen-tation ofh may be decomposed as a direct sum of irreducibleinvariant subspaces,H. %,H,, with each of theH, being inturn the direct sum of its weight spaces. Every irrep appear-ing in the decomposition has a highest(or lowest) weight,and for each of these irreps, there is a manifold of GCSs forthe irrep constructed as the orbit of a highest weight state forthat irrep. The weight polytope for thereduciblerepresenta-tion will be the convex hull of those for all the irreps con-tained in it. Because of this, the GCSs for these irreps willnot, in general, all satisfy the extremality property that de-fines generalized unentangled states. This reflects the factthat even for a state belonging to a specifich irrep, GE is aproperty which depends in general on how the state relates tothe whole representation, not solely the irrep. Nor is therenecessarily a single weight, for one of the constituent irreps,that generates(as the convex hull of the Weyl group orbit)the weight polytope of the reducible representation. Indeed,the extremal weights in the weight polytope, which corre-spond to generalized unentangled states, need not all havethe same length. Since this squared length is theh-purity [asdefined in Eq.(5)] of the corresponding state, it is thus nolonger the case that all generalized unentangled states havemaximal Lie-algebraic purity. However, maximal purity re-mains asufficient,though no longer a necessary, conditionfor generalized unentanglement. If the algebra is reductive,the expectations of a maximal commutative subalgebra nowinclude ones for the Abelian part of the algebra, i.e., opera-tors that commute with the entire algebra. These must beproportional to the identity on each irrep, but may have dif-ferent eigenvalues(possibly degenerate) on different irreps.States on this algebra then involve not just weights for thesemisimple part of the algebra, but expectation values for the

Abelian part of the algebra as well. These can distinguishdifferent subsets of the irreps, and so irreps whose highestweight is not extremal for the semisimple part may becomeextremal(generalized unentangled) in the full reductive al-gebra. However, maximal quadratic purity will remain a suf-ficient, though in general still not necessary, condition for astate being generalized unentangled.

More intuition about GE, purity, and GCSs may be gainedfrom simple examples. Consider a physical system which iscomposed of two spin-1

2’s (namely, two qubits), and let thembe labeled byA, B, with H=HA ^ HB=C4, and correspond-ing sus2d generatorssa

A,saB,aP hx,y,zj. Consider GE rela-

tive to aglobal representation ofsus2d, whose total-spin gen-erators areJa=sa

A+saB. This representation splits into two

irreps, the one-dimensional singlet representation withJ=0and the three-dimensional triplet representation withJ=1.The generalized unentangled states relative to this represen-tation ofsus2d are those for which there exists ana such thatthe state is a ±1 eigenstate ofJa. With respect to the CSAc=hJzj, those are the statesu↑ , ↑ l , u↓ , ↓ l, which are alsoGCSs(with purity equal to 1). No generalized unentangledstate is contained in the singlet irrep. In particular, neither thespin-zero state in the triplet, nor that which spans the singlet,is generalized unentangled(they both have purity equal to 0),nor are they on highest-weight orbits(thus GCSs).

As another example, consider a single spin-1 system,whose state spaceH=C3 carries an irrep ofsus2d [11]. Inthis case, for any choice of spin direction(say z) only theJz= ±1 eigenstates are generalized coherent. There is also aone-dimensionalJz=0 eigenspace. The maximal-purity statesare also the highest-weight states; however, the pureJz=0eigenstate is not a GCS, has zero purity, and is generalizedentangled. If, for the same system, a distinguished algebrasos2d generated byJz alone is chosen, then the representationreduces as the direct sum of the three invariant one-dimensional subspaces corresponding toJz=1,0,−1. Inthiscase, three different orbits exist in the representation, each ofthem consisting of only one state up to phases. However,only the states withuJzu=1 are extremal, whereas the statewith Jz=0 is not: as one can easily verify from the fact thatthe reduced state is now just the expectation value ofJz, anequal mixture of aJz=1 and aJz=−1 state has the samereduced state as aJz=0 state, so the latter remains, as in theirreducible case, generalized entangled.

A generalization of the latter example, which is relevantto the LMG model we will study in Sec. V, is the case of aspin-J system with a distinguished Abelian subalgebra gen-erated byJz. Again, one can see that only the states withmaximal magnitude ofJz are generalized unentangled, andonly they have maximal purity.

By definition, note that the relative purity and the invari-ant uncertainty functionals as defined in the previous sectionrelate to each other via

sDFd2 = kC2l − Ph, s8d

whereC2 denotes the quadratic Casimir invariant of the Liealgebra andPh is given by Eq.(5) (prior to rescaling). Be-cause, by standard representation theory,C2=c,1, with c,


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PR within each irrep, relative purity and invariant uncer-tainty essentially provide the same information ifh acts irre-ducibly. This, however, is no longer true in general in thereducible case. In the above two-spin-1

2 example, for in-stance, the two measures agree on the singlet sector; for trip-let states,JsJ+1d=2, thus the invariant uncertainty value is 1(same asPh) for uJzu=1 (generalized unentangled) states,whereas it yields 2 for the(zero-purity) state withJz=0 inthe triplet sector.

C. Extension to mixed states

For mixed states onH, the direct generalization of thesquared length of the projection ontoh as in Eq.(2) doesnotgive a GE measure with well-defined monotonicity proper-ties under appropriate generalizations of the LOCC semi-group of transformations[15]. A proper extension of the qua-dratic purity measure defined in the previous section for purestates to mixed states may be naturally obtained via a stan-dard convex roof construction. Ifr=ospsucslkcsu, withosps=1 andosps

2,1, the latter is obtained by calculating themaximum h-purity (minimum entanglement) over all pos-sible convex decompositionshps, ucslj of the density operatorr as a pure-state ensemble. In general, similarly to what hap-pens for most mixed-state entanglement measures, the re-quired extremization makes the resulting quantity very hardto compute.

While a more expanded discussion of mixed-state GEmeasures is given in[15], we focus here on applying thenotion of GE to characterize QPTs in different lattice sys-tems. Because the latter take place in the limit of zero tem-perature, the ground state of the system may be assumed tobe pure under ideal conditions. Accordingly, Eq.(7) will suf-fice for our current purposes.

III. RELATIVE PURITY AS A MEASURE OFENTANGLEMENT IN DIFFERENT QUANTUM SYSTEMS

We now apply the concept of relative purity to differentphysical systems in order to understand its meaning as ameasure of entanglement for pure quantum states. First, wewill concentrate on spin systems, showing that for particularsubsets of observables, theh-purity can be reduced to theusual notion of entanglement: the pure quantum states thatcan be written as a product of states of each party will begeneralized unentangled. However, for other physically natu-ral choices of observable sets, this is no longer the case.Next, we study theh-purity as a measure of entanglement forfermionic systems, since this is a good starting point for theanalysis of the QPT present in the anisotropicXY model in atransverse magnetic field(Sec. IV). In particular, we showthat if a fermionic state can be represented as a single Slaterdeterminant, it is generalized unentangled relative to the LiealgebrausNd, which is built from bilinear products of fermi-onic operators. These examples illustrate how the conceptand measure of GE is applicable to systems described bydifferent operator languages, in preparation for the study ofQPTs.

Let us introduce the following representative quantumstates forN spins of magnitudeS:

uFSNl = uS,S,…,Sl,

uWSNl =

1ÎN

oi=1

N

uS,…,S,sS− 1di,S,…,Sl,

uGHZSNl =

1Î2S+ 1

ol=0

2S

uS− l,S− l,…,S− ll, s9d

where the product stateuS1,S2,… ,SNl= uS1l1 ^ uS2l2 ^ ¯

^ uSNlN, anduSili denotes the state of theith party with thezcomponent of the spin equal toSi (defining the relevant com-putational basis for theith subsystem).

A. Two-spin systems

For simplicity, we begin by studying the GE of a two-qubit system(two-spin-12), where the most general purequantum state can be written asucl=au 1

2 , 12l+bu 1

2 ,−12l

+cu−12 , 1

2l+du−12 ,−1

2l, with the complex numbersa,b,c, andd satisfyinguau2+ ubu2+ ucu2+ udu2=1. The traditional measuresof pure-state entanglement in this case are well understood,indicating that the Bell statesuGHZ1/2

2 l [36] (and its localspin rotations) are maximally entangled with respect to thelocal Hilbert space decompositionH1 ^ H2. On the otherhand, calculating the purity relative to the(irreducible) Liealgebra of all local observables h=sus2d1 % sus2d2

=hsai ; i :1 ,2 ;a=x,y,zj classifies the pure two-spin-1

2 statesin the same way as the traditional measures do(see Fig. 1).Here, the operatorssa

1 =sa ^ 1 andsa2 =1^ sa are the Pauli

operators acting on spin 1 and 2, respectively, and

1 =S1 0

0 1D, sx = S0 1

1 0D ,

sy = S0 − i

i 0D, sz = S1 0

0 − 1D , s10d

in the basis whereu+1/2l= u↑ l=s1

0 d andu−1/2l= u↓ l=s0

1 d. In

this case, Eq.(7) simply gives

Phsucld = 12o

i,aksa

i l2, s11d

where Bell’s states are maximally entangledsPh=0d andproduct states of the formucl= uf1l1 ^ uf2l2 (GCSs of thelocal algebrah above) are generalized unentangled, withmaximum purity. Therefore, the normalization factorK= 1

2may be obtained by settingPh=1 in such a product state. Asexplained in Sec. II,Ph is invariant under group operations,i.e., in this case, local rotations. Since all GCSs ofh belongto the same orbit generated by the application of group op-erations to a particular product state(a reference state likeu 12 , 1

2l= u↑ , ↑ l), they all consistently have maximumh-puritysPh=1d.


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Another important insight may be gained by calculatingthe purity relative to the algebra ofall observables for thesystem,h=sus4d=hsa

i ,sa1

^ sb2 ; i =1,2;a ,b=x,y,zj in this

case. One finds thatany two-spin-12 pure stateucl (includingBell’s states) is then generalized unentangled(Ph=1, see alsoFig. 1). This property is a manifestation of the relative natureof GE, as considering the set of all observables as beingphysically accessible is equivalent to not making any pre-ferred subsystem decomposition. Accordingly, in this caseany pure quantum state becomes a GCS ofsus4d.

In Fig. 1, we also show the GE for systems of two partiesof spin S relative to different algebras. We observe that thepurity reduces again to the traditional concept of entangle-ment for higher spin if it is calculated relative to the(irre-ducible) Lie algebra ofall local observablesh=sus2S+1d1

% sus2S+1d2. For example, if we are interested in distin-guishing product states from entangled states in a two-spin-1system, we need to calculate the purity relative to the(irre-ducible) algebrah=sus3d1 % sus3d2=hla

1^ 12,11 ^ la

2s1øaø8dj, where the 333 Hermitian and traceless matricesli

are the well-known Gell-Mann matrices[27],

l1 = 1Î210 1 0

1 0 0

0 0 02, l2 = 1

Î210 − i 0

i 0 0

0 0 02 ,

l3 = 1Î211 0 0

0 − 1 0

0 0 02, l4 = 1

Î210 0 1

0 0 0

1 0 02 ,

l5 = 1Î210 0 − i

0 0 0

i 0 02, l6 = 1

Î210 0 0

0 0 1

0 1 02 ,

l7 = 1Î210 0 0

0 0 − i

0 i 02, l8 = 1

Î611 0 0

0 1 0

0 0 − 22 ,

which satisfy Trflalbg=da,b. In this basis, the computationalspin-1 states are represented by the three-dimensional vec-tors

u1l = 11

0

02, u0l = 10

1

02, and u− 1l = 10

0

12 . s12d

Then, the relative purity for a generic pure stateucl becomes

Phsucld =3

4oa=1

8

oi=1

2

klai l2, s13d

whereklai l denotes the expectation value ofla

i in the stateucl. In this way, product states likeucl= uf1l1 ^ uf2l2 aregeneralized unentangledsPh=1d and states likeuGHZ1

2l (andstates connected through local spin unitary operations) aremaximally entangled in this algebrasPh=0d.

Different results are obtained if the purity is calculatedrelative to asubalgebra of local observables.For example,the two-spin-1 product stateu0,0l= u0l ^ u0l where both spins

FIG. 1. Purity relative to different possible al-gebras for a two-spin-S system. The quantumstatesuGHZS

2l and uFS2l are defined in Eqs.(9).


042311-6

have zero projection alongz becomes generalized entangledrelative to the(irreducible) algebrasus2d1 % sus2d2 of localspin rotations, which is generated byhSa

i ; i :1 ,2 ;a=x,y,zj,the spin-1 angular momentum operatorsSa for each spinbeing given by

Sx =1Î210 1 0

1 0 1

0 1 02 ,

Sy =1Î210 − i 0

i 0 − i

0 i 02 ,

Sz = 11 0 0

0 0 0

0 0 − 12 . s14d

Notice that access to local angular momentum observablessuffices to operationally characterize the system as describ-able in terms of two spin-1 particles(by imagining, for in-stance, performing a Stern-Gerlach type of experiment oneach particle). Thus, even when a subsystem decompositioncan be naturally identified from the beginning in this case,states which are manifestly separable(unentangled) in thestandard sense may exhibit GE(see also Appendix I). On theother hand, this is physically quite natural in the example,since there are no SUs2d3SUs2d group operations(local ro-tations) that are able to transform the stateu0,0l into theunentangled product stateu1,1l.

The examples described in this section together with otherexamples of states of bipartite quantum systems are shown inFig. 1. It is clear that calculating the purity relative to differ-ent algebras gives information about different types of quan-tum correlations present in the system.

B. N-spin systems

The traditional concept of pure multipartite entanglementin an N-spin-S quantum system refers to quantum states thatcannot be written as a product of states of each party. Theh-purity distinguishes pure product states from entangledones if it is calculated relative to the(irreducible) algebra oflocal observablesh= % i=1

N sus2S+1di (see Appendix I). ByEq. (6), the measurePh is invariant under local unitary op-erations as desired. In particular, the usual concept of en-tanglement in anN-qubit quantum state(N-spin-12) can berecovered if the purity is calculated relative to the local al-gebrah= % i=1

N sus2di =hsx1,sy

1,sz1,… ,s x

N,s yN,s z

Nj, where thePauli operatorss a

i sa=x,y,zd are now

s15d

and the 232 matricessa and 1 are given in Eq.(10). Then,the local purity becomes

Phsucld =1

No

a=x,y,zoi=1

N

ksai l2, s16d

where again the normalization factor 1/N is obtained by set-ting Ph=1 in any product state likeucl= uf1l1 ^ uf2l2 ^ ¯

^ ufNlN (a GCS in this algebra). With this definition, stateslike uGHZ1/2

N l, fsu↑ , ↓ l− u↓ , ↑ ld /Î2g^n (with obvious nota-tions), and the cluster statesuFlC introduced in Ref.[37] (seealso Appendix II), will be maximally entangledsPh=0d.

Remarkably, as announced in[11], after some algebraicmanipulations(see Appendix III), one can prove that

Phsucld = 1 −Qsucld, s17d

whereQ is the (pure-state) measure ofglobal entanglementfor N-spin-12 systems originally introduced by Meyer andWallach in [38]. A similar relation was independently de-rived in [39]. See also[40] for additional related consider-ations.

In Fig. 2, we display some examples of the purity relativeto the local algebrah= % i=1

N sus2di for anN-spin-Ssystem. Wealso show the purity relative to the algebra of all observables

FIG. 2. Purity relative to different algebras for anN-spin-S sys-tem. The quantum statesuGHZS

Nl, uWSNl, and uFS

Nl are defined inEqs.(9).


042311-7

susf2S+1gNd, where any pure quantum state is a GCS, thusgeneralized unentangledsPh=1d.

C. Purity relative to the u„N… algebra

We now apply the concept of GE to a physical systemconsisting ofN (spinless) fermion modesj , each mode beingdescribed in terms of canonical creation and annihilation op-eratorscj

† andcj, respectively, satisfying the following anti-commutation rules:

hci†,cjj = di,j, hci,cjj = 0. s18d

For instance, different modes could be associated with dif-ferent sites in a lattice, or to delocalized momentum modesrelated to the spatial modes through a Fourier transform. Ingeneral, for anyN3N unitary matrixU, any transformationcj °o jUijcj maps the original modes into another possibleset of fermionic modes. Using the above commutation rela-tions, one also finds that

fci†cj,ck

†clg = d jkci†cl − dilck

†cj . s19d

Thus, the set of bilinear fermionic operatorshcj†cj8 ; 1

ø j , j8øNj provides a realization of the unitary Lie algebrausNd in the 2N-dimensional Fock spaceHFock of the system.The latter is constructed as the direct sum of subspacesHncorresponding to a fixed fermion numbern=0,… ,N, withdimsHnd=N! / fn! sN−nd ! g. For our purposes, it is conve-nient to expressusNd as the linear span of a Hermitian, or-thonormal operator basis, which we choose as follows:

usNd = 5scj†cj8 + cj8

† cjd with 1 ø j , j8 ø N

iscj†cj8 − cj8

† cjd with 1 ø j , j8 ø N

Î2scj†cj − 1/2d with 1 ø j ø N

6 s20d

(we use henceforth the notational convention that the largeleft curly bracket means “is the span of”). The action ofusNdon HFock is reducible, because any operator inusNd con-serves the total number of fermionsn=ko j=1

N cj†cjl. It turns out

that the irrep decomposition ofusNd is identical to the directsum into fixed-particle-number subspacesHn, each irrep thusappearing with multiplicity 1.

Using Eq. (7), the h-purity of a generic pure many-fermion state relative tousNd becomes

Phsucld =2

No

j, j8=1

N

fkcj†cj8 + cj8

† cjl2 − kcj†cj8 − cj8

† cjl2g

+4

Noj=1

N

kcj†cj − 1/2l2. s21d

Here, we tookK=2/N, for reasons that will become clearshortly. In this case, the fermionic product states(Slater de-terminants) of the form ufl=plcl

†uvacl, with uvacl denotingthe reference state with no fermions andl labeling a particu-lar set of modes, are the GCSs of theusNd algebra[31,32].Because a Slater determinant carries a well-defined numberof particles, each GCS belongs to an irrep spaceHn for some

n, states with differentn belonging to different orbits underusNd. A fixed GCS has maximumh-purity when compared toany other state within the same irrep space. Remarkably, italso turns out that any GCS ofh=usNd gives rise to a re-duced state which is extremal(thus generalized unentangled)regardless ofn, theh purity assuming the same(maximum)value in each irrep. Using this property, the normalizationfactorK=2/N was calculated by settingPh=1 in an arbitrarySlater determinant. Thus, the purity relative to theusNd al-gebra is a good measure of entanglement in fermionic sys-tems, in the sense thatPh=1 in any fermionic product state,and Ph,1 for any other state, irrespective of whether thelatter has a well-defined number of fermions or not. Noticethat, thanks to the invariance ofPh under group transforma-tions [Eq. (6)], the property of a state being generalized un-entangled is independent of the specific set of modes that ischosen. This is an important difference between our GE andthe mode entanglement approach[25,41].

IV. ENTANGLEMENT AND QUANTUM PHASETRANSITIONS

As already mentioned, although many measures of en-tanglement have been defined in the literature, assessing theirability to help us better understand QPTs in quantum systemsremains largely an open problem. In the following two sec-tions, we attempt to characterize the QPTs present in theLMG model and in the anisotropicXY model in an externalmagnetic field through the GE notion, relative to a particularsubset of observables which will be appropriately chosen ineach case. Interestingly, for both of these models the groundstates can be computed exactly by mapping the set of observ-able operators involved in the system Hamiltonian to a newset of operators which satisfy the same commutation rela-tions, thus preserving the underlying algebraic structure. Inthe new operator language, the models are seen to containsome symmetries that make them exactly solvable, allowingone to obtain the ground-state properties in a number of op-erations that scales polynomially with the system size(seealso [42] for related discussions). It is possible then to un-derstand which quantum correlations give rise to the QPTs inthese cases.

Several issues should be considered when looking for analgebrah of observables that may make the correspondingrelative purity a good indicator of a QPT. A first relevantobservation is that in each of these cases, a preferred Liealgebra exists, where the respective ground state would havemaximumh-purity independently of the interaction strengthsin the Hamiltonian. The purity relative to such an algebraremains constant, therefore it does not identify the QPT.(Inthese cases, this algebra is in fact the Lie algebra generatedby the parametrized family of model Hamiltonians, as theparameters are varied.) Thus, one needs to extract a subalge-bra relative to which the ground state may be generalizedentangled, depending on the parameters in the Hamiltonian.A second, closely related observation is that the purity mustcontain information about quantum correlations which un-dergo a qualitative change as the transition point is crossed:thus, the corresponding degree of entanglement, as measured


042311-8

by the purity, must depend on the interaction strengths gov-erning the phase transition. Finally, whenever a degeneracyof the ground state exists or emerges in the thermodynamiclimit, a physical requirement is that the purity be the samefor all ground states.

Although these restrictions together turn out to be suffi-cient for choosing the relevant algebra of observables in thefollowing two models, they do not provide an unambiguousanswer when solving a nonintegrable model whose exactground-state solution cannot be computed efficiently. Typi-cally, in the latter cases the ground states are GCSs of Liealgebras each of whose dimension increases exponentiallywith the system size. Choosing the observable subalgebrathat contains the proper information on the QPTs(such asinformation on critical exponents) then becomes, in general,a difficult task.

On the other hand, a concept ofgeneralized mean-fieldHamiltonian emerges from these considerations. Given aHilbert spaceH of dimensionpN (with p an integer.1), wewill define a mean-field Hamiltonian as an operator

HMF = oa

eaAa, ea P R, s22d

that is an element of an irreducibly represented Lie algebraof Hermitian operatorsh=hA1,… ,ALj whose dimensionscales polynomially inN, that is, L=polysNd. When theground state of such anHMF is nondegenerate, it turns out tobe a GCS ofh [15], while the remaining eigenstates(some ofwhich may also be GCSs) and energies can be efficientlycomputed. The connection between Lie-algebraic mean-fieldHamiltonians and their efficient solvability deserves a carefulanalysis in its own right, which we will present elsewhere[43].

V. LIPKIN-MESHKOV-GLICK MODEL

Originally introduced in the context of nuclear physics[21], the Lipkin-Meshov-Glick(LMG) model is widely usedas a testbed for studying critical phenomena in(pseudo)spinsystems[31]. This model was shown to be exactly-solvablein Ref. [22]. In this section, we investigate the critical prop-erties of this model by calculating the purity relative to aparticular subset of observables, which will be chosen byanalyzing theclassicalbehavior of the ground state of thesystem. For this purpose, we first need to map the model to asingle spin, where it becomes solvable and where the stan-dard notion of entanglement is not immediately applicable.

The model is constructed by consideringN fermions dis-tributed in twoN-fold degenerate levels(termed upper andlower shells). The latter are separated by an energy gape,which will be set here equal to 1. The quantum numbers= ±1s↑or↓ d labels the level while the quantum numberkdenotes the particular degenerate state in the level(forboth shells, kP hk1,… ,kNj). In addition, we consider a“monopole-monopole” interaction that scatters pairs of par-ticles between the two levels without changingk. The modelHamiltonian may be written as

H = H0 + V + W

=1

2ok,s

scks† cks +

V

2No

k,k8,s

cks† ck8s

† ck8scks

+W

2No

k,k8,s

cks† ck8s

† ck8scks, s23d

where s=−s, and the fermionic operatorscks† scksd create

(annihilate) a fermion in the level identified by the quantumnumberssk,sd and satisfy the fermionic commutation rela-

tions given in Sec. III C. Thus, the interactionV scatters apair of particles belonging to one of the levels, and the inter-

action W scatters a pair of particles belonging to differentlevels. Note that the factor 1/N must be present in the inter-action terms for stability reasons, as the energy per particlemust be finite in the thermodynamic limit.

Upon introducing the pseudospin operators

J+ = ok

ck↑† ck↓, s24d

J− = ok

ck↓† ck↑, s25d

Jz = 12o

k,sscks

† cks = 12sn↑ − n↓d, s26d

which satisfy thesus2d commutation relations of the angularmomentum algebra,

fJz,J±g = ± J±, s27d

fJ+,J−g = Jz, s28d

the Hamiltonian of Eq.(23) may be rewritten as

H = Jz +V

2NsJ+

2 + J−2d +

W

2NsJ+J− + J−J+d. s29d

As defined by Eq.(29), H is invariant under theZ2 inversionsymmetry operationK that transformssJx,Jy,Jzd° s−Jx,−Jy,Jzd, and it also commutes with the(Casimir) total angu-lar momentum operatorJ2=Jx

2+Jy2+Jz

2. Therefore, the nonde-generate eigenstates ofH are simultaneous eigenstates ofboth K and J2, and they may be obtained by diagonalizingmatrices of dimension 2J+1 (whereby the solubility of themodel). Notice that, by definition ofJz as in Eq.(26), themaximum eigenvalue ofJz and J= uJu is N/2. In particular,for a system withN fermions as assumed, both the groundstateugl and first excited stateuel belong to the largest pos-sible angular momentum eigenvalueJ=N/2 [21] (so-calledhalf-filling configurations); thus, they can be computed bydiagonalizing a matrix of dimensionN+1.

The Hamiltonian(29) does not exhibit a QPT for finiteN.It is important to remark that some critical properties of theLMG model in the thermodynamic limitN→` can be un-derstood by using a semiclassical approach[44] (note thatthe critical behavior is essentially mean-field): first, we re-place the angular momentum operators inH /N [with H


042311-9

given in Eq.(29)] by their classical components(Fig. 3); thatis,

J = sJx,Jy,Jzd → sJ sinu cosf,J sinu sinf,J cosud,

s30d

H/N → hcs j ,u,fd, s31d

wherehc is the resulting classical Hamiltonian andj =J/N,j =0,… , 1

2. In this way, one can show that in the thermody-namic limit (see Appendix IV)

limN→`

kguHuglN

= limN→`

Eg

N= min

j ,u,fhcs j ,u,fd, s32d

so the ground-state energy per particleEg/N can be easilyevaluated by minimizing

hcs j ,u,fd = j cosu +V

2j2 sin2 u coss2fd + Wj2 sin2 u.

s33d

As mentioned, the ground and first excited states have maxi-mum angular momentumj = 1

2. In Fig. 4, we show the orien-tation of the angular momentum in the ground states of theclassical Hamiltonianhc, represented by the vectorsJ ,J1,and J2, for different values ofV and W. When D= uVu−Wø1, we haveu=p and the classical angular momentum isoriented in the negativez direction. However, whenD.1 wehave cosu=−D−1 and the classical ground state becomestwofold degenerate(notice that hc is invariant under thetransformationf°−f). In this region and forV,0 the an-gular momentum is oriented in thexzplanesf=0d, while forV.0 it is oriented in theyz plane sf= ±p /2d. The modelhas a gauge symmetry in the lineV=0, W,−1, wheref cantake any possible value.

A. First- and second-order QPTs, and critical behavior

Going back to the original Hamiltonian of Eq.(23), thequantum system undergoes a second-order QPT at the criti-cal boundaryDc= uVcu−Wc=1, where forD.Dc the ground

and first excited statesugl and uel become degenerate in thethermodynamic limit and the inversion symmetryK breaks.The order parameter is given by the mean number of fermi-ons in the upper shellkn↑l=1/2+kJzl /N, which in the ther-modynamic limit converges to its classical value,

limN→`

kn↑l =1 + cosu

2. s34d

Obviously, for DøDc we havekn↑l=0, andkn↑l.0 other-wise (see Fig. 4). The critical exponents of the order param-eter are easily computed by making a Taylor expansion nearthe critical pointssD→1+d. Defining the quantitiesx=Vc

−V andy=Wc−W, we obtain

limD→1+

kn↑l = Hsya − xb d/2 for V . 0

sya + xb d/2 for V , 0,J

where the critical exponents area=1 andb=1.In Fig. 5, we show the exact ground-state energy per par-

ticle Eg/N (with Eg=kguHugl) as a function ofV andW in thethermodynamic limit[Eqs.(32)]. One can see that also in thebroken symmetry regionsD.1d the system undergoes afirst-order QPT atV=0; that is, the first derivative of theground-state energy with respect toV is not continuous inthis line.

B. Purity as an indicator of the QPTs in the LMG model

The standard notion of entanglement is not directly appli-cable to the LMG model as described by Eq.(29), for this isa single spin system and no physically natural partition intosubsystems is possible. Therefore, using theh-purity as ameasure of entanglement becomes an advantage from thispoint of view, since the latter only depends on a particular

FIG. 3. Angular momentum coordinates in the three-dimensional space.

FIG. 4. Representation of the classical ground state of the LMGmodel.


042311-10

subset of observables and no partition of the system is nec-essary. The first required step is the identification of a rel-evant Lie algebra of observables relative to which the purityhas to be calculated.

Since both the ground and first excited states of the quan-tum LMG model may be understood as states of a systemcarrying total angular momentumJ=N/2, a first possible al-gebra to consider is thesusN+1d algebra acting on the rel-evant sN+1d-dimensional eigenspace. Relative to this alge-bra, ugl is generalized unentangled for arbitrary values ofV,W, thus the corresponding purity remains constant and doesnot signal the QPTs. However, the family of Hamiltonians(29) do not generate this Lie algebra, but rather ansus2dalgebra, so perhapssusN+1d is not a natural choice physi-cally [45].

Thus, a natural choice suggested by the commutation re-lationships of Eqs.(27) and (28), is to study the purity rela-tive to the spin-N/2 representation of the angular momentumLie algebrah=sus2d=hJx,Jy,Jzj,

Phsucld =4

N2fkJxl2 + kJyl2 + kJzl2g , s35d

where the normalization factorK=N2/4 is chosen to ensurethat the maximum ofPh is equal to 1. With this normaliza-tion factor, Ph can be calculated exactly in the thermody-namic limit by relying on the semiclassical approach de-scribed earlier[see Appendix IV and Eq.(30)]. For V=0 andarbitrary W.0, ugl= uJz=−N/2l, which is a GCS ofsus2dand hasPh=1. For generic interaction values such thatDø1, the classical angular momentum depicted in Fig. 4 isoriented along thez direction and is not degenerate: becausekJxl=kJyl=0, only kJzl contributes toPh; by recalling thatlimN→`kJz/Nl=−1

2, this givesPh=1, so thatas far as relativepurity is concerned, the ground state behaves asymptoticallylike a coherent state in the thermodynamic limit.Physically,this means that with respect to the relevant fluctuations,GCSs ofsus2d are a good approximation of the quantumground state for large particle numbers, as is well establishedfor this model[46]. However, in the regionD.1, the groundstate(both classical and quantum) is twofold degenerate in

the N→` limit, and the value ofPh depends in general onthe particular linear combination of degenerate states. Thiscan be understood from Fig. 4, where different linear com-binations of the two degenerate vectorsJ1 andJ2 imply dif-ferent values ofkJxl for V,0 and different values ofkJyl forV.0, while kJzl remains constant. With these features, thepurity relative to thesus2d algebra will not be a good indi-cator of the QPT.

An alternative option is then to look at a subalgebra ofsus2d. In particular, if we only consider the purity relative tothe single observableh=sos2d=hJzj [i.e., a particular CSA ofsus2d], and retain the same normalization as above, we have

Phsucld =4

N2kJzl2. s36d

This new purity will be a good indicator of the QPT, sincePh=1 only for Dø1 in the thermodynamic limit, and inaddition Ph does not depend on the particular linear combi-nation of the twofold-degenerate states in the regionD.1,wherePh,1. Obviously, in this casePh is straightforwardlyrelated to the order parameter[Eq. (34)]; the critical expo-nents ofPh−1 are indeed the same(a=1 andb=1).

Note that the purity defined by Eq.(36) does not alwaystake its maximum value for GCSs ofh=sos2d (eigenstates ofJz). In the regionD,1 wherePh=1, the quantum groundstate of the LMG model[Eq. (29)] does not have a well-definedz component of angular momentum except atV=0(fH ,JzgÞ0 if VÞ0), thus in general it does not lie on acoherent orbit of this algebra for finiteN. However, as dis-cussed above, it behaves asymptotically(in the infinite-Nlimit ) as a GCS(in the sense thatPh→1). Moreover, in Sec.II we showed that forJz eigenstates with eigenvaluesuJzu,N/2, we also obtainPh,1.

In Fig. 6, we show the behavior ofPh as a function of theparametersV andW. Interestingly, the purity relative toJz isa good indicator not only of the second-order QPT but alsoof the first-order QPT(the lineV=0, W,−1).

FIG. 5. Ground-state energy per particle in the LMG model. FIG. 6. Purity relative to the observableJz in the ground state ofthe LMG model.


042311-11

VI. ANISOTROPIC XY MODEL IN A TRANSVERSEMAGNETIC FIELD

In this section, we exploit the purity relative to theusNdalgebra(introduced in Sec. III C) as a measure able to iden-tify the paramagnetic to ferromagnetic QPT in the aniso-tropic one-dimensional spin-1

2 XY model in a transversemagnetic field and classify its universality properties.

The model Hamiltonian for a chain ofN sites is given by(see Fig. 7)

H = − goi=1

N

fs1 + gds xi s x

i+1

+ s1 − gds yi s y

i+1g + oi=1

N

s zi , s37d

where the operatorssai sa=x,y,zd are the Pauli spin-12 opera-

tors on sitei [defined in Eqs.(10) and(15)], g is the param-eter one may tune to drive the QPT, and 0,gø1 is theamount of anisotropy in thexy plane. In particular, forg=1 Eq. (37) reduces to the Ising model in a transverse mag-netic field, while for g→0 the model becomes isotropic.Periodic boundary conditions were considered here, that is,sa

i+N=s ai , for all i anda.

Wheng@1 andg=1, the model is Ising-like. In this limit,the spin-spin interactions are the dominant contribution tothe Hamiltonian(37), and the ground state becomes degen-erate in the thermodynamic limit, exhibiting ferromagneticlong-range-order correlations in thex direction: Mx

2

= limN→`ksx1s x

N/2l.0, whereMx is the magnetization in thex direction. In the opposite limit whereg→0, the externalmagnetic field becomes important, the spins tend to align inthez direction, and the magnetization in thex direction van-ishes:Mx

2= limN→`ksx1s x

N/2l=0. Thus, in the thermodynamiclimit the model is subject to a paramagnetic-to-ferromagneticsecond-order QPT at a critical pointgc that will be deter-mined later, with critical behavior belonging to the two-dimensional(2D) Ising model universality class.

This model can be exactly solved using the Jordan-Wigner transformation[47], which maps the Pauli(spin-12)algebra into the canonical fermion algebra through

cj† = p

l=1

j−1

s− szl ds +

j , s38d

where the fermionic operatorscj†scjd have been introduced in

Sec. III C ands +j =ss x

j + is yj d /2 is the raising spin operator.

In order to find the exact ground state, we first need towrite the Hamiltonian given in Eq.(37) in terms of thesefermionic operators,

H = − 2goi=1

N−1

sci†ci+1 + gci

†ci+1† + H.c.d

+ 2gKscN†c1 + gcN

†c1† + H.c.d + 2N, s39d

whereK=p j=1N s−s z

jd is an operator that commutes with the

Hamiltonian, andN=oi=1N ci

†ci is the total number operator(here, we chooseN to be even). Then, the eigenvalue ofK is

a good quantum number, and noticing thatK=eipN we obtainK= +1 s−1d whenever the(nondegenerate) eigenstate ofH isa linear combination of states with an even(odd) number offermions. In particular, the numerical solution of this modelin finite systems(with N even) indicates that the ground statehas eigenvalueK= +1, implying antiperiodic boundary con-ditions in Eq.(39).

The second step is to rewrite the Hamiltonian in terms ofthe fermionic operatorsck

†sckd, defined by the Fourier trans-form of the operatorscj

†scjd,

ck† =

1ÎN

oj=1

N

e−ikjcj†, s40d

where the setV of possiblek is determined by the antiperi-odic boundary conditions in the fermionic operators:

V = V+ + V− = F±p

N, ±

3p

N,…, ±

sN − 1dpN

G .

Therefore, we rewrite the Hamiltonian as

H + N = − 2okeV

s− 1 + 2g coskdck†ck

+ igg sinksc−k† ck

† + c−kckd. s41d

The third and final step is to diagonalize the Hamiltonianof Eq. (41) using the Bogoliubov canonical transformation

gk = ukck − ivkc−k† ,

g−k† = ukc−k

† − ivkck,

where the real coefficientsuk andvk satisfy the relations

uk = u−k, vk = − v−k anduk2 + vk

2 = 1, s42d

where

FIG. 7. Anisotropic one-dimensionalXY model in an externaltransverse magnetic fieldB.


042311-12

uk = cosSfk

2D, vk = sinSfk

2D , s43d

with fk given by

tansfkd =2gg sink

− 1 + 2g cosk. s44d

In this way, the quasiparticle creation and annihilation op-eratorsgk

† andgk satisfy the canonical fermionic anticommu-tation relations of Eq.(18), and the Hamiltonian may befinally rewritten as

H = okeV

jksgk†gk − 1/2d, s45d

wherejk=2Îs−1+2g coskd2+4g2g2 sin2 k is the quasiparti-cle energy. Since in generaljk.0, the ground state is thequantum state with no quasiparticles(BCS state[48]), suchthat gkuBCSl=0. Thus, one finds

uBCSl = pkeV+

suk + ivkck†c−k

† duvacl, s46d

whereuvacl is the state with no fermionssckuvacl=0d.Excited states with an even number of fermionssK=

+1d can be obtained applying pairs of quasiparticle creationoperatorsgk

† to the uBCSl state. However, one should bemore rigorous when obtaining excited states with an oddnumber of particles, sinceK=−1 implies periodic boundaryconditions in Eq.(39), and the new set of possiblek’s (wavevectors) is

V = F− p,…,−2p

N,0,

2p

N, ¯ ,

2sN − 1dpN

G(different fromV).

A. QPT and critical point

In Fig. 8, we show the order parameterMx

2= limN→`ksx1sx

N/2l as a function ofg in the thermody-namic limit and for different anisotropiesg [4]. We observethat Mx

2=0 for gøgc and Mx2Þ0 for g.gc, so the critical

point is located atgc= 12, regardless of the value ofg. The

value ofgc can also be obtained by settingjk=0 in Eq.(45),where the gap vanishes.

Notice that the Hamiltonian of Eq.(37) is invariant underthe transformation that mapsss x

i ;s yj ;s z

kd° s−s xi ;−s y

j ;s zkd

(Z2 symmetry), implying that ksxi l=0 for all g. However,

since in the thermodynamic limit the ground state becomestwofold degenerate, forg.gc it is possible to build up aground state where the discreteZ2 symmetry is broken, i.e.,ksx

i lÞ0. This statement can be easily understood if we con-sider the case ofg=1, where for 0øg,gc the ground statehas no magnetization in thex direction: Forg=0, the spinsalign with the magnetic field, while an infinitesimal spin in-teraction disorders the system andMx=0. On the other hand,

for g→` the states ug1l=s1/Î2dfu→ ,… , → l+ u← ,… ,← lg and ug2l=s1/Î2dfu→ ,… , → l− u← ,… , ← lg, withu→ l=s1/Î2dfu↑ l+ u↓ lg and u← l=s1/Î2dfu↑ l− u↓ lg, becomedegenerate in the thermodynamic limit, and a ground statewith ksx

i lÞ0 can be constructed from a linear combination.Remarkably, this paramagnetic–to-ferromagnetic QPT

does not exist in the isotropic limitsg=0d. In this case, theHamiltonian of Eq.(37) has a continuousus1d symmetry;that is, it is invariant under anyz rotation of the formexpfiuo jsz

jg. Since the model is one-dimensional, this sym-metry cannot be spontaneously broken, regardless of themagnitude of the coupling constants. Nevertheless, a simplecalculation of the ground state energy indicates a divergencein its second derivative at the critical pointgc=1/2, thus asecond order nonbroken symmetry QPT. Forg.gc all thespins(in the ground state) are aligned with the external mag-netic field, with total magnetization in thez direction Mz=o jksz

jl=−N, and the quantum phase is gapped. Forg,gc,the total magnetization in thez direction isMzù−N, the gapvanishes, and the quantum phase becomes critical(i.e., thespin-spin correlation functions decay with a power law), withan emergentus1d gauge symmetry[17]. Then, in terms offermionic operators[Eq. (39)], an insulator-metal(or super-fluid) -like second order QPT exists atgc for the isotropiccase, with no symmetry order parameter. It is a Lifshitz tran-sition.

B. u„N…-purity in the BCS state, and critical behavior

The uBCSl state of Eq.(46) is a GCS of the algebra ofobservablesh=sos2Nd, spanned by an orthonormal Hermit-ian basis which is constructed by adjoining to the basis ofusNd given in Eq. (20) the following set r of number-nonconserving fermionic operators:

FIG. 8. (Color) Order parameterMx2 in the thermodynamic limit

as a function ofg for different anisotropiesg. The critical point is atgc= 1

2.


042311-13

r =H scj†cj8

† + cj8cjd with 1 ø j , j8 ø N

iscj†cj8

† − cj8cjd with 1 ø j , j8 ø NJ sos2Nd = usNd % r. s47d

Then, theuBCSl state is generalized unentangled with re-spect to thesos2Nd algebra and its purityPh [Eq. (7)] con-tains no information about the phase transition:Ph

=1 ∀ g,g. Therefore, in order to characterize the QPT weneed to look at the possible subalgebras ofsos2Nd. A naturalchoice is to restrict to operators which preserve the totalfermion number, that is, to consider theusNd algebra definedin Sec. III C, relative to which theuBCSl state may becomegeneralized entangled.[Note that, as mentioned in Sec. III C,theusNd algebra can also be written in terms of the fermionicoperatorsck

† and ck, with k belonging to the setV.]In the uBCSl state,kck

†ck8lÞ0 only if k=k8, thus using Eq.(21) the purity relative toh=usNd is

PhsuBCSld =4

NokeV

kck†ck − 1/2l2

=4

NokeV

svk2 − 1/2d2, s48d

where the coefficientsvk can be obtained from Eqs.(43) and(44). In particular, forg=0 the spins are aligned with themagnetic field and the fully polarized uBCSlg=0= u↓ , ↓ ,¯ , ↓ l state is generalized unentangled in this limit[a GCS ofusNd with Ph=1]. In the thermodynamic limit, thepurity relative to theusNd algebra can be obtained by inte-grating Eq.(48),

PhsuBCSld =2

pE

0

2p

svk2 − 1/2d2dk, s49d

leading to the following result:

PhsuBCSld =51

1 − g2F1 −g2

Î1 − 4g2s1 − g2dG if g ø12

1

1 + gif g .

12 .6

s50d

Although this function is continuous, its derivative is not andhas a drastic change atg= 1

2, where the QPT occurs. More-over,Ph is minimum forg.

12, implying maximum entangle-

ment at the transition point and in the ordered(ferromag-netic) phase. Remarkably, forg.

12 and N→`, where the

ground state of the anisotropicXY model in a transversemagnetic field is twofold degenerate,Ph remains invariantfor arbitrary linear combinations of the two degeneratestates.

As defined, for largeg the purityPh approaches a constantvalue which depends ong. It is convenient to remove such adependence in the ordered phase by introducing a new quan-tity Ph8=Ph−f1/s1+gdg (shifted purity). We thus obtain

Ph8suBCSld = 5 g

1 − g2F1 −g

Î1 − 4g2s1 − g2dG if g ø12

0 if g .12 .6

s51d

The new functionPh8 behaves like adisorder parameterforthe system, being zero in the ferromagnetic(ordered) phaseand different from zero in the paramagnetic(ordered) one.The behavior ofPh8 as a function ofg in the thermodynamiclimit is depicted in Fig. 9 for different values ofg. In thespecial case of the Ising model in a transverse magnetic fieldsg=1d, one has the simple behaviorPh8= 1

2 −2g2 for gø12 and

Ph8=0 if g.12.

The critical behavior of the system is characterized by apower-law divergence of thecorrelation lengthe, which isdefined such that for g,

12, limui−j u→`uks x

i s xj lu

,exps−ui − j u /ed. Thus,e→` signals the emergence of long-range correlations in the ordered regiong.

12. Near the criti-

cal point sg→gc−d, the correlation length behaves ase,sgc

−gd−n, where n is a critical exponent and the valuen=1corresponds to the Ising universality class. Let the parameterl2=e−1/e. The fact that the purity contains information aboutthe critical properties of the model follows from the possi-

FIG. 9. (Color) Shifted purityPusNd8 of the uBCSl as a function ofg for different anisotropiesg, Eq.(51). PusNd8 behaves like a disorder

parameter for this model, sharply identifying the QPT atgc= 12.


042311-14

bility of expressingPh8 for g,12 as a function of the correla-

tion length,

Ph8suBCSld =g

1 − g2F1 +g

2gl2s1 − gd − 1G , s52d

where a known relation betweeng, g, andl2 has been ex-ploited [4]. Performing a Taylor expansion of Eq.(52) in theregion g→gc

−, we obtain Ph8,2sgc−gdn /g with n=1 andg.0 (Fig. 10). Thus, the name disorder parameter forPh8 isconsistent.

Some physical insight into the meaning of the ground-state purity may be gained by noting that Eq.(48) can bewritten in terms of the fluctuations of the total fermion op-

eratorN,

PhsuBCSld = 1 −2

NskN2l − kNl2d, s53d

where theuBCSl propertykck†ck8l=dk,k8vk

2 has been used. Ingeneral, the purity relative to a given algebra can be writtenin terms of fluctuations of observables[15]. Since fluctua-tions of observables are at the root of QPTs, it is not surpris-ing that this quantity succeeds at identifying the criticalpoint. Interestingly, by recalling thatPsos2NdsuBCSld=1, theusNd-purity can also be formally expressed as

PusNdsuBCSld = 1 − oAaPr

kAal2, s54d

where the sum only extends to the non-number-conservingsos2Nd generators belonging to the setr specified in Eq.(47). Thus, the purity is entirely contributed by expectationsof operators connecting differentusNd irreps, the net effectof correlating representations with a different particle num-ber resulting in the fluctuation of asingleoperator, given by

N=okck†ck. In Fig. 11, we show the probabilityVsnd of hav-

ing n fermions in a chain ofN=400 sites forg=1. We ob-

serve that forg.12, the fluctuations remain almost constant,

and so does the purity.Again, the isotropic casesg=0d is particular in the sense

that Ph=1 or Ph8=0, see Fig. 9, without identifying the cor-responding metal-insulator QPT. The reason is that in thislimit, the Hamiltonian of Eq.(39) contains only fermionicoperators that preserve the number of particles[i.e., HPusNd], and the ground state of the system is always a GCSof theusNd algebra. Therefore, in order to obtain informationabout this QPT, one should look into algebras other thanusNd, relative to which the ground state is generalized en-tangled. For example, in Sec. VI D we study the purity rela-tive to the local algebra of observables and in Fig. 13 weshow that it successfully identifies the QPT in the isotropiccase, being maximum forgøgc (thus implying generalizedunentanglement).

C. Comparison with concurrence

As mentioned, the critical behavior of theXY model in atransverse field has also been investigated by looking at vari-ous quantities related to the concurrence, which is intrinsi-cally a measure of bipartite entanglement. For a genericmixed stater of two qubits, the latter is calculated as[14]

Csrd = maxhl1 − l2 − l3 − l4,0j,

wherel1ù . . .ùl4 are the square roots of the eigenvalues ofthe matrixR=rr and r=sy ^ syr* sy ^ sy. The concurrencefor the reduced density operatorr,,m of two nearest-neighborqubitssu,−mu=1d and next-nearest-neighborsu,−mu=2d qu-bits on a lattice has been investigated in detail in Ref.[8].Since, thanks to translational invariance,r,,m depends on thequbit indexes only via their distance, we will use the notationCs1d, Cs2d for the resulting quantities as in[8]. While theresults reported in the above work agree nicely with the scal-ing behavior expected for this model, the emerging picturebased on concurrence cannot be regarded as fully satisfac-tory. As stressed in[8], the entanglement as quantified by thenearest-neighbor concurrence isnot directly an indicator ofthe QPT, in this model, showing maximum entanglement at a

FIG. 10. Scaling properties of the disorder parameter for aniso-tropy g=1. The exponentn=1 belongs to the Ising universalityclass.

FIG. 11. Distribution of the fermion number in theuBCSl statefor a chain ofN=400 sites and anisotropyg=1.


042311-15

point which is not related to the QPT. However, the deriva-tive ]Cs1d /]g of the concurrence with respect to the spin-spin coupling parameter can be seen to diverge logarithmi-cally at the critical point forg.0, and with a power law forthe isotropic case[Fig. 12], identifying the critical point inthis model. Such a divergence is not found when analyzing,at the isotropic point, other QPTs in models of interest, likethe one-dimensional anisotropic Heisenberg chain(see, forinstance, Ref.[51]). Therefore, it suggests that the identifi-cation of a critical point using concurrence could be a hardtask in general.

D. Purity of the BCS state relative to the local algebra

Finally, we have also investigated the behavior of the pu-rity of the uBCSl state relative to the algebra of local observ-ablesh= % i=1

N sus2di. Using Eq.(16), this is physically relatedto the total magnetizationMz

2 alongz. The resulting behavioris plotted in Fig. 13 as a function ofg andg. As explained inSec. III B, this is a measure of the usual notion of entangle-ment in theN-spin-12 system. In particular, theuBCSl state isunentangled forg→0 (where uBCSl,u−1

2l1 ^ ¯ ^ u−12lNd,

thus Ph→1 in this limit. Moreover, for g→` we haveuBCSl,uGHZ1/2

N l (up to local rotations), thus uBCSl be-comes maximally entangled, andPh→0.

Compared to the purity relative to theusNd algebra, thepurity relative toh= % i=1

N sus2di is not as good an indicator ofthe phase transition wheng.0, in the sense that it does notpresent(similar to the concurrence) any drastic change in itsbehavior. However, its derivative with respect to the spin-spin coupling parameter diverges at the critical point in thismodel[Fig. 14]. Only in the isotropic casesg=0d the purityrelative to the local algebra presents a drastic change at thecritical point (see Fig. 12). In this case, the operatorMz=s1/Ndko jsz

jl for g→gc+ scales as

Mz + 1 , sg − gcdx

with the exponent beingx=1/2. On theother hand, thisexponent can also be obtained from the purity relative to thelocal algebra, in the same limit:

1 − Ph , sg − gcdx.

Therefore, this measure of entanglement is also a good indi-cator of the QPT for the isotropic case.

VII. CONCLUSIONS

In this paper, we have explored the usefulness of general-ized entanglement(GE) for characterizing the broken(and

FIG. 12. Nearest-neighbor concurrence and its derivative for theuBCSl state as a function ofg in the isotropicXY model,g=0. Bothcurves correspond to the exact solution in the thermodynamic limit.The value of]Cs1d /]g below gc is also zero asCs1d (not shown). FIG. 13. (Color) Purity of the uBCSl state relative to the local

algebra % i=1N sus2di, as a function ofg for different anisotropies

gsgc= 12

d. The number of sitesN=400 as in Fig. 11.

FIG. 14. (Color) Derivative of the purity of theuBCSl staterelative to the local algebra as a function ofg for g=0.5 and dif-ferent lattice sizes.


042311-16

one case of nonbroken) symmetry quantum phase transitions(QPTs) present in different lattice systems. As we focused onsituations where the physically relevant observables form aLie algebra, a natural GE measure provided by the relativepurity of a state relative to the algebra has been used toidentify and characterize these transitions.

In Secs. III A and III B, we showed using several illustra-tive examples how the concept ofh-purity can be useful fordifferent spin systems, by encompassing the usual notion ofentanglement if the family of all local observables is distin-guished. In addition, the possibility to directly apply the GEnotion to arbitrary quantum systems, including indistinguish-able particles, was explicitly shown in Sec. III C, using fer-mionic systems as a relevant case study. Depending on thesubset of observables chosen, theh-purity contains informa-tion about differentn-body correlations present in the quan-tum state, allowing for a more general and complete charac-terization of entanglement. Finally, in Secs. V and VI weshowed that theh-purity successfully distinguishes betweenthe different phases present in two lattice systems, where thecritical points are characterized by a broken symmetry andthe usual notion of entanglement cannot be straightforwardlyapplied. As also discussed in Sec. IV, the most critical step isto determine which subset of observables may be relevant ineach case, since theh-purity must contain information aboutthe quantum correlations that play a dominant role in theQPT. In particular, the ground state of the two models weconsidered can be exactly calculated and the relevant quan-tum correlations in the different phases are well understood,thus choosing this subset of observables becomes relativelyeasy.

Applying these concepts to a more general case, wherethe ground state of the system cannot be exactly computed,can be done in principle by following the same strategy.However, determining in a systematic way the minimal sub-set of observablesh whose purity is able to signal and char-acterize the QPT, thereby providing the relevant correlations,requires an elaborate analysis. Even more interesting, per-haps, is the open question of finding the minimal number ofGE measures, possibly including measures of GE relative todifferent observable sets, needed to unambiguously charac-terize the universality class of a transition, obtaining all of itscritical exponents. Finally, a fascinating direction for furtherinvestigation is to explore the significance of the GE notionwithin topological quantum-information settings[50] and tounderstand what generalizations might be needed to handletopological QPTs.

ACKNOWLEDGMENTS

It is a pleasure to thank James Gubernatis, Leonid Gur-vits, Juan Pablo Paz, and Wojciech Zurek for discussions. Weacknowledge support from the U.S. DOE through ContractNo. W-7405-ENG-36. H.B. and L.V. gratefully acknowledgefinancial support from the Los Alamos Office of the Director.

APPENDIX A: SEPARABILITY, GENERALIZEDUNENTANGLEMENT, AND LOCAL PURITIES

Given a quantum systemS whose statesucl belong to aHilbert spaceH of dimension dimsHd=d, the purity relative

to the (real) Lie algebra of all traceless observablesh=susdd spanned by an orthogonal, commonly normalizedHermitian basishA1¯ALj, L=d2−1, is, according to Eq.(7),given by

Phsucld = Koa=1

L

kAal2. sA1d

The normalization factorK depends ond and is determinedso that the maximum purity value is 1. If TrsAaAbd=da,b (asfor the standard spin-1 Gell-Mann matrices), then K=d/sd−1d, whereas in the case TrsAaAbd=dda,b (as for ordinaryspin-12 Pauli matrices), K=1/sd−1d. Recall that any quantumstateuclPH can be obtained by applying a group operatorUto a reference stateurefl [a highest or lowest weight state ofsusdd]; that is,

ucl = Uurefl, sA2d

with U=eioataAa, and ta real numbers. Therefore, any quan-tum stateucl is a GCS ofsusdd, thus generalized unentangledrelative to the algebra of all observables:Phsucld=1 for allucl.

Let now assume thatS is composed ofN distinguishablesusbsytems, corresponding to a factorizationH= ^ j=1

N H j,with dimsH jd=dj, d=p jdj. Then the set of alllocal observ-ables onS becomesh=hloc= % jsusdjd. An orthonormal basiswhich is suitable for calculating the local purityPh may beobtained by considering a collection of orthonormal baseshAa1

j¯AaLj

j j, Lj =dj2−1, each acting on thej th subsystem,

that is,

sA3d

where 1j =1/Îdj. Then for any pure stateuclPH one maywrite

Phsucld = K8oj=1

N Foa j=1

Lj

kAa j

j l2G . sA4d

By letting h j =spanhAa jj be the Lie algebra of traceless Her-

mitian operators acting onH j alone, the above equation alsois naturally rewritten as

Phsucld = K8oj=1

N1

K jPh j

sucld, K j =dj

dj − 1. sA5d

The h j-purity Ph jmay be simply related to the conventional

subsystem purity. Letr j =TriÞ jshuclkcujd be the reduced den-sity operator describing the state of thej th subsystem. Be-cause the latter can be represented as


042311-17

r j =1

dj+ o

a j=1

Lj

kAa jlAa j

= oa j=1

Lj

kAa j

j lAa j, sA6d

one can also equivalently express Eq.(A4) as

Phsucld = K8oj=1

N FTrr j2 −

1

djG , sA7d

that is,Ph jsucld=sdjTrr j

2−1d / sdj −1d. Clearly, the maximumvalue of either Eq.(A5) or Eq. (A7) will be attained when,and only when, each of the conventional purities Trr j

2

=1↔Ph j=1 for all j , which allows determining the

K8-normalization factor as

K8 =1

oj

1

K j

=1

N − oj

1

dj

=1

NS1 −1

No

j

1

djD . sA8d

Accordingly,

Phlocsucld = max = 1↔ ucl = uf1l ^ ¯ ^ ufNl, sA9d

and the equivalence with the standard notions of separabilityand entanglement are recovered. Note that for the case ofNqubits considered in Sec. III B, the above value simplifies toK8=2/N, which in turn gives the purity expression of Eq.(16) once the standard unnormalized Pauli matrices are used(Aa j

j =s a j

j /Î2, thus removing the overall factor 2).

APPENDIX B: CLUSTER AND VALENCE BOND SOLIDSTATES ARE MAXIMALLY ENTANGLED

In Ref. [37], Briegel and Raussendorf introduced the so-called cluster states for a system ofN qubits in D spacedimensions which, in the computational basis, are expressedas

uFlC =1

2N/2 ^jPC

Su↑l j pgPG

szs j+gd + u↓l jD , sB1d

where C defines the clustersC,ZDd and g denotes somenearest-neighbor qubits in the cluster:G=h1j for D=1, G=hs1,0d ,s0,1dj for D=2, G=hs1,0,0d ,s0,1,0d ,s0,0,1dj forD=3, etc. We considersz

s j+gd;1 when j +g is not in C.The usual notion of entanglement, as applied to a cluster

state, is recovered when theh-purity is calculated relative to

the local algebrah= % jPCsus2d j (see Appendix I). For thispurpose, we first calculate the expectation valuesks a

j lC, witha=x,y,z. One can immediately realize thatks y

j lC=0,∀ j,sincesy

j is a Hermitian operator(i.e., ks yj lPR) that acting

on the j th qubit’s state(in the natural basis) introduces aphase factor ±i, and the coefficients of Eq.(B1) are all real.Moreover,ks z

jlC=0,∀ j, since the weight of every state of thenatural basis is the same in Eq.(B1). In other words, we havea linear combination of basis states where each single qubithas the same probability of pointing up or down. Finally, onecan also prove thatks x

j lC=0,∀ j. This can be done by usingthe eigenvalue equationsKjuFlC= ± uFlC, for the family of

operatorsKj =s xj pgPG s z

s j+gd, where G=Gø−G denotes theset of all nearest-neighbor qubits to thej th qubit. Therefore,ks x

j lC= ± ks xj KjlC= ± kpgPGsz

s j+gdlC. Again, since Eq.(B1) isa combination of all the states of the computational basiswith the same probability, we obtainks x

j lC=0. In this way,the h-purity [Eq. (16)] is Ph=0, and the cluster states aremaximally entangled relative to the local seth= % jPCsus2d j.

Another important class of spin states is the one definedby the so-calledvalence bond solid(VBS) states. Thesestates have been introduced in the context of Heisenberg-likemagnets, and have been recently revisited in the context ofquantum computation[49]. Their general form is

uFlVBS = pki,jl

sai†bj

† − bi†aj

†dMu0l, sB2d

where ki , jl represent nearest-neighbor bonds of aD-dimensional lattice of coordinationz, aj

† and bj† are

Schwinger-Wigner boson(creation) operators on sitejwhose relation tosus2d spin-S generators is

Sxj = 1

2saj†bj + bj

†ajd,

Syj = 1

2i saj†bj − bj

†ajd,

Szj = 1

2saj†aj − bj

†bjd, sB3d

with the constraintaj†aj +bj

†bj =2S, andM =2S/z. M being aninteger causes the possible values ofS to depend upon theconnectivity of the lattice, which is defined byz.

We start by showing that the bond operatorsai†bj

†−bi†aj

†

are invariant under global spin rotations. The Schwinger-Wigner boson operators transform as vectors for rotations

Saj†

bj†D → UjSaj

†

bj†DUj

† =1 cosu2

2eisu3+u1d/2 sin

u2

2eisu3−u1d/2

− sinu2

2e−isu3−u1d/2 cos

u2

2e−isu3+u1d/22Saj

†

bj†D sB4d


042311-18

under an arbitrary spin rotation on lattice sitej , defined by

Uj = eiu1Szjeiu2Sy

jeiu3Sz

j, UjUj

† = Uj†Uj = 1. sB5d

Then, we can use this result to prove that

UjUisai†bj

† − bi†aj

†dUi†Uj

† = ai†bj

† − bi†aj

†, sB6d

implying, for U†=p jUj†,

U†uFlVBS = uFlVBS. sB7d

Therefore,uFlVBS belongs to the singlet irrep of the totalspin Ja=o jSa

j (i.e., kJalVBS=0).We want to show now thatkSa

j lVBS=0, ∀ j . We first ob-serve thatkSz

jlVBS=0,∀ j , because the transformation thatmapsaj

†°bj† andbj

†°−aj† [i.e., a global spin rotation about

the y axis, settingu1=u3=0 andu2=p in Eq. (B4)] implieskaj

†ajlVBS=kbj†bjlVBS. Then, from the invariance under global

spin rotations and the singlet nature ofuFlVBS, we obtainkSx

j lVBS=0=kSyj lVBS , ∀ j . In other words, the purity relative

to the algebrah= % jsus2d j vanishes, meaning thatuFlVBS ismaximally generalized entangled relative to this algebra.

However, in order to make contact with the standard no-tion of entanglement(Appendix I), we need to address theGE relative to the algebrah= % jsus2S+1d j, that is, relative tothe set of all local observables. For simplicity, we only dis-cuss the 1D case forS=1 [i.e., M =1 in Eq. (B2)] but thereader could use the same techniques to obtain results inhigher dimensions and spin magnitudeS.

The algebrah= % jsus3d j =hSmnj j,

fSmm8j ,Snn8

j8 g = d j j 8sdm8n Smn8j − dmn8Snm8

j d, sB8d

can be written in terms of thesus2d generators as[17]

S00j = 2

3 − sSzjd2, S11

j =Sz

jsSzj + 1d2

− 13 ,

S10j =

1

2Î2fS+

j + hS+j ,Sz

jjg,

S01j =

1

2Î2fS−

j + hS−j ,Sz

jjg,

S20j =

1

2Î2fS−

j − hS−j ,Sz

jjg,

S02j =

1

2Î2fS+

j − hS+j ,Sz

jjg,

S12j =

i

2hSx

j ,Syj j + sSx

j d2 + 12sSz

jd2 − 1,

S21j =

1

2ihSx

j ,Syj j + sSx

j d2 + 12sSz

jd2 − 1, sB9d

with S±j =Sx

j ± iSyj . From the spin-rotational invariance of the

state uFlVBS, we get ksSxj d2lVBS=ksSy

j d2lVBS=ksSzjd2lVBS

=fSsS+1dg /3 and, sinceS=1, we obtainkS00j lVBS=kS11

j lVBS

=0. Moreover, the spin-rotational invariance also impliesthat kSa

j Sa8j lVBS remains the same constant∀ aÞa8. Then,

for example, applying a globalp rotation about they axis touFlVBS (i.e., the operation that mapsSz

j °−Szj and Sy

j °Syj )

we obtain kSyj Sz

jlVBS=−kSyj Sz

jlVBS=0, hence, kSmnj lVBS=0.

Therefore, the stateuFlVBS sS=1,M =1d is maximally en-tangled when using the standard notion of entanglement[Ph=0, for the algebra of all local observablesh= % jsus3d j].

APPENDIX C: RELATION BETWEEN PURITY IN THELOCAL ALGEBRA AND THE MEYER-WALLACH

MEASURE OF ENTANGLEMENT

In Ref. [38], Meyer and Wallach define a measure of en-tanglementQ for pure states of qubit systems that is invari-ant under local unitary operations(local rotations). For thispurpose, they first define the mappingl jsbd acting on productstates as

l jsbdub1,…,bNl = dbbjub1,…,bj,…,bNl, sC1d

whereb andbj are either the statesu 12l or u−1

2l, andbj denotesthe absence of thej th qubit. On the other hand, anyN-qubitspure quantum state can be written in the natural basis(zcomponent of the spin equal to ±1

2) as

ucl = oi=1

2N−1

fgiju 1

2l j + hiju− 1

2l jgufil, sC2d

wheregij andhi

j are complex coefficients, and the orthonor-mal statesufil of N−1 qubits(absence of thej th qubit) arealso written in the natural basis. Therefore, the action ofl jsbdon ucl is

l js 12ducl = o

i=1

2N−1

gijufil,

l js− 12ducl = o

i=1

2N−1

hijufil.

Then, they define the entanglementQsucld as

Qsucld =4

Noj=1

N

D„l js 12ducl,l js− 1

2ducl… , sC3d

where the distance between two quantum statesuul=ouiufil and uvl=oviufil is

Dsu,vd = 12o

i,juuiv j − ujviu2. sC4d

Therefore,


042311-19

D„l js 12ducl,l js− 1

2ducl… = 12o

i,i8

ugijhi8

j − gi8j hi

ju2

= oi,i8

fugiju2uhi8

j u2 − sgijhi8

j dshijgi8

j d* g,

sC5d

where * denotes complex conjugate. After some simple cal-culations, we obtain the following relations:

oi=1

2N−1

ugiju2 = kcuS1 + s z

j

2Ducl, sC6d

oi=1

2N−1

uhiju2 = kcuS1 − sz

j

2Ducl, sC7d

oi=1

2N−1

gijshi

jd* = kcus−j ucl, sC8d

and the distance becomesD(l js 12

ducl , l js−12

ducl)= 14f1−ksz

jl2

−ksxj l2−ksy

j l2g. SinceQsucld contains a sum over all qubits[see Eq.(C3)], we finally obtain

Qsucld = 1 −1

Noj=1

N

fkszjl2 + ksx

j l2 + ksyj l2g

= 1 − Phsucld, sC9d

where Ph is the purity relative to the local algebrahloc= % j=1

N sus2d j defined in Sec. III B.

APPENDIX D: CLASSICAL LIMIT IN THE LMG MODEL

As we mentioned in Sec. V, some critical properties of theLMG, such as the order parameter or the ground-state energyper particle in the thermodynamic limit, may be obtainedusing a semiclassical approach. In this appendix, we sketch arough analysis of why such an approximation is valid(for amore extensive analysis, see Ref.[44]).

We first define the collective operators

Ess,s8d = ok=1

N

cks† cks8, sD1d

where s, s8=↑ or ↓ and the fermionic operatorscks† scksd

have been defined in Sec. V. The collective operators satisfythe us2d commutation relations(Sec. III C); that is,

fEss,s8d,Ess9,s-dg = ds8s9Ess,s-d − dss-Ess9,s8d. sD2d

If the number of degenerate levelsN is very large, it isuseful to define the intensive collective operatorsEss,s8d=Ess,s8d /N, with commutation relations

fEss,s8d,Ess9,s-dg =1

Nsds8s9Ess,s-d − dss-Ess9,s8dd.

sD3d

Therefore, the intensive collective operators commute in thelimit N→`; they are effectively classical and can be simul-taneously diagonalized. Similarly, the intensive angularmomentum operatorsJx/N=sEs↑,↓d+Es↓,↑dd /2 ,Jy/N=sEs↑,↓d

−Es↓,↑dd /2i, and Jz/N=sEs↑,↑d−Es↓,↓dd /2 [with Ja defined inEqs. (24), (25), and (26)] commute with each other in thethermodynamic limit, so they can be thought of as the angu-lar momentum operators of a classical system.

Since the intensive LMG HamiltonianH /N, with H givenin Eq. (29), can be written in terms of the intensive angularmomentum operators, it can be regarded as the Hamiltoniandescribing a classical system. The ground state of the LMGmodel ugl is then an eigenstate of such intensive operatorswhen N→`: sJa /Ndugl= jaugl, ja being the correspondingeigenvalue. In other words, when obtaining some expectationvalues of intensive operators such asJa /N or H /N, theground stateugl can be pictured as a classical angular mo-mentum with fixed coordinates in the three-dimensionalspace(see Fig. 3).

This point of view makes it clear why such operatorsought to be intensive. Otherwise, such a classical limit is notvalid and terms of order 1 would be important for the calcu-lations of the properties of the LMG model. Obviously, allthese concepts can be extended to more complicated Hamil-tonians such as the extended LMG model, or even Hamilto-nians including interactions of higher orders as in[44].

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