RevModPhys_80_000517Entanglement in many-body systems

Entanglement in many-body systems

Luigi Amico

MATIS-CNR-INFM and Dipartimento di Metodologie Fisiche e Chimiche(DMFCI),

viale A.Doria6,95125Catania,Italy

Rosario Fazio

International School for Advanced Studies(SISSA)via Beirut2-4,I-34014Trieste,Italy

and NEST-CNR-INFM and Scuola Normale Superiore,I-56126Pisa,Italy

Andreas Osterloh

Institut für Theoretische Physik,Leibniz Universit?t Hannover,30167Hannover,Germany

Vlatko Vedral

The School of Physics and Astronomy,University of Leeds,Leeds LS29JT,

United Kingdom and Center for Quantum Technologies,National University of Singapore,

3Science Drive2,Singapore117543,Singapore

?Published6May2008?

Recent interest in aspects common to quantum information and condensed matter has prompted a ?urry of activity at the border of these disciplines that were far distant until a few years ago.Numerous interesting questions have been addressed so far.Here an important part of this?eld,the properties of the entanglement in many-body systems,are reviewed.The zero and?nite temperature properties of entanglement in interacting spin,fermion,and boson model systems are discussed.Both bipartite and multipartite entanglement will be considered.In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram.The behavior of entanglement can be related, via certain witnesses,to thermodynamic quantities thus offering interesting possibilities for an experimental test.Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

DOI:10.1103/RevModPhys.80.517PACS number?s?:05.30.?d,03.65.Ud

CONTENTS

I.Introduction518

II.Measures of Entanglement519

A.Bipartite entanglement in pure states519

B.Pairwise qubit entanglement in mixed states520

C.Localizable entanglement521

D.Entanglement witnesses521

E.Multipartite entanglement measures522

F.Indistinguishable particles523

1.Two fermion entanglement524

2.Multipartite entanglement for fermions525

3.Entanglement of particles525

4.Entanglement for bosons526

G.Entanglement in harmonic systems526 III.Model Systems527

A.Spin models527

1.Spin-1/2models with short range

interactions527

2.Spin-1/2models with in?nite range

interaction528

3.Frustrated spin-1/2models529

4.Spin-1models529

B.Strongly correlated fermionic models530

C.Spin-boson models530

D.Harmonic lattices531 IV.Pairwise Entanglement531

A.Pairwise entanglement in spin chains532

1.Concurrence and magnetic order532

2.Pairwise entanglement and quantum phase

transitions533

3.Entanglement versus correlations in spin

systems536

4.Spin models with defects536

B.Two-and three-dimensional systems537

C.Pairwise entanglement in fermionic models537

1.Noninteracting fermions537

2.Pairing models538

3.Kondo models539

D.Entanglement in itinerant bosonic systems540

E.Entanglement of particles540 V.Entanglement Entropy540

A.One-dimensional spin systems541

1.Spin chains541

2.XY chains and free fermion models543

3.Disordered chains544

4.Boundary effects544

B.Harmonic chains545

C.Systems in d?1and the validity of the area law546

1.Fermi systems547

2.Harmonic systems547

D.LMG model548

E.Spin-boson systems548

F.Local entropy in Hubbard-type models549

REVIEWS OF MODERN PHYSICS,VOLUME80,APRIL–JUNE2008

0034-6861/2008/80?2?/517?60??2008The American Physical Society

517

G.Topological entanglement entropy551

H.Entanglement along renormalization group?ow552 VI.Localizable Entanglement552

A.Localizable entanglement and quantum criticality552

B.Localizable entanglement in valence bond ground

states553 VII.Thermal Entanglement554

A.Thermal pairwise entanglement554

B.Pairwise entanglement in the T 0critical region555

C.Thermal entanglement witnesses557

D.Experimental results558 VIII.Multipartite Entanglement559

A.Multipartite entanglement in spin systems560

B.Global entanglement561

C.Generalized entanglement561

D.Renormalization group for quantum states562

E.Entanglement distribution for Gaussian states563 IX.Dynamics of Entanglement564

A.Propagation of entanglement564

1.Pairwise entanglement564

2.Dynamics of the block entropy566

3.Chaos and dynamics of entanglement567

B.Generation of entanglement567

C.Extraction of entanglement568

D.Time evolution of the entanglement in Gaussian

states568 X.Conclusions and Outlook569 Acknowledgments570 References570

I.INTRODUCTION

Entanglement expresses the“spooky”nonlocality in-herent to quantum mechanics?Bell,1987?.Because of that,it gave rise to severe skepticisms since the early days of quantum mechanics.It was only after the semi-nal contribution of John Bell that the fundamental ques-tions related to the existence of entangled states could be tested experimentally.In fact,under fairly general assumptions,Bell derived a set of inequalities for corre-lated measurements of two physical observables that any local theory should obey.The overwhelming majority of experiments done so far are in favor of quantum me-chanics thus demonstrating that quantum entanglement is physical reality?Peres,1993?.1

Entanglement has gained renewed interest with the development of quantum information science?Nielsen and Chuang,2000?.In its framework,quantum entangle-ment is viewed as a resource in quantum information processing.It is,e.g.,believed to be the main ingredient of the quantum speed-up in quantum computation and communication.Moreover,several quantum protocols, such as teleportation?Bennett et al.,1993?just to men-tion an important example,can be realized exclusively with the help of entangled states.

The role of entanglement as a resource in quantum information has stimulated intensive research trying to

unveil both its qualitative and quantitative aspects?Ple-

nio and Vedral,1998;Wootters,2001;Bru?,2002;Ve-

dral,2002;Bengtsson and Zyczkowski,2006;Eisert,

2006;Horodecki et al.,2007;Plenio and Virmani,2007?.

To this end,necessary criteria for any entanglement

measure to be ful?lled have been elaborated and lead to

the notion of an entanglement monotone?Vidal,2000?

allowing one to attach a precise number to the entangle-

ment encoded in a given state.There is a substantial

bulk of work for bipartite systems,in particular for the

case of qubits.Many criteria have been proposed to dis-

tinguish separable from entangled pure states,as,for

example,the Schmidt rank and the von Neumann en-

tropy.The success in the bipartite case for qubits asked

for extensions to the multipartite case,but the situation

proved to be far more complicated:Different classes of

entanglement occur,which are inequivalent not only un-

der deterministic local operations and classical commu-

nication,but even under their stochastic analog?Bennett

et al.,2001?.

In the last few years it has become evident that quan-

tum information may lead to further insight into other

areas of physics as statistical mechanics and quantum

?eld theory?Preskill,2000?.The attention of the quan-

tum information community to systems studied in con-

densed matter has stimulated an exciting cross-

fertilization between the two areas.Methods developed

in quantum information have proved to be extremely

useful in the analysis of the state of many-body systems. At T=0many-body systems are often described by a

complex ground state wave function which contains all

correlations that give rise to the various phases of matter ?superconductivity,ferromagnetism,quantum Hall sys-tems,etc.?.Traditionally many-body systems have been

studied by looking,for example,at their response to ex-

ternal perturbations,various order parameters,and ex-

citation spectrum.The study of the ground state of

many-body systems with methods developed in quantum

information may unveil new properties.At the same

time experience built up over the years in condensed

matter is helping in?nding new protocols for quantum computation and communication:A quantum computer is a many-body system where,different from traditional ones,the Hamiltonian can be controlled and manipu-lated.

The amount of work at the interface between statisti-cal mechanics and quantum information has grown dur-ing the last few years,shining light on many different aspects of both subjects.In particular,there has been extensive analysis of entanglement in quantum critical models?Osborne and Nielsen,2002;Osterloh et al., 2002;Vidal et al.,2003?.Tools from quantum informa-tion theory also provided support for numerical meth-ods,such as the density matrix renormalization group or the design of new ef?cient simulation strategies for many-body systems?see,for example,Verstraete,Por-ras,and Cirac,2004;Vidal,2003,2004?.Spin networks have been proposed as quantum channels?Bose,2003?by exploiting the collective dynamics of their low lying

1There are states that do not violate Bell inequalities and

nevertheless are entangled?Methot and Scarani,2000?.

518Amico et al.:Entanglement in many-body systems Rev.Mod.Phys.,V ol.80,No.2,April–June2008

excitations for transporting quantum information.

Despite being at its infancy,this new area of research

has grown so fast that a description of the whole?eld is

beyond the scope of a single review.Many interesting

facets of this branch of research will therefore remain

untouched here.In this review we only discuss the prop-

erties of entanglement in many-body systems.The mod-

els which will be considered include interacting spin net-

works,itinerant fermions,harmonic and bosonic

systems.All of them are of paramount interest in con-

densed matter physics.

This review is organized as follows.In the next section

we give a brief overview on the concepts and measures

of entanglement,with particular attention to those mea-

sures that we use later.In Sec.III we then proceed with

an introduction to several models of interacting many-

body systems which will be the subject of the review.We

discuss various aspects of quantum correlations starting

from the pairwise entanglement,Sec.IV,we then pro-

ceed with the properties of block entropy,Sec.V,and

localizable entanglement,Sec.VI.In these three sec-

tions it is especially relevant the connection between en-

tanglement and quantum phase transitions.The effect of

a?nite temperature is considered in Sec.VII.The char-

acterization of entanglement in many-body systems re-

quires also the understanding of multipartite entangle-

ment.This topic will be reviewed in Sec.VIII.From the

point of view of quantum information processing,dy-

namical properties of entanglement are important as

well.They will be addressed in Sec.IX.The conclusions,

the outlook,and a very short panorama of what is left

out from this review are presented in the concluding

section.

II.MEASURES OF ENTANGLEMENT

The problem of measuring entanglement is a vast and

lively?eld of research of its own.Numerous different

methods have been proposed for its quanti?cation.In

this section we do not attempt to give an exhaustive

review of the?eld.Rather we introduce those measures

that are largely being used to quantify entanglement in

many-body https://www.sodocs.net/doc/1e17759554.html,prehensive overviews of en-

tanglement measures can be found in Plenio and Vedral ?1998?;Wootters?2001?;Bru??2002?;Vedral?2002?; Bengtsson and Zyczkowski?2006?;Eisert?2006?;Horo-

decki et al.?2007?;Plenio and Virmani?2007?.In this

context,we also outline a method of detecting entangle-

ment,based on entanglement witnesses.

A.Bipartite entanglement in pure states

Bipartite entanglement of pure states is conceptually

well understood,although quantifying it for local dimen-

sions higher than two still bears theoretical challenges ?Virmani and Plenio,2000;Horodecki et al.,2007?.A pure bipartite state is not entangled if and only if it can be written as a tensor product of pure states of the parts. Moreover,for every pure bipartite state??AB??with the two parts,A and B?,two orthonormal bases???A,i??and ???B,j??exist such that??AB?can be written as

??AB?=?

i

?i??A,i???B,i?,?1?

where?i are positive coef?cients.This decomposition is called the Schmidt decomposition and the parti-cular basis coincides with the eigenbasis of the corre-sponding reduced density operators?B/A=tr A/B???AB??=?i?i2??B/A,i???B/A,i?.The density operators?A and?B have a common spectrum,in particular,they are equally mixed.Since only product states lead to pure reduced density matrices,a measure for their mixedness points a way towards quantifying entanglement in this case. Given the state??AB?,we can thus take its Schmidt de-composition,Eq.?1?,and use a suitable function?i to quantify the entanglement.

An entanglement measure E is?xed uniquely after imposing the following conditions:?1?E is invariant un-der local unitary operations??E is indeed a function of the?i’s only?;?2?E is continuous?in a certain sense also in the asymptotic limit of in?nite copies of the state;see, e.g.,Plenio and Virmani?2007??;and?3?E is additive, when several copies of the system are present:E???AB? ??AB??=E???AB??+E???AB??.The unique measure of entanglement satisfying the above conditions is the von Neumann entropy of the reduced density matrices:

S??A?=S??B?=??i?i2log??i2?,?2?

this is just the Shannon entropy of the moduli squared of the Schmidt coef?cients.In other words:under the above regularity conditions,the answer to the question of how entangled a bipartite pure state is,given by the von Neumann entropy of?either of?the reduced density matrices.The amount of entanglement is generally dif-?cult to de?ne once we are away from bipartite states, but in several cases we can still gain some insight into many-party entanglement if one considers different bi-partitions of a multipartite system.

It is worth noticing that a variety of purity measures are admissible when the third condition on additivity is omitted.In principle,there are in?nitely many measures for the mixedness of a density matrix;two of them will typically lead to a different ordering when the Hilbert space of the parts has a dimension larger than2.

In contrast,if we trace out one of two qubits in a pure state,the corresponding reduced density matrix?A con-tains only a single independent and unitarily invariant parameter:its eigenvalue?1/2.This implies that each monotonic function?0,1/2???0,1?of this eigenvalue can be used as an entanglement measure.Though,also here an in?nity of different mixedness measures exists, all lead to the same ordering of states with respect to their entanglement,and in this sense all are equivalent.

A relevant example is the?one-?tangle?Coffman et al., 2000?

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Amico et al.:Entanglement in many-body systems Rev.Mod.Phys.,V ol.80,No.2,April–June2008

?1??A ?=4det ?A .

?3?

By expressing ?A in terms of spin expectation values,it follows that ?1??A ?=1?4??S x ?2+?S y ?2+?S z ?2?,where

?S ??=tr A ??A S ??and S ?=12

??

,????=x ,y ,z ?being the Pauli matrices.For a pure state ??AB ?of two qubits the

relation ?1????*??A y ?B y ????2?C ???AB ??2

??2applies,where C is the concurrence ?Hill and Wootters,1997;Wootters,1998?for pure states of two qubits,a measure of pairwise entanglement ?see the next section ?,and the asterisk indicates the complex conjugation in the eigen-basis of ?z .The von Neumann entropy can be expressed

as a function of the ?one-?tangle S ??A ?=h ?12

?1+?1??1??A ???,where h ?x ???x log 2x ??1?x ?log 2?1?x ?is the binary entropy.

B.Pairwise qubit entanglement in mixed states

Subsystems of a many-body ?pure ?state will generally

be in a mixed state.In this case a different way of quan-tifying entanglement can be introduced.Three impor-tant representatives are the entanglement cost E C ,the distillable entanglement E D ?both de?ned by Bennett,Bernstein,et al.?1996??,and the entanglement of forma-tion E F ?Bennett,Di Vincenzo,et al.,1996?.Whereas E D and E C are asymptotic limits of multicopy extraction probabilities of Bell states and creation from such states,the entanglement of formation is the amount of pure state entanglement needed to create a single copy of the mixed state.Although recent progress has been achieved ?Paz-Silva and Reina,2007?,the full additivity of the E F for bipartite systems has not been established yet ?see,e.g.,Vidal et al.?2002??.

The conceptual dif?culty behind the calculation of E F lies in the in?nite number of possible decompositions of a density matrix.Therefore even knowing how to quan-tify bipartite entanglement in pure states,we cannot simply apply this knowledge to mixed states in terms of an average over mixtures of pure state entanglement.The problem is that two decompositions of the same density matrix usually lead to a different average en-tanglement.Which one do we choose?It turns out that we must take the minimum over all possible decompo-sitions,simply because if there is a decomposition where the average is zero,then this state can be created locally without the need of any entangled pure state,and there-fore E F =0.The same conclusion can be drawn from the requirement that entanglement must not increase on av-erage by means of local operations including classical communication ?LOCC ?.

The entanglement of formation of a state ?is there-fore de?ned as

E F ???amin ?j

p j S ??A ,j ?,

?4?

where the minimum is taken over all realizations of the state ?AB =?j p j ??j ???j ?,and S ??A ,j ?is the von Neumann entropy of the reduced density matrix ?A ,j atr B ??j ???j ?.Equation ?4?is the so-called convex roof ?also the ex-pression convex hull is found in the literature ?of the entanglement of formation for pure states,and a decom-position leading to this convex roof value is called an optimal decomposition.

For systems of two qubits,an analytic expression for E F is given by

E F ???=?

??=±

?1+?C 2???

2

ln

?1+?C 2???

2

,?5?

where C ???is the so-called concurrence ?Wootters,1998,

2001?,the convex roof of the pure state concurrence,which has been de?ned in the previous section.Its con-vex roof extension is encoded in the positive Hermitian

matrix R ????

???=????y ?y ??*??y ?y ???,with eigen-values ?12?ˉ??42

in the following way:

C =max ??1??2??3??4,0?.

?6?

As the entanglement of formation is a monotonous function of the concurrence,also C itself or its square ?2—called also the 2-tangle—can be used as entangle-ment measures.This is possible due to a curious pecu-liarity of two-qubit systems:namely,that a continuous variety of optimal decompositions exist ?Wootters,1998?.The concurrence C and the tangle ?1both range from 0?no entanglement ?to 1.

By virtue of Eq.?6?,the concurrence in a spin-1/2chain can be computed in terms of up to two-point spin correlation functions.As an example we consider a case where the model has a parity symmetry,it is transla-tional invariant,and the Hamiltonian is real;the concur-rence in this case reads

C ij =2max ?0,C ij I ,C ij II

?,

?7?

where C ij I =?g ij xx +g ij yy ????1/4+g ij zz ?2?M z 2and C ij II =?g ij

xx ?g ij yy ?+g ij zz ?1/4,with g ij ??=?S i ?S j ??and M z =?S z

?.A state with dominant ?delity of parallel and antiparallel Bell states is characterized by dominant C I and C II ,respec-tively.This was shown by Fubini et al.?2006?,where the concurrence was expressed in terms of the fully en-tangled fraction as de?ned by Bennett,DiVincenzo,et al.?1996?.Systematic analysis of the relation between the concurrence ?together with the 3-tangle,see Sec.II.E ?and the correlation functions has been presented by Glaser et al.?2003?.

The importance of the tangle and concurrence is due to the monogamy inequality derived in Coffman et al.?2000?for three qubits.This inequality has been proven to hold also for n -qubits system ?Osborne and Verstra-ete,2006?.In the case of many-qubits ?the tangle may depend on the site i that is considered ?it reads

?j i

C ij 2??1,i .?8?

The so-called residual tangle ?1,i ??j i C ij 2is a measure for multipartite entanglement not stored in pairs of qu-bits only.We ?nally mention that the antilinear form of the pure state concurrence was the key for the ?rst ex-plicit construction of a convex roof,and hence its exten-

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Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

sion to mixed states?Hill and Wootters,1997;Wootters, 1998;Uhlmann,2000?.

Another measure of entanglement is the relative en-tropy of entanglement?Vedral et al.,1997?.It can be applied to any number of qubits in principle?or any dimension of the local Hilbert space?.It is formally de-?ned as E???amin??D S?????,where S?????=tr??ln??ln??is the quantum relative entropy.This relative en-

tropy of entanglement quanti?es the entanglement in?by its distance from the set D of separable states?since D is compact,the minimum is assumed always?.The main dif?culty in computing this measure is to?nd the disentangled state closest to?.This is in general an open problem,even for two qubits.In the presence of certain symmetries—which is the case for,e.g.,eigenstates of certain models—an analytical access is possible.In these cases,the relative entropy of entanglement becomes a very useful tool.The relative entropy reduces to the en-tanglement entropy in the case of pure bipartite states; this also means that its convex roof extension coincides with the entanglement of formation,and is readily de-duced from the concurrence?Wootters,1998?.

We close this summary on the pairwise entanglement by commenting on the quantum mutual information. Groisman et al.quanti?ed the work necessary to erase the total correlations existing in a bipartite system?Gro-isman et al.,2005?.The entanglement can be erased by a suitable random ensemble of unitary transformations acting on one of the parts,but a certain amount of clas-sical correlation among the two partners may survive. The work necessary to erase all correlations is given by the quantum mutual information

I AB=S??A?+S??B??S??AB?.?9?

C.Localizable entanglement

A different approach to entanglement in many-body systems arises from the quest to swap or transmute dif-ferent types of multipartite entanglement into pairwise entanglement between two parties by means of general-ized measures on the rest of the system.In a system of interacting spins on a lattice one could then try to maxi-mize the entanglement between two spins?at positions i and j?by performing measurements on all others.The system is then partitioned in three regions:the sites i,j and the rest of the lattice.This concentrated pairwise entanglement can then be used,e.g.,for quantum infor-mation processing.A standard example is that the three qubit Greenberger-Horne-Zeilinger?GHZ?state ?1/?2???000?+?111??after a projective measure in the x direction on one of the sites is transformed into a Bell state.

The concept of localizable entanglement has been in-troduced by Verstraete?2004?;Popp et al.?2005?.It is de?ned as the maximal amount of entanglement that can be localized,on average,by doing local measure-ments in the rest of the system.In the case of N parties, the possible outcomes of the measurements on the re-maining N?2particles are pure states??s?with corre-sponding probabilities p s.The localizable entanglement E loc on sites i and j is de?ned as the maximum of the average entanglement over all possible outcome states ??s?ij,

E loc?i,j?=sup E?s p s E???s?ij?,?10?

where E is the set of all possible outcomes?p s,??s??of measurements and E is the chosen measure of entangle-ment of a pure state of two qubits?e.g.,the concur-rence?.Although dif?cult to compute,lower and upper bounds have been found which allow one to deduce a number of consequences for this quantity.

An upper bound to the localizable entanglement is given by the entanglement of assistance?Laustsen et al., 2003?obtained from localizable entanglement when also global and joint measurements were allowed on the N ?2spins.A lower bound of the localizable entanglement comes from the following theorem?Verstraete,Martin-Delgado,and Cirac,2004?:Given a?pure or mixed?state of N qubits with reduced correlations Q ij?,?=?S i?S j????S i???S j??between the spins i and j and directions?and ?then there always exist directions in which one can measure the other spins such that this correlation does not decrease,on average.It then follows that a lower bound to localizable entanglement is?xed by the maxi-mal correlation function between the two parties?one of the various spin-spin correlation functions Q ij?,??.2

D.Entanglement witnesses

It is important to realize that not just the quanti?ca-tion of many-party entanglement is a dif?cult task;it is an open problem to tell,in general,whether a state of n parties is separable or not.It is therefore of great value to have a tool that is able to merely certify if a certain state is entangled.An entanglement witness W is a Her-mitian operator which is able to detect entanglement in a state.The basic idea is that the expectation value of the witness W for the state?under consideration ex-ceeds certain bounds only when?is entangled.An ex-pectation value of W within this bound,however,does not guarantee that the state is separable.Nonetheless, this is an appealing method also from an experimental point of view,since it is sometimes possible to relate the presence of the entanglement to the measurement of a few observables.

Simple geometric ideas help to explain the witness op-erator W at work.Let T be the set of all density matrices and let E and D be the subsets of entangled and sepa-

2It has been argued?Gour,2006;Gour and Spekkens,2006?that in order to extend the entanglement of assistance and localizable entanglement to an entanglement monotone?Vidal, 2000?one should admit also local operations including classical communication on the extracted two spins,this was named entanglement of collaboration.

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rable states,respectively.The convexity of D is a key property for witnessing entanglement.The entangle-ment witness is then an operator de?ning a hyperplane which separates a given entangled state from the set of separable states.The main scope of this geometric ap-proach is then to optimize the witness operator?Lewen-stein et al.,2000?or to replace the hyperplane by a curved manifold,tangent to the set of separable states ?Gühne,2004??for other geometric aspects of entangle-ment see Klyachko?2002?,Bengtsson and Zyczkowski

?2006?,and Leinaas et al.?2006??.We have the freedom to choose W such that tr??D W??0for all disentangled states?D?D.Then,tr??W??0implies that?is en-tangled.A caveat is that the concept of a witness is not invariant under local unitary operations?see,e.g.,Cav-alcanti and Terra-Cunha?2005??.

Entanglement witnesses are a special case of a more general concept,namely that of positive maps.These are injective superoperators on the subset of positive opera-tors.When we now think of superoperators that act non-trivially only on part of the system?on operators that act nontrivially only on a sub-Hilbert space?,then we may ask the question whether a positive map on the subspace is also positive when acting on the whole space.Maps that remain positive also on the extended space are called completely positive maps.The Hermitian time evolution of a density matrix is an example for a com-pletely positive map.Positive but not completely posi-tive maps are important for entanglement theory.There is a remarkable isomorphism between positive maps and Hermitian operators?Jamiolkowski,1972?.This can be used to prove a key theorem?Horodecki et al.,1996?:A state?AB is entangled if and only if a positive map?exists?not completely positive?such that?1A ?B??AB ?0.For a two-dimensional local Hilbert space the situ-ation simpli?es considerably in that any positive map P can be written as P=CP1+CP2T B,where CP1and CP2 are completely positive maps and T B is a transposition operation on subspace B.This decomposition tells us that for a system of two qubits the lack of complete positivity in a positive map is due to a partial transposi-tion.This partial transposition clearly leads to a positive operator if the state is a tensor product of the parts.In fact,also the opposite is true:a state of two qubits?AB is separable if and only if?AB T B?0that is,its partial trans-position is positive.This is very simple to test and it is known as the Peres-Horodecki criterion?Horodecki et al.,1996;Peres,1996?.The properties of entangled states under partial transposition lead to a measure of en-tanglement known as the negativity.The negativity N AB of a bipartite state is de?ned as the absolute value of the sum of the negative eigenvalues of?AB T A.The logarithmic negativity is then de?ned as

E N=log22?2N AB+1?.?11?For bipartite states of two qubits,?AB T A has at most one negative eigenvalue?Sanpera et al.,1998?.For general multipartite and higher local dimension this is only a suf?cient condition for the presence of entanglement.There exist entangled states with a positive partial trans-

pose known as bound entangled states?Horodecki et al.,

1998;Acin et al.,2001?.

E.Multipartite entanglement measures

Both the classi?cation of entanglement and its quan-

ti?cation are at a preliminary stage even for distinguish-

able particles?see,however,Dür et al.?2000?;Miyake

and Wadati?2002?;Verstraete et al.?2002?;Briand et al.?2003,2004?,Luque and Thibon?2005?;Osterloh and Siewert?2005,2006?;Mandilara et al.?2006?,and refer-

ences therein?.We restrict ourselves to those approaches

which have been applied so far for the study of con-

densed matter systems discussed in this review.It has

already been mentioned that several quantities are use-

ful as indicators for multipartite entanglement when the

whole system is in a pure state.The entropy of entangle-

ment is an example for such a quantity and several

works use multipartite measures constructed from and

related to it?see,e.g.,Coffman et al.?2000?;Meyer and

Wallach?2002?;Barnum et al.?2003?;Scott?2004?;de

Oliveira,Rigolin,and de Oliveira?2006a?;Love et al.?2006??.These measures are of“collective”nature—in contrast to“selective”measures—in the sense that they give indication of a global correlation without discerning among the different entanglement classes encoded in the state of the system.

The geometric measure of entanglement quanti?es

the entanglement of a pure state through the minimal

distance of the state from the set of pure product states ?Vedral et al.,1997;Wei and Goldbart,2003?

E g???=?log2max

?

???????2,?12?

where the maximum is on all product states?.As dis-cussed in detail by Wei and Goldbart?2003?,the previ-ous de?nition is an entanglement monotone if the con-vex roof extension to mixed states is taken.It is zero for separable states and rises up to unity for,e.g.,the maxi-mally entangled n-particle GHZ states.The dif?cult task in its evaluation is the maximization over all possible separable states and of course the convex roof extension to mixed states.Despite these complications,a clever use of the symmetries of the problem renders this task accessible by substantially reducing the number of pa-rameters?see Sec.VIII?.

Another example for the collective measures of mul-tipartite entanglement as mentioned in the beginning of this section are the measures introduced by Meyer and Wallach?2002?and by Barnum et al.?2003,2004?.In the case of qubit system the Q measure of Meyer and Wallach is the average purity?which is the average one-tangle in Coffman et al.?2000??of the state?Meyer and Wallach,2002;Brennen,2003;Barnum et al.,2004?

E gl=2?

2

N

?

j=1

N

Tr?j2.?13?

522Amico et al.:Entanglement in many-body systems Rev.Mod.Phys.,V ol.80,No.2,April–June2008

The notion of generalized entanglement introduced by Barnum et al.?2003,2004?relaxes the typically cho-sen partition into local subsystems in real space.The generalized entanglement measure used by Barnum et al.is the purity relative to a distinguished Lie algebra A of observables.For the state???it is de?ned as P A=Tr??P A???????2?,?14?

where P A is the projection map?→P A???.If the set of observables is de?ned by the operator basis ?A1,A2,...,A L?then P A=?i=1L?A i?2from which the re-duction to Eq.?13?in the case of all local observables is evident.This conceptually corresponds to a rede?nition of locality as induced by the distinguished observable set,beyond the archetype of partition in the real space. It de?nes an observer dependent concept of entangle-ment adapted to, e.g.,experimentally accessible or physically relevant observables.In this case,the gener-alized entanglement coincides with the global entangle-ment of Meyer and Wallach.

Another approach pursued is the generalization of the concurrence.For the quanti?cation of pairwise entangle-ment in higher dimensional local Hilbert spaces,the concept of concurrence vectors has been formulated ?Audenaert et al.,2001;Badziag et al.,2002?besides the I-concurrence?Rungta et al.,2001?.A concurrence vec-tor was also proposed for multipartite systems of qubits ?Akhtarshenas,2005?.It consists in applying the pure state concurrence formula to a mixed two-site reduced density matrix.It coincides with the true concurrence if and only if the eigenbasis of the density matrices provide optimal decompositions.

The n-tangle is a straightforward extension of the con-currence to multipartite states as the overlap of the state with its time reversed?Wong and Christensen,2001?.It vanishes identically for an odd number of qubits,but an entanglement monotone is obtained for an even number of qubits.It detects products of even-site entangled states in addition to certain genuine multipartite en-tangled states:it detects the multipartite GHZ or cat state,but not,for example,the four qubit cluster state. Three classes of states inequivalent under SLOCC ?stochastic LOCC?exist for four qubits?Osterloh and Siewert,2005,2006?.Representatives are the GHZ state, the celebrated cluster state,and a third state,which is also measured by the4-qubit hyperdeterminant.Class selective measures are constructed from two basic ele-ments,namely the operator?y employed for the concur-rence,and the operator??·??a1·1??x·?x??z·?z where the dot is a tensor product indicating that the two operators are acting on different copies of the same qu-bit.Both are invariant under sl?2,C?operations on the qubit.The3-tangle is then expressed as?3???=??*?·??*??? ?? ??·?? ?? ?????·???.The multi-linearity,however,makes it problematic to employ the procedure of convex roof construction presented by Wootters?1998?and Uhlmann?2000?for general mix-tures.

Finally we mention the approach pursued by Gühne et al.?2005??see also Sharma and Sharma,2006?where different bounds on the average energy of a given sys-

tem were obtained for different types of n-particle quan-tum correlated states.A violation of these bounds then implies the presence of multipartite entanglement in the system.The starting point of Gühne et al.is the notion

of n-separability and k-producibility which admit to dis-

criminate particular types of n-particle correlations

present in the system.A pure state???of a quantum system of N parties is said to be n-separable if it is pos-

sible to?nd a partition of the system for which???=??1???2?ˉ??n?.A pure state???can be produced by k-party entanglement?i.e.,it is k-producible?if we can

write???=??1???2?ˉ??m?,where??i?are states of maxi-mally k parties;by de?nition m?N/k.It implies that it

is suf?cient to generate speci?c k-party entanglement to construct the desired state.Both these indicators for multipartite entanglement are collective,since they are based on the factorizability of a given many particle state into smaller parts.k-separability and k-producibility both do not distinguish between differ-ent k-particle entanglement classes?as, e.g.,the k-particle W-states and different k-particle graph states ?Hein et al.,2004?,like the GHZ state?.

F.Indistinguishable particles

For indistinguishable particles the wave function is ?anti?symmetrized and therefore the de?nition of en-tangled states as given in the previous section does not apply.In particular,it does not make sense to consider each individual particle as parts of the partition of the system.Having agreed upon a de?nition of entangle-ment,concepts as entanglement cost or distillation re-main perfectly valid.Following Ghirardi et al.?2002?and Ghirardi and Marinatto?2003?one can address the problem of de?ning entanglement in an ensemble of in-distinguishable particles by seeing if one can attribute to each of the subsystems a complete set of measurable properties, e.g.,momenta for free pointless particles. Quantum states satisfying the above requirement repre-sent the separable states for indistinguishable particles. There is another crucial difference between the en-tanglement of?indistinguishable?spin-1/2particles and that of qubits.We therefore consider two fermions on two sites.Whereas the Hilbert space H s of a two-site spin lattice has dimension dim H s=4,the Hilbert space H f for two fermions on the same lattice has dimension dim H f=6.This is due to the possibility that both fermi-ons,with opposite spins,can be located at the same lattice site.When choosing the following numbering of the states?1?=f1??0??c L,↑??0?,?2?=f2??0??c L,↓??0?,?3?=f3??0??c R,↑??0?,?4?=f4??0??c R,↓??0?,and the de?nition ?i,j?=f i?f j??0?,there are Bell states analogous to those oc-curring for distinguishable particles??1,3?±?2,4??/?2 and??1,4?±?2,3??/?2.There are,however,new en-tangled states,as??1,2?±?3,4??/?2,where both fermions take the same position.The local Hilbert space is made

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of four states labeled by the occupation number and the spin,if singly occupied.The site entanglement of indis-tinguishable particles is then de?ned as the entangle-ment of the corresponding Fock states.It can be mea-sured, e.g.,by the local von Neumann entropy.This quantity is the analog to the one-tangle for qubits,but the local Hilbert space dimension is four due to the pos-sibility of having empty and doubly occupied sites.Also the quantum mutual information?Groisman et al.,2005?, see Eq.?9?,can be de?ned in this way,quantifying the total amount?classical and quantum?of correlations stored in a given state of a second quantized system. Although from a mathematical point of view the en-tanglement of indistinguishable particles can be quanti-?ed,the major part of the literature on second quantized systems that we discuss in this review considers the site entanglement described above or the entanglement of degrees of freedom,singled out from a suitable set of local quantum numbers?e.g.,the spin of the particle at site i?.In both cases,entanglement measures for distin-guishable particles?see Secs.IV.C.1and V.F?can be used.With this respect,this section has a different scope than the others on the quanti?cation of entanglement; although most of the discussion which follows will not be used later on,we believe that it will be of interest for further studies of entanglement in itinerant many-body systems.

1.Two fermion entanglement

Due to the antisymmetry under particle exchange, there is no Schmidt decomposition for fermions.Never-theless,a fermionic analog to the Schmidt rank which classi?es entanglement in bipartite systems of distin-guishable particles does exist:the so-called Slater rank.

A generic state of two electrons on two lattice sites can be written as???a?i,j=1

4?i,j?i,j?,where?is a4?4matrix which can be assumed antisymmetric and normalized as tr???=12.Since the local entities whose entanglement studied here are particles,unitary transformations act on the four-dimensional single-particle Hilbert space.Due to the indistinguishability of particles,the transforma-tion must be the same for each particle.Given a unitary transformation U?SU?4?such that f j?aU jk f k,the trans-formed state is given by????,where??aU?U T.The above unitary transformation preserves the antisymme-try of?and can transform every pure state of two spin-1/2particles on two sites into a state corresponding to the normal form of?.In fact,every two-particle state within a D-dimensional single-particle Hilbert space can be transformed into the normal form?s =diag?Z1,...,Z r,Z0?,where Z j=iz j?y and?Z0?ij=0for i,j??1,...,D?2r?.In the previous expression r is then called the Slater rank of the pure fermion state?Schli-emann,Cirac,et al.,2001;Schliemann,Loss,and Mac-Donald,2001;Eckert et al.,2002?.A pure fermion state is entangled if and only if its Slater rank is larger than1. It is important to note that the above concept of en-tanglement only depends on the dimension of the Hil-bert space accessible to each of the particles?this in-cludes indistinguishable particles on a single D-level system?.

For electrons on an L-site lattice the“local”Hilbert space dimension is2L,and the question whether a pure state living in a2L-dimensional single particle Hilbert space has full Slater rank can be answered by consider-ing the Pfaf?an of??Caianello and Fubini,1952;Muir, 1960?:

?

??S2L?

sgn????j=1

L

???2j?1?,??2j?,?15?

which is nonzero only if?has full Slater rank L.In the above de?nition S2L?denotes those elements?of the symmetric group S2L with ordered pairs,i.e.,??2m?1????2m?for all m?L and??2k?1????2m?1?for k ?m.Note that relaxing the restriction to S2L?leads to a combinatorial factor of2L L!by virtue of the antisymme-try of?and hence can we write

pf???=

1

2L L!

?

j1,...,j2L=1

2L

?j1,...,j2L?j

1

,j2

...?j

2L?1

,j2L

,?16?

where?j1,...,j2L is the fully antisymmetric tensor with ?1,2,...,2L=1.There is a simple relation between the Pfaff-ian and the determinant of an antisymmetric even-dimensional matrix:pf???2=det???.

For the simplest case of two spin-1/2fermions on two lattice sites the Pfaf?an reads pf???=?1,2?3,4??1,3?2,4 +?1,4?2,3.Normalized in order to range in the interval ?0,1?this has been called the fermionic concurrence C??????Schliemann,Cirac,et al.,2001;Schliemann,Loss, and MacDonald,2001;Eckert et al.,2002?:

C?????=????????=8?pf????,?17?

where??a1

2

?ijkl?

k,l

*has been termed the dual to?.Then,?????D???is the analog to the conjugated state in?Hill and Wootters,1997;Wootters,1998;Uhlmann,2000?leading to the concurrence for qubits.It is important to note that the Pfaf?an in Eq.?15?is invariant under the complexi?cation of su?2L?,since it is the expectation value of an antilinear operator,namely the conjugation D for state???.Since this invariant is a bilinear expres-sion in the state coef?cients,its convex roof is readily obtained?Uhlmann,2000?by means of the positive eigenvalues?i2of the6?6matrix R=??D?D??. The conjugation D expressed in the basis ??1,2?,?1,3?,?1,4?,?2,3?,?2,4?,?3,4??takes the form D0C,where C is the complex conjugation and the only nonzero ele-ments of D are D16=D61=D34=D43=1and D25=D52=1. Notice that the center part of this matrix is precisely ?y ?y and indeed corresponds to the Hilbert space of two qubits.The remaining part of the Hilbert space gives rise to an entanglement of different values for the occupation number.This type of entanglement has been referred to as the“?uffy bunny”?Verstraete and Cirac, 2003;Wiseman et al.,2003?,in the literature.

For a single-particle Hilbert space with dimension larger than four one encounters similar complications as

524Amico et al.:Entanglement in many-body systems Rev.Mod.Phys.,V ol.80,No.2,April–June2008

for two distinguishable particles on a bipartite lattice and local Hilbert space dimension larger than two,i.e., for two qudits.This is because different classes of en-tanglement occur,which are characterized by different Slater rank as opposed to their classi?cation by different Schmidt rank for distinguishable particles.The Slater rank can be obtained by looking at Pfaf?an minors ?Muir,1960?:if the Slater rank is r,all Pfaf?an minors of dimension larger than2r are identically zero.

2.Multipartite entanglement for fermions

For indistinguishable particles the only classi?cation available up to now is to check whether or not a pure state has Slater rank one.Eckert et al.formulated two recursive lemmata?Eckert et al.,2002?that can be sum-marized as follows:Let an N-electron state be con-tracted with N?2arbitrary single electron states en-coded in the vectors a j as a k j f k??0??j=1,...,N?2and sum convention?to a two-electron state.Then the Pfaf?an of the two-electron state is zero if and only if the original state?and hence all intermediate states in a successive contraction?has Slater rank one.This means that all four-dimensional Pfaf?an minors of?are zero. Instead of the Pfaf?an of?,the single-particle re-duced density matrix can also be considered,and its von Neumann entropy as a measure for the quantum en-tanglement has been analyzed by Li et al.?2001?and Pa?kauskas and You?2001?.It is important to remember that for distinguishable particles the local reduced den-sity matrix has rank one if and only if the original state were a product.This is no longer true for indistinguish-able particles.For an N-particle pure state with Slater rank one the rank of the single-particle reduced density matrix coincides with the number of particles N.A subtlety is that a measure of entanglement is obtained after subtraction of the constant value of the von Neu-mann entropy of a disentangled state.This must be taken into account also for the extension of the measure to mixed states.

3.Entanglement of particles

Entanglement in the presence of superselection rules ?SSR?induced by particle conservation has been dis-cussed by Bartlett and Wiseman?2003?,Schuch et al.?2003,2004?,and Wiseman and Vaccaro?2003?.The main difference in the concept of entanglement of par-ticles?Wiseman and Vaccaro,2003?from the entangle-ment of indistinguishable particles as described in the preceding section?but also to that obtained from the reduced density matrix of,e.g.,spin degrees of freedom of indistinguishable particles?consists in the projection of the Hilbert space onto a subspace of?xed particle numbers for either part of a bipartition of the system. The bipartition is typically chosen to be spacelike,as motivated from experimentalists or detectors sitting at distinct positions.For example,two experimentalists in order to detect the entanglement between two indistin-guishable particles must have one particle each in their laboratory.

This difference induced by particle number superse-lection is very subtle and shows up if multiple occupan-cies occur at single sites for fermions with some inner degrees of freedom,such as spin.Their contribution is ?nite for?nite discrete lattices and will generally scale to zero in the thermodynamic limit with vanishing lattice spacing.Therefore both concepts of spin entanglement of two distant particles coincide in this limit.Signi?cant differences are to be expected only for?nite nondilute systems.It must be noted that the same restrictions im-posed by SSR which change considerably the concept of entanglement quantitatively and qualitatively,on the other hand,enable otherwise impossible protocols of quantum information processing?Schuch et al.,2003, 2004?which are based on variances about the observable ?xed by superselection.

Wiseman and Vaccaro projected an N-particle state ??N?onto all possible subspaces,where the two parties have a well de?ned number?n A,n B=N?n A?of particles ?Wiseman and Vaccaro,2003?.Let???n A??be the respec-tive projection,and let p n

A

be the weight ???n A????n A??/??N??N?of this projection.Then the en-tanglement of particles E p is de?ned as

E p???n??=?n p n E M???n A??,?18?

where E M is some measure of entanglement for distin-guishable particles.Although this certainly represents a de?nition of entanglement appealing for experimental issues,it is sensitive only to situations where,e.g.,the two initially indistinguishable particles eventually are separated and can be examined one by one by Alice and Bob.Consequently,“local operations”have been de-?ned by Wiseman and Vaccaro?2003?as those per-formed by Alice and Bob in their laboratory after hav-ing measured the number of particles.3

Verstraete and Cirac pointed out that the presence of SSR gives rise to a new resource which has to be

3As a potential difference between the entanglement of pho-tons as opposed to that of massive bosonic particles,it has been claimed that certain superselection rules may hold for massive particles only.One such claim is that we would in practice not be able to build coherent superpositions of states containing a different number of massive particles?for a recent discussion see Bartlett et al.?2007??.This superselection rule would,for instance,prohibit creating a superposition of a hy-drogen atom and molecule.However,the origin and validity of any superselection rule remains a debated subject.The argu-ments pro superselection rules usually involve some symmetry considerations,or some decoherence mechanism.On the other hand,it turns out that if we allow most general operations in quantum mechanics,we no longer encounter any superselec-tion restrictions.Recent work?Dowling,Bartlett,et al.,2006; Terra Cunha et al.,2007?has shown that it should be possible to coherently superpose massive particles and to observe a vio-lation of certain Bell inequalities?Terra Cunha et al.,2007?also for this case.

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quanti?ed.They have proposed to replace the quantity E p with the SSR entanglement of formation.This is de-?ned as

E f?SSR????N??=min

p n,?n ?

n

p n E M??n?,

where the minimization is performed over all those de-composition of the density matrix where the???n are eigenstates of the total number of particles?Schuch et

al.,2003,2004?.

4.Entanglement for bosons

The quanti?cation and classi?cation of boson en-

tanglement is very close in spirit to that of fermions as described in Sec.II.F.1.In the bosonic case the matrix?introduced in the previous section is symmetric under

permutations of the particle numbers.Consequently,for any two-particle state of indistinguishable bosons,?can be diagonalized by means of unitary transformations of

the single-particle basis.This leads to the Schmidt de-

composition for bosons?Eckert et al.,2002?.A curious

feature distinguishing this case from the entanglement

measures of distinguishable particles is that the Schmidt

decomposition is not unique.In fact,any two equal

Schmidt coef?cients admit for a unitary transformation

of the two corresponding basis states,such that the su-

perposition of the two doubly occupied states can be

written as a symmetrized state of two orthogonal states ?Li et al.,2001;Ghirardi and Marinatto,2005?.This is the reason why it is not directly the Schmidt rank,but rather

the reduced Schmidt rank—obtained after having re-

moved all double degeneracies of the Schmidt

decomposition—that determines whether or not a state

is entangled.This nonuniqueness of the Schmidt rank is

also responsible for the ambiguity of the von Neumann

entropy or other purity measures of the single-particle

reduced density matrix as an entanglement measure for

bosons?Ghirardi and Marinatto,2005?.

With z i the Schmidt coef?cients with degeneracy g i, the reduced Schmidt rank is at most g i/2+2?g i/2?,where ?·?denotes the noninteger part.As a consequence,a Schmidt rank larger than2implies the presence of en-tanglement.A Schmidt rank2with degenerate Schmidt coef?cients can be written as a symmetrized product of orthogonal states and consequently is disentangled ?Ghirardi and Marinatto,2005?.This feature is also present in the N-boson case,where in the presence of up to N-fold degenerate Schmidt coef?cients the corre-sponding state can be rewritten as a symmetrization of a product.

For bipartite systems?has full Schmidt rank if det? 0.A Schmidt rank1can be veri?ed by the same contraction technique described for the fermion case in the previous section,where the Pfaf?an must be re-placed by the determinant.This applies to both the bi-partite and the multipartite case?Eckert et al.,2002?.G.Entanglement in harmonic systems

In this section we concentrate on the entanglement between distinct modes of harmonic oscillators?see Braunstein and van Loock?2005?;Adesso and Illuminati ?2007?for recent reviews on the subject?.The entangle-

ment in this case is termed as continuous variable en-tanglement in the literature?to be distinguished from the entanglement of indistinguishable bosonic particles; see Sec.II.F?.

Dealing with higher dimensional space of the local degrees of freedom generally involves complications which are not tamable within the current knowledge about entanglement.The Peres-Horodecki criterion,just to mention an important example,is not suf?cient al-ready for two three-level systems,3?3.The situation simpli?es if only so-called Gaussian states of the har-monic oscillator modes are considered.This restriction makes the in?nite dimensional case even conceptually simpler than the?nite dimensional counterparts.In or-der to explain what Gaussian states are,we introduce the Wigner distribution function W?p,q??Wigner,1932?. For a single degree of freedom it is de?ned from the density operator?as

W?r,p?a

1

??

?

??

?

dr??r+r????r?r??e?2i/??pr?,?19?

where r and p are conjugate position and momentum variables of the degree of freedom.The connection be-tween bosonic operators a?,a??and phase space operators r?,p?is a?=?r?+ip??/?2,a??=?a???=?r??ip??/?2.More degrees of freedom are taken into account in a straightforward manner.A?mixed?state?is then called Gaussian when its Wigner distribution function is Gaussian.Examples for such states are coherent pure states???,a????=????with??C,and arbitrary mixtures of coherent states?=?d2?P?????????,determined by the so-called P-distribution P?z?.Such states are termed classical if the Wigner function and the P-distribution are non-negative?see Simon?2000??.

The key quality of Gaussian states is that they are completely classi?ed by second moments,which are en-coded in the symmetric so-called?co-?variance matrix with the uncertainties of the phase space coordinates as entries.For two bosonic modes the phase space is four dimensional and the covariance matrix V is de?ned as

V??a??????,??????=?d4???????W??????,?20?where the curly brackets on the left-hand side indicate the anticommutator.The components of???,?=1,...,4 are?r?1,p?1,r?2,p?2?and????a?????????;the average?·?is taken with respect to the given two-mode density matrix ?,or,equivalently,using the Wigner distribution of?. Then,the canonical commutation relations assume the compact form????,????=i???,?with?=i?y 1.When ex-pressed in terms of V,the Heisenberg uncertainty rela-tion can be invoked in invariant form with respect to

526Amico et al.:Entanglement in many-body systems Rev.Mod.Phys.,V ol.80,No.2,April–June2008

canonical transformations as det V ??2/4?see,e.g.,Si-mon et al.?1994??.The set of the real linear canonical transformation generates the symplectic group Sp ?2n ,R ?that plays an important role in the theory.Being a sym-plectic matrix,V can be brought in its diagonal form V n by means of symplectic transformations.The elements on the diagonal are then called the symplectic eigenval-ues of V .An analysis of V n ?has unveiled an even more powerful invariant form of the Heisenberg uncertainty principle,V +?i ?/2???0,where the positive semide?-niteness means that all symplectic eigenvalues are non-negative.The uncertainty relation can hence be cast di-rectly in terms of the symplectic eigenvalues of the covariance matrix V ,which are the absolute values of the eigenvalues of ?i ?V .

Some aspects of the harmonic systems can be dis-closed by recognizing that the Gaussian structure of the bosonic states can be thought of as a certain limit of the algebraic structure of the qubits in the sense that Sp ?2,R ??SL ?2,C ?.The latter is the invariance group relevant for qubit entanglement classi?cation and quan-ti?cation ?Dür et al.,2000;Verstraete et al.,2003;Oster-loh and Siewert,2005?.

We now introduce the notion of bipartite entangle-ment for Gaussian states.In complete analogy to the ?nite-dimensional case,a state is termed separable if it is a mixture of product states.In particular,all classical states,i.e.,

?=

?

d 2z 1d 2z 2P ?z 1,z 2??z 1??z 1? ?z 2??z 2?,?21?

with positive P ?z 1,z 2?are separable.

It was Simon ?2000?that ?rst proved the Peres-Horodecki positive partial transpose criterion being nec-essary and suf?cient for entanglement of two harmonic oscillator modes,again in complete analogy to a system of two qubits.The effect of transposition of the density matrix is a sign change in the momentum variables of the Wigner function ?19?.Consequently,a partial trans-position induces a sign change of those momenta in the phase space vector,where the transposition should act

on.For an entangled state,the partial transposition V ?of its covariance matrix V might then have symplectic ei-genvalues smaller than ?/2.This can be detected by the

logarithmic negativity as de?ned from the symplectic

?doubly degenerate ?eigenvalues ?c

?i ;i =1,...,n ?of V ?/??Vidal and Werner,2002?:

E N ?V ?=??i =1n

log 2?2c

?i ?.?22?

These important results paved the way towards a sys-tematic analysis of multipartite systems of distinguish-able bosonic modes.

III.MODEL SYSTEMS

This section is devoted to the basic properties of the

model systems that will be analyzed in the rest of the

review ?in several cases we concentrate on one-dimensional systems ?.

A.Spin models

Interacting spin models ?Auerbach,1998;Schollw?ck et al.,2004?provide a paradigm to describe a wide range of many-body systems.They account for the effective interactions in a variety of different physical contexts ranging from high energy to nuclear physics ?Polyakov,1977;Belitsky et al.,2004?.In condensed matter beside describing the properties of magnetic compounds ?see Matsumoto et al.?2004?for a recent survey ?,they cap-ture several aspects of high-temperature superconduct-ors,quantum Hall systems,and heavy fermions,just to mention few important examples.Hamiltonians for in-teracting spins can be realized arti?cially in Josephson junctions arrays ?Fazio and van der Zant,2001?or with neutral atoms loaded in optical lattices ?Duan et al.,2003;Janéet al.,2003;Porras and Cirac,2004?.Interact-ing spins are paradigm systems for quantum information processing ?Nielsen and Chuang,2000?.

1.Spin-1?2models with short range interactions

A model Hamiltonian for a set of localized spins in-teracting with nearest-neighbor exchange coupling on a d -dimensional lattice is

H ??,?,h z /J ?=

J 2??i ,j ?

??1+??S i x S j x +?1???S i y S j y

?+J ???i ,j ?

S i z S j z ?h z ?i

S i z .

?23?

In the previous expression i ,j are lattice points,?·?con-straints the sum over nearest neighbors,and S i ???=x ,y ,z ?are spin-1/2operators.A positive ?negative ?ex-change coupling J favors antiferromagnetic ?ferromag-netic ?ordering in the x -y plane.The parameters ?and ?account for the anisotropy in the exchange coupling in the z direction,and h z is the transverse magnetic ?eld.There are only very few exact results concerning H ??,?,h z /J ?in dimension d ?1.The ground state of Eq.?23?is in general entangled.It exists,however,for any value of the coupling constants ?and ?,J ?0a point in d =1,2?for bipartite lattices ?where the ground state is factorized ?Kurmann et al.,1982;Roscilde et al.,2005b ?.It occurs at the so-called factorizing ?eld h f given by h f =?z /2?J ??1+??2???/2?2,where z is the coordina-tion number.

In d =1the model is exactly solvable in several impor-tant cases.In the following we discuss the anisotropic quantum XY model ??=0and 0???1?and the XXZ model ??=0?.Also the XYZ model in zero ?eld,? 0,? 0can be solved exactly but it will not be dis-cussed here ?see Takahashi ?1999?for a review ?.

?a ??=0:Quantum XY model .The quantum Ising model corresponds to ?=1while the ?isotropic ?XX model is obtained for ?=0.In the isotropic case the

Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

model possesses an additional symmetry resulting in the conservation of magnetization along the z axis.For any value of the anisotropy the model can be solved exactly ?Lieb et al.,1961;Pfeuty,1970;Barouch and McCoy,

1971?.By?rst applying the Jordan-Wigner transforma-tion c k=e i??j=1k?1?j+?j??k??with?±=?1/2???x±i?y??the XY model can be transformed onto a free fermion Hamil-tonian

H=?i,j

?c i?A i,j c j+12?c i?B i,j c j?+H.c.??+12?i A i,i.?24?

In the previous equation c i,c i?are the annihilation and creation operators for the spinless Jordan-Wigner fermi-ons.The two matrices A,B are de?ned as A j,k =?J??k,j+1+?j,k+1??h z?j,k and B j,k=??J??k,j+1??j,k+1?.For the case of periodic boundary conditions on the spins,an extra boundary term appears in the fermionic Hamil-tonian which depends on the parity of the total number of fermions N F.Note that although N F does not com-mute with the Hamiltonian the parity of N F is con-served.A generic quadratic form,like Eq.?24?,can be diagonalized in terms of the normal-mode spinless Fermi operators by?rst going to the Fourier space and then performing a Bogoliubov transformation.

The properties of the Hamiltonian are governed by the dimensionless coupling constant?=J/2h.In the in-terval0???1the system undergoes a second order quantum phase transition at the critical value?c=1.The order parameter is magnetization in the x direction,?S x?, which is different from zero for??1.In the ordered phase the ground state has a twofold degereracy re?ect-ing a global phase?ip symmetry of the system.Magne-tization along the z direction,?S z?is different from zero for any value of?,but presents a singular behavior of its ?rst derivative at the transition.In the interval0???1the transition belongs to the Ising universality class. For?=0the quantum phase transition is of the Berezinskii-Kosterlitz-Thouless type.

As discussed in Secs.II.A and II.B one-and two-site-entanglement measures can be related to various equal-time spin correlation functions?in some cases the block entropy can be reduced to the evaluation of two-point correlators?M l??t?=???S l??t????and g lm???t?=???S l??t?S m??t????.These correlators have been calcu-lated for this class of models in the case of thermal equi-librium?Lieb et al.,1961;Pfeuty,1970;Barouch and Mc-Coy,1971?.These can be recast in the form of Pfaf?ans that for stationary states reduce to Toeplitz determi-nants?i.e.,determinants,whose entries depend only on the difference of their row and column number?.It can be demonstrated that the equal-time correlation func-tions can be expressed as a sum of Pfaf?ans?Amico and Osterloh,2004?.

?b??=0:XXZ model.The two isotropic points?=1 and?=?1describe the antiferromagnetic and ferromag-netic chains,respectively.In one dimension the XXZ Heisenberg model can be solved exactly by the Bethe ansatz technique?Bethe,1931??see, e.g.,Takahashi ?1999??and the correlation functions can be expressed in terms of certain determinants?see Bogoliubov et al.?1993?for a review?.Correlation functions,especially for intermediate distances,are in general dif?cult to evalu-ate,although important steps in this direction have been made?Kitanine et al.,1999;G?hmann and Korepin, 2000?.

The zero temperature phase diagram of the XXZ model in zero magnetic?eld shows a gapless phase in the interval?1???1.Outside this interval the excita-tions are gapped.The two phases are separated by a Berezinskii-Kosterlitz-Thouless phase transition at?=1 while at?=?1the transition is?rst order.In the pres-ence of the external magnetic?eld a?nite energy gap appears in the spectrum.The universality class of the transition is not affected as a result of the conservation of the total spin-z component?Takahashi,1999?. When one moves away from one dimension,exact re-sults are rare.Nevertheless it is now established that the ground state of a two-dimensional antiferromagnet pos-sesses Néel long range order?Manousakis,1991;Dag-otto,1994?.

2.Spin-1?2models with in?nite range interaction

In this case each spin interacts with all other spins in the system with the same coupling strength H =??J/2??ij?S i x S j x+?S i y S j y???i h i·S i.For site-independent magnetic?eld h i?=h??i,this model was originally pro-posed by Lipkin,Meshkov,and Glick?LMG??Lipkin et al.,1965;Meshkov,Glick,and Lipkin,1965;Meshkov, Lipkin,and Glick,1965?to describe a collective motion in nuclei.In this case the dynamics of the system can be described in terms of a collective spin S?=?j S j?.The pre-vious Hamiltonian reduces to

H=?J

2

??S x?2+??S y?2??h·S.?25?

Since the Hamiltonian commutes with the Casimir op-erator S2the eigenstates can be labeled by the represen-tation S of the collective spin algebra,at most linear in the number N of spins;this reduces?from2N to N/2?the complexity of the problem.A further simpli?cation is achieved at the supersymmetric point corresponding to J2?=4h z,where the Hamiltonian can be factorized in two terms linear in the collective spin?Unanyan and Fleischhauer,2003?;then the ground state can be ob-tained explicitly.For a ferromagnetic coupling?J?0?and h x=h y=0the system undergoes a second order quantum phase transition at?c=1,characterized by mean?eld critical indices?Bottet et al.,1982?.The aver-age magnetization?for any??m z=?S z?/N saturates for ???c while it is suppressed for???c.For h y=0,h z?1, and?=0the model exhibits a?rst order transition at h x=0?Vidal et al.,2006?while for an antiferromagnetic coupling and h y=0a?rst order phase transition at h z =0occurs,where the magnetization saturates abruptly at the same value m z=1/2for any?’s.

Rev.Mod.Phys.,V ol.80,No.2,April–June2008

The model Hamiltonian introduced at the beginning of this section embraces an important class of interacting fermion systems as well.By interpreting the nonhomo-geneous magnetic ?eld as a set of energy levels ?h z ?i ???i ,for h x =h y =0and ?=1,it expresses the BCS model.This can be realized by noting that the operators

S j ?ac j ↑c j ↓,S j +a?S j ???,S j z

a?c j ↑?c j ↑+c j ↓?c j ↓?1?/2span the su ?2?algebra in the representation 1/2.In the fermion language the Hamiltonian reads H BCS =?j ,?=?↑,↓??j c j ??c j ???J /2??ij c j ↑?c j ↓?c i ↓c i ↑.

Both the LMG and BCS type models can be solved exactly by Bethe ansatz ?Richardson,1963;Richardson and Sherman,1964??see also Dukelsky et al.?2004?for a review ?as they are quasiclassical descendants of the six vertex model ?Amico et al.,2001;Di Lorenzo et al.,2002;Ortiz et al.,2005?.

3.Frustrated spin-1?2models

Frustration arises in systems where certain local con-straints prevent the system from reaching a local energy minimum.The constraints can be of geometric nature ?for example,the topology of the underlying lattice ?or of dynamical nature ?two terms in the Hamiltonian tend-ing to favor incompatible con?gurations ?.A classical ex-ample of the ?rst type is that of an antiferromagnet in a triangular lattice with Ising interaction.At a quantum mechanical level this phenomenon can result in the ap-pearance of ground state degeneracies.The equilibrium and dynamical properties of frustrated systems have been extensively studied in the literature ?Diep,2005?in both classical and quantum systems.

A prototype of frustrated models in one dimension is the antiferromagnetic Heisenberg model with nearest-and next-nearest-neighbor interactions.This class of models was discussed originally to study the spin-Peierls transition ?Schollw?ck et al.,2004?.The Hamiltonian reads

H ?=J ?i =1N

?S i ·S i +1+?S i ·S i +2?.

?26?

Analytical calculations ?Haldane,1982?corroborated by

numerical result ?Okamoto and Nomura,1992?indicate that at ??1/4there is a quantum phase transition to a dimerized two-fold degenerate ground state,where sin-glets are arranged on doubled lattice constant distances.Such a phase is characterized by a ?nite gap in the low lying excitation spectrum.

The Majumdar-Ghosh model ?Majumdar and Ghosh,1969a ,1969b ;Majumdar,1970?is obtained from Eq.?26?for ?=1/2.The exact ground state can be solved by means of matrix product states ?see next section ?and it is shown to be disordered.It is a doubly degenerate va-lence bond state made of nearest-neighbor spin singlets.Although all two-point correlation functions vanish,a ?nite four-spin correlation function does re?ect an or-dered dimerization.

4.Spin-1models

Spin-1systems were originally considered to study the quantum dynamics of magnetic solitons in antiferromag-nets with single ion anisotropy ?Mikeska,1995?.In one dimension,half-integer and integer spin chains have very different properties ?Haldane,1983a 1983b ?.Long range order which is established in the ground state of systems with half-integer spin ?Lieb et al.,1961?may be washed out for integer spins.In this latter case,the sys-tem has a gap in the excitation spectrum.A paradigm model of interacting spin-1systems is

H =?i =0N

S i ·S i +1+??S i ·S i +1?2.

?27?

The resulting gapped phase arises because of the pres-ence of zero as an eigenvalue of S i z ;the corresponding eigenstates represent a spin excitation that can move freely in the chain,ultimately disordering the ground state of the system ?Gomez-Santos,1991;Mikeska,1995?.The so-called string order parameter was pro-posed to capture the resulting “?oating”Néel order,made of alternating spins ?↑?,?↓?with strings of ?0?’s in between ?den Nijs and Rommelse,1989?,

O string ?=

lim

R →?

?S i ?

?

?k =i +1

i +R ?1

e i ?S k ?

?

S i +R ??

.

?28?

The ground state of physical systems described by Hamiltonians of the form of Eq.?27?has been studied in great detail ?Schollw?ck et al.,2004?.Various phase tran-sitions have been found between antiferromagnetic phases,Haldane phases,and a phase characterized by a large density of vanishing weights ?S i z =0?along the chain.

The Af?eck-Kennedy-Lieb-Tasaki (AKLT)model .Some features of the phenomenology leading to the de-struction of the antiferromagnetic order can be put on a ?rm ground for ?=1/3?AKLT model ?,where the ground state of the Hamiltonian in Eq.?27?is known exactly ?Af?eck et al.,1988?.In this case it was proven that the ground state is constituted by a sea of nearest-neighbor valence bond states,separated from the ?rst excitation by a ?nite gap with exponentially decaying correlation functions.Such a state is sketched in Fig.1.In fact it is a matrix product state ?MPS ?,i.e.,it belongs to the class of states which can be expressed in the

form

FIG.1.Schematic of the nearest-neighbor valence bond state,exact ground state of the spin-1model in Eq.?27?for ?=1/3?AKLT-model ?.The ground state is constructed regarding ev-ery S =1in the lattice sites as made of a pair of S =1/2,and projecting out the singlet state.The singlets are then formed taking pairs of S =1/2in nearest-neighbor sites.Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

??MPS ?=

?

s 1,...,s N

D

Tr

A 1

s 1ˉ

A N s N ?s 1,

...,s N ?,?29?

where the matrices ?A k

s i

?lm parametrize the state;?s i ?de-notes a local basis of the D -dimensional Hilbert space;the trace contracts the indices l ,m labeling bond states of the auxiliary system ?namely the spin-1/2for the AKLT model ?.The dimensions of A depend on the par-ticular state considered,if the state is only slightly en-tangled then the dimension of A is bounded by some D MPS .MPS,?rst discussed by Fannes et al.?1992?,ap-pear naturally in the density matrix renormalization group ?DMRG ??Ostlund and Rommer,1995?.In one-dimensional noncritical systems they describe faithfully the ground state.In fact,as shown by Vidal,matrix product states constitute an ef?cient representation of slightly entangled states ?Vidal,2003?.

B.Strongly correlated fermionic models

The prototype model of interacting fermions on a lat-tice is the Hubbard model ?Essler et al.,2004?

H =?t ??ij ?

?c i ,??c j ,?+H.c.?+U ?i

n i ,↑n i ,↓??N ,

?30?

where c i ,?,c i ,??are fermionic operators:?c i ,?,c j ,??

??

=?i ,j ????.The coupling constant U describes the on-site repulsion and t is the hopping amplitude.

The Hubbard model possesses a u ?1? su ?2?symme-try expressing the charge conservation u ?1?=span ?N =?j ?n j ??and the invariance under spin rotation su ?2?=span ?S z =?j ?n j ↑?n j ↓?,S +=?j c j ,↑?c j ,↓,S ?=?S +???.Such a symmetry allows one to employ the total charge and magnetization as good quantum numbers.At half ?lling n =N /L =1??=U /2?the symmetry is enlarged to so ?4?=su ?2? su ?2?by the generator ?=?j ???j c j ,↑c j ,↓together with its Hermitian conjugate ?Yang and Zhang,1990?.It was demonstrated that ???=???N ?gs ?are eigenstates of the Hubbard model ?in any dimension ?,characterized by off-diagonal long-range order via the mechanism of the so-called ?pairing ?Yang,1989?.

In one dimension the Hubbard model undergoes a Mott transition at U =0of the Berezinskii-Kosterlitz-Thouless type.By means of the Bethe ansatz solution ?Lieb and Wu,1968?it can be demonstrated how bare electrons decay in charge and spin excitations.The phe-nomenon of spin-charge separation occurs at low ener-gies away from half ?lling.For a repulsive interaction the half-?lled band is gapped in the charge sector;while spin excitations remain gapless.The mirror-inverted situation occurs for an attractive interaction where the gap is in spin excitations instead ?see Essler et al.,2004for a recent review ?.

The Hubbard model in a magnetic ?eld was proven to

exhibit two quantum critical points at h c a ±

=4??U ?±1?and

half ?lling for U ?0,while there is one at h c r =4??t 2+U

2?U ?for U ?0?Yang et al.,2000?.

If a nearest-neighbor Coulomb repulsion V ??,??,j n j ?n j +1??is taken into account in Eq.?30?,a spin density wave and a charge density wave phase appear.A transition to a phase separation of high density and low density regions ?see, e.g.,Clay et al.?1999??is also present.

The bond charge extended Hubbard model,originally proposed in the context of high T c superconductivity ?Hirsch,1989?,include further correlations in the hop-ping process already involved in Eq.?30?.The Hamil-tonian reads

H =U ?i

L n i ,↑n i ,↓?t ?1?x ?n i ,??+n i +1,????c i ,??c i +1,?

+H.c.

?31?

?for x =0the Eq.?31?coincides with the Hubbard model ?30??.For x 0the hopping amplitudes are modulated by the occupancy of sites involved in the processes of tunneling.Because of the particle-hole symmetry,x can be restricted in ?0,1?,without loss of generality.For x =1the correlated hopping term commutes with the in-teraction.In this case the exact ground state was shown to exhibit a variety of quantum phase transitions be-tween insulators and superconducting regimes,con-trolled by the Coulomb repulsion parameter U .For x =1the phase diagram is shown in Sec.IV ,Fig.15.At U /t =4and n =1,a superconductor-insulator quantum phase transition occurs;for ?4?U /t ?4the ground state is characterized by off-diagonal long-range order;the low lying excitations are gapless.For U /t =?4a further quantum critical point projects the ground state into the Hilbert subspace spanned by singly and doubly occupied states ?Arrachea and Aligia,1994;Schadschneider,1995?.For intermediate x the model has not been solved exactly.Numerical calculations indicate a superconducting-insulator transition controlled by U and parametrized by x .Speci?cally,for 0?x ?1/2the phase is gapped at any nonvanishing U ;for 1/2?x ?1the onset to a superconducting phase was evidenced at some ?nite U ?Anfossi,Degli Esposti Boschi,et al.,2006?.

C.Spin-boson models

A prototype model in this class is that of a quantum system coupled to a bath of harmonic oscillators ?see Weiss ?1999?for a review of open quantum mechanical systems ?known also as the Caldeira-Leggett model.In this case the quantum system is a two level system.This class of models was investigated to study the quantum-to-classical transition and the corresponding loss of quantum coherence ?Zurek,2003?.

The spin-boson Hamiltonian has the form

Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

H sb=??

2

S x+?n?n

?a n?a n+1

2?+12S z?n?n?a n?+a n?,?32?

it can be demonstrated to be equivalent to the aniso-tropic Kondo model?Anderson et al.,1970;Guinea, 1985?.The coupling constants??n??x the spectral den-sity of the bath:J???=??/2??n?n2?????n?/?n.At low en-

ergy the spectral function can be represented as a power law:J????2??s?01?s,where?is the parameter control-ling the spin-boson interaction and?0is a ultraviolet cutoff frequency.The power s characterizes the bath. For s=1the bath is called Ohmic,in this case the model has a second order quantum phase transition at?=1 from underdamped to overdamped oscillations?where the value of the spin is frozen?.The value?=1/2iden-ti?es a crossover regime where the two-level system is driven from coherent to incoherent oscillations.If the bath is super-Ohmic?s?1?,the quantum critical point is washed out,while a crossover occurs at??log??0/??. For sub-Ohmic baths?s?1?,several studies indicate the existence of a quantum critical point?Spohn and Dümcke,1985?.The question,however,is not com-pletely settled?Kehrein and Mielke,1996;Stauber and Mielke,2002;Bulla et al.,2003?.

An interesting case is also that of a spin interacting with a single bosonic mode,?n=??n,0:

H=??

2

S x+?0?a0?a0+12

?+?02S z?a0?+a0?.?33?

Such a model describes,for example,an atom interact-

ing with a monochromatic electromagnetic?eld?Cohen-

Tannoudji et al.,1992?via a dipole force?Jaynes and

Cummings,1963?.Recently,the dynamics corresponding

to Eq.?33?was studied in relation to ion traps?Cirac et

al.,1992?and quantum computation?Hughes et al., 1998?.The model de?ned in Eq.?33?with S=1/2?Jaynes-Cummings model?was generalized and solved exactly to consider generic spin?Tavis and Cummings,

1969?in order to discuss the super-radiance phenom-

enon in cavity QED.

D.Harmonic lattices

The Hamiltonian for a lattice of coupled harmonic

oscillators?harmonic lattice?can be expressed in terms of the phase space vector??T=?q1,...,q n;p1,...,p n?as

H=?T?m2?2U0012m1n??,?34?

where U is the n?n interaction matrix for the coordi-nates.If the system is translational invariant the matrix U is a Toeplitz matrix with periodic boundaries,also called circulant matrix?Horn and Johnson,1994?.In the case of?nite range interaction of the form ?r?k=1n K r?q k+r?q k?2and assuming periodic boundary conditions,its entries are U j,j=1+2?r?r and U j,j+r=??r with?r=2K r/m?2.Since the Hamiltonian?34?is qua-dratic in the canonical variables its dynamical algebra is sp?2n,R?.Then the diagonalization can be achieved by RHR?1,where R= ?=1

n exp?i??G??with G?the generic Hermitian element of sp?2,R?.

As discussed in Sec.II.G the key quantity that characterizes the properties of harmonic systems is the covariance matrix de?ned in Eq.?20?.For the resulting decoupled harmonic oscillators it is diag??m??1???1,...,?m??n???1;m??1?,...,m??n???, where?j are the eigenvalues of U.Employing the virial theorem for harmonic oscillators,the covariance matrix for a thermal state with inverse temperature?=1/k B T can be calculated as well,

V=??m??U??1N?0

0?m??U?N?

?,?35?

where N?=1n+2/?exp????U??1n?.The range of the po-sition or momentum correlation functions is related to the low lying spectrum of the Hamiltonian.For gapped systems the correlations decay exponentially.The ab-sence of a gap?some eigenvalues of U tend to zero for an in?nite system?leads to critical behavior of the sys-tem and characteristic long ranged correlations.A rigor-ous and detailed discussion of the relations between the gap in the energy spectrum and the properties of the correlations can be found in Cramer and Eisert?2006?.

IV.PAIRWISE ENTANGLEMENT

At T=0many-body systems are most often described by complex ground state wave functions which contain all correlations that give rise to the various phases of matter?superconductivity,ferromagnetism,quantum Hall systems,etc?.Traditionally many-body systems have been studied by looking,for example,at their re-sponse to external perturbations,various order param-eters,and excitation spectrum.The ground state of many-body systems studied with methods developed in quantum information unveil new properties.In this sec-tion we classify the ground state properties of a many-body system according to its entanglement.We concen-trate on spin systems.Spin variables constitute a good example of distinguishable objects,for which the prob-lem of entanglement quanti?cation is most developed. We discuss various aspects starting from the pairwise en-tanglement,and proceed with the properties of block entropy and localizable entanglement.Most calculations are for one-dimensional systems where exact results are available.Section IV.B overviews the status in the d-dimensional case.Multipartite entanglement in the ground state will be discussed in Sec.VIII.

Rev.Mod.Phys.,V ol.80,No.2,April–June2008

A.Pairwise entanglement in spin chains

1.Concurrence and magnetic order

The study of entanglement in interacting spin systems was initiated with the works on isotropic Heisenberg rings?Arnesen et al.,2001;Gunlycke et al.,2001; O’Connors and Wootters,2001?.O’Connors and Woot-ters aimed at?nding the maximum pairwise entangle-ment that can be realized in a chain of N qubits with periodic boundary conditions.Starting from the assump-tion that the state maximizing the nearest-neighbor con-currence C?1?was an eigenstate of the z component of the total spin?Ishizaka and Hiroshima,2000;Munro et al.,2001;Verstraete et al.,2001?the problem was recast to an optimization procedure similar in spirit to the co-ordinate Bethe ansatz?Bethe,1931?:4the search for the optimal state was restricted to those cases which ex-cluded the possibility to?nd two nearest-neighbor up spins.For?xed number of spins N and p spins up,the

state can be written as???=?1?i

1?ˉ?i p?N b i1ˉi p?i1ˉi p??b

are the coef?cients and i j are the positions of the up-spins?therefore mapping the spin state onto a particle state such that the positions of the p particles corre-spond to those of the up spins.The maximum concur-rence within this class of states can be related to the ground state gas of free spinless particles with the result

C?1?=?1

N

E gs=?

2sin

?p

N?p

sin

?

N?p

.?36?

Equation?36?gives a lower bound for the maximal at-

tainable concurrence.The isotropic antiferromagnetic

chain was considered as the physical system closest to a

perfectly dimerized system?classically,with alternating

up and down spins?.It was noticed,however,that the

concurrence of the ground state of the antiferromagnetic

chain is actually smaller than the value of the ferromag-

netic chain,indicating that the situation is more compli-

cated?O’Connors and Wootters,2001?.In order to

clarify this point,a couple of simple examples are useful. For a system of N=2spins the ground state is a singlet. However for general N?with an even number of sites?

the ground state is not made of nearest-neighbor singlets ?resonant valence bond?RVB?state?.For example,the N=4ground state is?gs?=?1/?6??2?0100?+2?1000???1001???0110???0011???1100??,different from the

product of two singlets.It can be seen that the effect of

the last two components of the state is to reduce the

concurrence with respect to its maximum attainable

value.Given the simple relation Eq.?36?between the

nearest-neighbor concurrence and the ground state en-ergy,the deviation from the RVB state can be quanti?ed by looking at the difference from the exact ground state energy corresponding to the maximum concurrence. This maximum value is reached within the set of eigen-states with zero total magnetization?the“balanced”states in O’Connors and Wootters?2001??,indicating that the concurrence is maximized only on the restricted Hilbert space of z-rotationally invariant states.Indica-tions on how to optimize the concurrence were dis-cussed by Meyer et al.?2004?and Hiesmayr et al.?2006?. The solution to the problem for N→?was given by Poulsen et al.?2006?.It turns out that the states with nearest-neighbor aligned spins?not included in O’Connors and Wootters?2001??correspond to a “density-density”interaction in the gas of the spinless particles considered above,that hence are important for the analysis.?in the analogy of the coordinate Bethe an-satz method,this provides the“interacting picture”?. Following the ideas of Wolf et al.?2003?,the problem to ?nd the optimum concurrence was shown to be equiva-lent to that of?nding the ground state energy of an effective spin Hamiltonian,namely,the XXZ model in an external magnetic?eld.The optimal concurrence is found in the gapless regime of the spin model with a magnetization M z=1?2p/N.It was further demon-strated that states considered by O’Connors and Woot-ters?2001?maximize the concurrence for M z?1/3?for 0?M z?1/3the states contain nearest-neighbor up spins?.

The concurrence,beyond nearest neighbors,in isotro-pic Heisenberg antiferromagnets in an external mag-netic?eld was discussed by Arnesen et al.?2001?;Wang ?2002a?;and Fubini et al.?2006?.The combined effect of the magnetic?eld and the anisotropy in Heisenberg magnets was studied by Jin and Korepin?2004a?making use of the exact results existing for the one-dimensional XXZ model.It turns out that the concurrence increases with the anisotropy??Kartsev and Karshnukov,2004?. For strong magnetic?elds the entanglement vanishes ?the order is ferromagnetic?;for large values of the an-isotropy?the state is a classical Neel state with Ising order.Except for these cases,quantum?uctuations in the ground state lead to entangled ground states.

As discussed in Sec.III,in low-dimensional spin sys-tem there exists a particular choice of the coupling con-stants for which the ground state is factorized?Kurmann et al.,1982?.This is a special point also from the perspec-tive of investigating the entanglement in the ground state.Several works were devoted to the characteriza-tion of the entanglement close to the factorizing point.It turns out that the point at which the state of the system becomes separable marks an exchange of the parallel and antiparallel sector in the ground state concurrence, see Eq.?7?.As this phenomenon involves a global?long-range?reorganization of the state of the system,the range of the concurrence diverges.?We notice that sev-eral de?nitions of characteristic lengths associated with entanglement decay exist.?The concurrence is often ob-served to vanish when the two sites are farther than R

4Such a method relies on the existence of a“noninteracting picture”where the wave function of the system can be written as a?nite sum of plane waves;the ansatz is successful for a special form of the scattering among such noninteracting pictures.

Rev.Mod.Phys.,V ol.80,No.2,April–June2008

sites apart:the distance R is then taken as the range of the concurrence.For the XY model it was found that this range is

R ??

ln

1??

1+?

?

?1

ln ???1??f ?1??1

.

?37?

The divergence of R suggests,as a consequence of the

monogamy of the entanglement ?Coffman et al.,2000;Osborne and Verstraete,2006?,that the role of pairwise entanglement is enhanced while approaching the sepa-rable point ?Roscilde et al.,2004,2005a ,2005b ?.In fact,for the Ising model ?i.e.,?=1?,one ?nds that the ratio ?2/?1→1when the magnetic ?eld approaches the factor-izing ?eld h f ?Amico et al.,2006?.For ? 1and h f ?h z ?h c it was found that ?2/?1monotonically increases

for h z →h f

+

and that the value ??2/?1??h f

+increases with ?→1.The existence of factorizing has been also pointed out in other one-dimensional systems for both short-?Amico et al.,2006;Roscilde et al.,2004,2005a ?and long-range interactions ?Dusuel and Vidal,2005?.In all these cases the range of the two-site entanglement di-verges.The range of the concurrence was also studied for the XXZ ?Jin and Korepin,2004a ?where it was shown to vary as

R =

?

2A ?h z ?1?4M z

2?

?

.?38?

The exponent ?=2for ?nite ?elds,while it is ?=1for

h =0;the coef?cient A ?h ?is known exactly in the para-magnetic phase ?Lukyanov and Zamolodchikov,1997;Lukyanov,1999;Lukyanov and Terras,2003??vanishing magnetization ?and in the saturation limit ?Vaidya and Tracy,1979a ,1979b ?.For generic h it was calculated nu-merically by Hikihara and Furusaki ?2004?.For the iso-tropic Heisenberg antiferromagnet,R =1?Gu et al.,2003?.

In all previous cases the increase in the range of the pairwise entanglement means that all pairs at distances smaller than R share a ?nite amount of entanglement ?as quanti?ed by the concurrence ?.There are one-dimensional spin systems where the pairwise entangle-ment has qualitative different features as a function of the distance between sites.An example is the long-distance entanglement observed by Campos Venuti,De-gli Esposti Buschi,and Roncaglia ?2006?.Given a mea-sure of entanglement E ??ij ?,Campos Venuti,Degli Esposti Buschi,and Roncaglia showed that it is possible that E ??ij ? 0when ?i ?j ?→?in the ground state.Long-distance entanglement can be realized in various one-dimensional models as in the dimerized frustrated Heisenberg models or in the AKLT model.For these two models the entanglement is highly nonuniform and it is mainly concentrated in the end-to-end pair of the chain.

Spontaneous symmetry breaking can in?uence the en-tanglement in the ground state.To see this,it is conve-nient to introduce the thermal ground state ?0

=1

2??gs o ??gs o ?+?gs e ??gs e ??=12??gs ???gs ??+?gs +??gs +??which

is the T →0limit of the thermal state.In the previous expression gs +and gs ?are the symmetry broken states which give the correct order parameter of the model.They are superpositions of the degenerate parity eigen-states gs o and gs e .Being convex,the concurrence in gs ±will be larger than in gs o /e ?Osterloh et al.,2006?.The opposite is true for the concave entropy of entanglement ?see Osborne and Nielsen ?2002?for the single spin von Neumann entropy ?.The spontaneous parity symmetry breaking does not affect the concurrence in the ground state as long as it coincides with C I ,Eq.?7?:that is,if the spins are entangled in an antiferromagnetic way ?Syl-ju?sen,2003b ?.For the quantum Ising model,the con-currence coincides with C I for all values of the magnetic ?eld,and therefore the concurrence is unaffected by the symmetry breaking,the hallmark of the present quan-tum phase transition.For generic anisotropies ?instead,also the parallel entanglement C II is observed precisely for magnetic ?elds smaller than the factorizing ?eld ?Os-terloh et al.,2004?;this interval excludes the critical point.This changes at ?=0,where the concurrence in-deed shows an in?nite range.Below the critical ?eld,the concurrence is enhanced by the parity symmetry break-ing ?Osterloh et al.,2006?

2.Pairwise entanglement and quantum phase transitions

A great number of papers have been devoted to study

entanglement close to quantum phase transition ?QPTs ?.QPT occur at zero temperature.They are induced by the change of an external parameter or coupling con-stant ?Sachdev,1999?.Examples are transitions occur-ring in quantum Hall systems,localization,and the superconductor-insulator transition in two-dimensional systems.Close to the quantum critical point the system is characterized by a diverging correlation length ?which is responsible for the singular behavior of differ-ent physical observables.The behavior of correlation functions,however,is not necessarily related to the be-havior of quantum correlations present in the system.This behavior seems particularly interesting as quantum phase transitions are associated with drastic modi?ca-tions of the ground state.

The critical properties in the entanglement we sum-marize below allow for a screening of the qualitative change of the system experiencing a quantum phase transition.In order to avoid possible confusion,it is worth noting that the study of entanglement close to quantum critical points does not provide new under-standing to the scaling theory of quantum phase transi-tions.Rather it may be useful in a deeper characteriza-tion of the ground state wave function of the many-body system undergoing a phase transition.In this respect it is important to understand,for instance,how the entangle-ment depends on the order of the transition,or what is the role of the range of the interaction to establish the entanglement in the ground state.In this section we dis-cuss exclusively the pairwise entanglement while in the

Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

next section we approach the same problem by looking at the block entropy.5

Pairwise entanglement close to quantum phase transi-tions was originally analyzed by Osborne and Nielsen ?2002?,and Osterloh et al.?2002?for the Ising model in one dimension.Below we summarize their results in this speci?c case.The concurrence tends to zero for ??1and ??1,the ground state of the system is fully polar-ized along the x axes ?z axes ?.Moreover,the concur-rence is zero unless the two sites are at most next-nearest-neighbors,we therefore discuss only the nearest neighbor concurrence C ?1??see,however,Sec.IV .A.1for cases where there is a longer-range pairwise en-tanglement ?.The concurrence itself is a smooth function of the coupling with a maximum close to the critical point ?see the right inset of Fig.2?;it was argued that the maximum in the pairwise entanglement does not occur at the quantum critical point because of the monogamy property ?it is the global entanglement that should be maximal at the critical point ?.The critical properties of the ground state are captured by the derivatives of the concurrence as a function of ?.The results for systems of different size ?including the thermodynamic limit ?are shown in Fig.2.For the in?nite chain ??C ?1?diverges on approaching the critical value as

??C ?1??

8

3?2

ln ????c ?.?39?

For a ?nite system the precursors of the critical behavior can be analyzed by means of ?nite size scaling.In the critical region the concurrence depends only on the combination N 1/?????m ?,where ?is the critical expo-nent governing the divergence of the correlation length and ?m is the position of the minimum ?see the left inset of Fig.2?.In the case of log divergence the scaling ansatz has to be adapted and takes the form ??C ?1??N ,?????C ?1??N ,?0??Q ?N 1/??m ???Q ?N 1/??m ?0?,where ?0is some noncritical value,?m ???=???m ,and Q ?x ??Q ???ln x ?for large x ?.Similar results have been ob-tained for the XY universality class ?Osterloh et al.,2002?.Although the concurrence describes short-range properties,nevertheless scaling behavior typical of con-tinuous phase transition emerges.

For this class of models the concurrence coincides with C I in Eq.?7?indicating that the spins can only be entangled in an antiparallel way ?this is a peculiar case of ?=1;for generic anisotropies the parallel entangle-ment is also observed ?.The analysis of the ?nite size scaling in the,so-called,period-2and period-3chains where the exchange coupling varies every second and third lattice sites,respectively,leads to the same scaling laws in the concurrence ?Zhang and Burnett,2005?.The concurrence was found to be discontinuous at the ?rst order ferromagnetic transition ?=?1in the XXZ chain ?Gu et al.,2003??see Glaser et al.?2003?for ex-plicit formulas relating the concurrence and correlators for the XXZ model in various regimes ?.This result can be understood in terms of the sudden change of the wave function occurring because of the level crossing characterizing these types of quantum critical points.The behavior of the two-site entanglement at the con-tinuous quantum critical point of the Kosterlitz-Thouless type ?=1separating the XY and the antiferro-magnetic phases is more complex.In this case the nearest-neighbor concurrence ?that is the only nonvan-ishing one ?reaches a maximum as shown in Fig.3.Fur-ther understanding of such behavior can be achieved by analyzing the symmetries of the model.At the antiferro-magnetic point the ground state is an su ?2?singlet where nearest-neighbor spins tend to form singlets;away from ?=1,this behavior is “deformed”and the system has the

tendency to reach a state of the type ??q j

?made of q -deformed singlets corresponding to the quantum alge-bra su q ?2?with 2?=q +q ?1?Pasquier and Saleur,1990?.This allows one to rephrase the existence of the maxi-mum in the concurrence as the loss of entanglement as-sociated to the q -deformed symmetry of the system away from ?=1?note that q -singlets are less entangled than undeformed ones ?.This behavior can be traced back to the properties of the ?nite size spectrum ?Gu et al.,2007?.In fact,at ?=1the concurrence can be related to the eigenenergies.The maximum arises since both the transverse and longitudinal orders are power law decay-

5

QPTs were also studied by looking at quantum ?delity ?Cozzini et al.,2006;Zanardi et al.,2006?or the effect of single bit operations ?Giampaolo,Illuminati,and Sienga,2006;Giam-paolo et al.,2008?.

0.7

0.80.9

1

1.1

1.2

1.3

λ

-1.5-1

-0.5

d C (1)/d λ

00.5

1 1.52λ0

0.050.10.150.20.25C (1)

N=infinite

4

5

678

ln N

-14

-12-10-8-6-4-2l n (λm -1)

FIG.2.?Color online ?The derivative of the nearest-neighbor concurrence as a function of the reduced coupling strength.The curves correspond to different lattice sizes.On increasing the system size,the minimum gets more pronounced and the position of the minimum tends ?see the left-hand side inset ?towards the critical point where for an in?nite system a loga-rithmic divergence is present.The right-hand side inset shows the behavior of the concurrence for the in?nite system.The maximum is not related to the critical properties of the Ising model.From Osterloh et al.,2002.

Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

ing at this critical point,and therefore the excited states contribute to C ?1?maximally.

Studies of ?nite size energy spectrum of other models like the dimerized Heisenberg chain ?Sun et al.,2005?and Majumdar-Ghosh model ?Eq.?26?with ?=1/2?show how level crossings in the energy spectrum affect the behavior of the bipartite entanglement occurring at the quantum phase transition ?Gu et al.,2007?.

?a ?LMG model .Because of the symmetry of the LMG models ?see Eq.?25??any two spins are entangled in the same way.The concurrence C is independent on the two site indices,it can be obtained by exploiting the explicit expression of the eigenstates.Due to the mo-nogamy of entanglement the result must be rescaled by the coordination number,C R =?N ?1?C to have a ?nite value in the thermodynamic limit.For the ferromagnetic model ?Vidal,Mosseri,and Dukelsky,2004?,it was proven that close to the continuous QPT,?=1character-izing the ferromagnetic LMG model,the derivative of the concurrence diverges,but,differently from Ising case,with a power law.It is interesting that C R can be related to the so-called spin squeezing parameter ?=2??S n ??Wang and Burnett,2003?,measuring the spin ?uctuations in a quantum correlated state ?the subscript n ?indicates a perpendicular axes to ?S ??.The relation reads ?=?1?C R .According to Lewenstein and Sanpera ?1998?the two-spin reduced density operator can be de-composed into a separable part and a pure entangled state ?e with a certain weight ?.Such a decomposition leads to the relation C ???=?1???C ??e ?.Critical spin ?uctuations are related to the concurrence of the pure state C ??e ?while the diverging correlation length is re-lated to the weight ??Shimizu and Kawaguchi,2006?.Analysis of critical entanglement at the ?rst order quan-tum critical point of the antiferromagnetic LMG model shows that ?Vidal,Mosseri,and Dukelsky,2004?the dis-continuity is observed directly in the concurrence for spin interacting with a long range;see Fig.4.

?b ?Pairwise entanglement in spin-boson models .We

?rst discuss the Tavis-Cummings model de?ned in Eq.?33?.In this model the spin S is proportional to the num-ber of atoms,all interacting with a single mode radiation ?eld.The pairwise entanglement between two different atoms undergoing the super-radiant quantum phase transition ?Lambert et al.,2004,2005;Reslen et al.,2005?can be investigated through the rescaled concurrence C N =NC ,see Fig.5,similar to what has been discussed above for the LMG models.In the thermodynamic limit the spin-boson model can be mapped onto a quadratic bosonic system through a Holstein-Primakoff transfor-mation ?Emary and Brandes,2003?.Many of the prop-erties of the Tavis-Cummings model bear similarities with the ferromagnetic LMG model.In the thermody-namic limit the concurrence reaches a maximum value 1??2/2at the super-radiant quantum phase transition with a square root singularity ?see also Schneider and Milburn,2002?.The relationship between the squeezing of the state and entanglement was highlighted by S?-

?

C (1)

FIG. 3.Nearest-neighbor concurrence of the XXZ model.From Gu et al.,2003

.

0.10.20.3

0.40.50.6

0.7

0.80.910

0.2

0.4

0.6

0.8

1 1.

2 1.4 1.6 1.82

C R

h z

FIG.4.The rescaled concurrence of the antiferromagnetic LMG model.The ?rst order transition occurs at h =0.From Vidal,Mosseri,and Dukelsky,2004.

0.050.10.150.2

0.250.300.20.40.60.81 1.2 1.4 1.6

λ/λc

C

N

N=8N=16N=32N →∞

?6

?4?22

345

Log 2[N]

Log 2FIG.5.?Color online ?The rescaled concurrence between two atoms in the Dicke mode.The concurrence is rescaled both for ?nite N and in the thermodynamic limit.The inset shows the ?nite size scaling.From Lambert et al.,2004.

Rev.Mod.Phys.,V ol.80,No.2,April–June 2008

rensen and M?lmer?2001?and analyzed in more details by Stockton et al.?2003?where it was also suggested how to deal with entanglement between arbitrary splits of symmetric Hilbert spaces?such as the Dicke state span?. Entanglement between the qubits and a single mode and between two spins with a Heisenberg interaction of the XXZ type,additionally coupled to a single bosonic ?eld,was considered by Liberti et al.?2006a,2006b?and He et al.?2006?,respectively.

3.Entanglement versus correlations in spin systems

From the results summarized above it is clear that the anomalies characterizing the quantum critical points are re?ected in the two-site entanglement.At a qualitative level this arises because of the formal relation between the correlation functions and the entanglement.A way to put this observation on a quantitative ground is pro-vided by a generalized Hohenberg-Kohn theorem?Wu et al.,2006?.Accordingly,the ground state energy can be considered as a unique function of the expectation val-ues of certain observables.These,in turn,can be related to?the various derivatives of?a given entanglement measure?Wu et al.,2004;Campos Venuti,Degli Esposti Boschi,et al.,2006?.

Speci?cally,for a Hamiltonian of the form H=H0 +?l?l A l,with control parameters?l associated with op-erators A l,it can be shown that the ground state reduced operators of the system are well-behaved functions of ?A l?.Then,any entanglement measure related to re-duced density operators M=M???is a function of?A l??in absense of ground state degeneracy?by the Hellmann-Feynman theorem:?E/??l=??E/??l?=?A l?. Therefore it can be proven that

M??A l??=M??E??l?,?40?

where E is the ground state energy.From this relation emerges how the critical behavior of the system is re-?ected in the anomalies of the entanglement.In particu-lar,?rst order phase transitions are associated with the anomalies of M while second order phase transitions correspond to a singular behavior of the derivatives of M.Other singularities,like those in the concurrence for models with three-spin interactions?Yang,2005?,are due to the nonanalyticity intrinsic in the de?nition of the concurrence as a maximum of two analytic functions and the constant zero.

The relation given in Eq.?40?was constructed explic-itly for the quantum Ising,XXZ,and LMG models?Wu et al.,2006?.For the Ising model:?l?A l=h?l S l z;the di-vergence of the?rst derivative of the concurrence is then determined by the nonanalytical behavior of?S x S x??Wu et al.,2004?.For the XXZ model:?l?l A l =??l S l z S l+1z.At the transition point?=1both the purity and the concurrence display a maximum.It was proven that such a maximum is also re?ected in a stationary point of the ground state energy as a function of?S i z S i+1z?; the concurrence is continuous since the Berezinskii-Kosterlitz-Thouless transition is of in?nite order.A rel-

evant caveat to Eq.?40?is constituted by the uniaxial LMG model in a transverse?eld?with h y=0and?=0?

that displays a?rst order QPT for h x=0.The concur-

rence is continuous at the transition since it does not

depend on the discontinuous elements of the reduced

density matrix?Vidal,Palacios,and Mosseri,2004?.

The relation between entanglement and criticality was also studied in the spin-1XXZ with single ion aniso-

tropy.It was established that the critical anomalies in the entropy experienced at the Haldane large D?if an axial anisotropy D?i?S i z?2is added to the Hamiltonian in Eq.?27??transition fans out from the singularity of the local order parameter??S z?2??Campos Venuti,Degli Esposti Boschi,et al.,2006a?.

A way to study the general relation between entangle-

ment and critical phenomena was pursued by Hasel-

grove et al.?2004?.It was argued how for systems with

?nite range interaction a vanishing energy gap in the

thermodynamic limit is an essential condition for the

ground state to have nonlocal quantum correlations be-

tween distant subsystems.

4.Spin models with defects

The problem of characterizing entanglement in chains with defects was addressed?rst for the quantum XY models with a single defect in the exchange interaction term of the Hamiltonian?Osenda et al.,2003?.It was found that the effect of the impurity is to pin the en-tanglement.Moreover,the defect can induce a pairwise entanglement on the homogeneous part of the system that was disentangled in the pure system.Even at the quantum critical point the?nite size scaling of the criti-cal anomaly of the concurrence is affected by the dis-tance from the impurity.This basic phenomenology was observed in a variety of different situations that we re-view below.

The presence of two defects has been analyzed in the XXZ chain.It turns out?Santos,2003?that various types of entangled states can be created in the chain by spin ?ip excitations located at defect positions.The entangle-ment oscillates between defects with a period that de-pends on their distance.The anisotropy?of the chain is a relevant parameter controlling the entanglement be-tween defects.Small anisotropies can suppress the en-tanglement?Santos and Rigolin,2005?.This kind of lo-calization,which can be exploited for quantum algorithms,was studied by Santos et al.?2005?.The en-tanglement was also studied in systems with defects in the presence of an external magnetic?eld?Apollaro and Plastina,2006?.It was demonstrated that such a defect can lead to an entanglement localization within a typical length which coincides with the localization length.

A possible way to mimic a defect is to change the

boundary conditions.The concurrence was studied for the ferromagnetic spin-1/2XXZ chain with an antipar-allel boundary magnetic?eld which gives rise to a term in the Hamiltonian of the form H boundary=h?S1?S N??Al-caraz et al.,2004?.The boundary?eld triggers the pres-

Rev.Mod.Phys.,V ol.80,No.2,April–June2008