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An entanglement-area law for general bosonic harmonic lattice systems

M. Cramer1, J. Eisert2,3,1, M. B. Plenio2,3, and J. Dreißig1

1 Institut fur Physik, Universitat Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany2 QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, UK

3 Institute for Mathematical Sciences, Imperial College London, Exhibition Road, London, SW7 2BW, UK

We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respectto the rest of a general harmonic lattice system in the groundor a thermal state scale at most as the boundaryarea of the region. This area law is rigorously proven to holdtrue in non-critical harmonic lattice system ofarbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape and statesof nonzero temperature. For nearest-neighbor interactions – corresponding to the Klein-Gordon case – upper andlower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizingthe findings of [Phys. Rev. Lett.94, 060503 (2005)]. These higher dimensional analogues of theanalysis ofblock entropies in the one-dimensional case show that undergeneral conditions, one can expect an area lawfor the entanglement in non-critical harmonic many-body systems. The proofs make use of methods fromentanglement theory, as well as of results on matrix functions of block banded matrices. Disordered systems arealso considered. We moreover construct a class of examples for which the two-point correlation length diverges,yet still an area law can be proven to hold. We finally considerthe scaling of classical correlations in a classicalharmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment ona general relationship between criticality and area laws for the entropy of entanglement.

PACS numbers: 03.67.Mn, 05.50.+q, 05.70.-a

I. INTRODUCTION

Ground states of quantum systems with many constituentsare typically entangled. In a similar manner as one can iden-tify characteristic length scales of correlation functions, quan-tum correlations are expected to exhibit some general scal-ing behavior, beyond the details of a fine-grained descrip-tion. Such characteristic features provide a physical picturethat goes beyond the specifics of the underlying microscopicmodel. A central question of this type is the following: If onedistinguishes a certain collection of subsystems, representingsome spatial region, of a quantum many-body system in a pureground state, the state of this part will typically have a posi-tive entropy, reflecting the entanglement between this regionand the rest of the system [1, 2, 3, 4, 5, 6, 7, 8]. This de-gree of entanglement is certainly expected to depend on thesize and also on the shape of the region. Yet, how does thedegree of entanglement specifically depend on the size of thedistinguished region? In particular, does it scale as the volumeof the interior – which is meant to be the number of degreesof freedom of the interior? Or, potentially as the area of theboundary, i.e., the number of contact points between the inte-rior and the exterior?

This paper provides a detailed answer to the scaling be-havior of the entanglement of regions with their exterior ina general setting of harmonic bosonic lattice systems and pro-vides a comprehensive treatment of upper bounds on thesequantities. We find that in arbitrary spatial dimensions thedegree of entanglement in terms of the von Neumann entropyscales asymptotically as the area of the boundary of the distin-guished region. This paper significantly extends the findingsof Ref. [9] on harmonic bosonic lattice systems: There, thearea-dependence of the geometric entropy has been provenfor cubic regions in non-critical harmonic lattice systemsofarbitrary dimension with nearest-neighbor interactions,cor-

responding to discrete versions of Klein-Gordon fields. Inthis paper we extend our analysis to a general class of finite-ranged harmonic interactions and also take regions of arbitraryshape into account. For thermal Gibbs states, the entropy ofa reduction is longer a meaningful measure of entanglement.Instead, an area-dependence for an appropriate mixed-stateentanglement measure, the distillable entanglement, is estab-lished. Also, an analogous statement holds for classical cor-relations in classical systems. The area-dependence is evenfound in certain cases where one can prove the divergence ofthe two-point correlation length. This demonstrates that thispreviously conjectured dependence between area and entan-glement is valid under surprisingly general conditions.

The presented analysis will make use of methods from thequantitative theory of entanglement in the context of quantuminformation science [10, 11, 12]. It has been become clearrecently that on questions about scaling of entropies and de-grees of entanglement – albeit often posed some time ago –new light can be shed with such methods [1, 2, 3, 4, 5, 6, 7, 8,9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].In this language, quantum correlations are sharply graspedinterms of rates that can be achieved in local physical transfor-mations. To assess quantum correlations using novel pow-erful tools from quantum information and to relate them toinformation-theoretical quantities constitutes an exciting per-spective.

In the context of quantum field theory, such questions ofscaling of entropies and entanglement have a long traditionunder the keyword of geometric entropy. In particular, workon the geometric entropy of free Klein-Gordon fields wasdriven in part by the intriguing suggested connection [28] tothe Bekenstein-Hawking black hole entropy [29, 30, 31]. Inseminal works by Bombelli et al. [32] and Srednicki [33] therelation between the entropy and the boundary area of the re-gion has been suggested and supplemented with numerical ar-

2

guments. This connection has been made more specific usinga number of different methods. In particular, for half spacesin general and intervals in the one-dimensional case, the prob-lem has been assessed employing methods from conformalfield theory, notably [34, 35], based on earlier work by Cardyand Peschel [36], and by Cardy and Calabrese [14].

In one-dimensional non-critical chains, one observes a sat-uration of the entanglement of a distinguished block, as wasproven analytically for harmonic chains [5, 9] and was laterobserved for non-critical spin chains numerically [6] and an-alytically [7, 13, 14]. In turn, in critical systems systems, oneoften – but not always – finds a logarithmically diverging en-tropy [6, 7, 13, 14]. In case of a model the continuum limit ofwhich leads to a conformal field theory, the factor of the log-arithmically diverging term is related to the central charge ofthe conformal field theory. The findings of the present paperand of the above-mentioned results motivate further questionsconcerning the general area-dependence of the degree of en-tanglement, for example in fermionic systems [24, 25]. In par-ticular, the connection between the geometric entropy fulfill-ing no area law and the correlated quantum many-body systembeing critical is not fully understood yet. This is particularlytrue for the interesting case of more than one-dimensionalquantum systems.

This paper is structured as follows. We start, in Section II,with a presentation of the major results of the paper. In Sec-tion III we define our notation and recall some basics on har-monic lattice systems. Section IV provides a general frame-work of upper and lower bounds for entanglement measures,expressed in terms of the spectrum of the Hamiltonian andtwo-point correlation functions. This analysis is performedboth for the case of the ground state as well as for mixedGibbs states. Of particular interest are Hamiltonians withfinite-ranged interactions in Section V. For such Hamilto-nians, we first study the behavior of the two-point correla-tion functions, then the entanglement bounds. In Section VI,discussing the Gibbs state case, we determine temperaturesabove which there is no entanglement left. A class of exam-ples of Hamiltonians which exhibit a divergent two-point cor-relation length in their ground state, but an area-dependenceof the entanglement is presented in Section IX and expressedin analytical terms. The specific case of Hamiltonians theinteraction part of which can be expressed as a square of abanded matrix is discussed in Section VIII. In this case, veryexplicit expressions for entanglement measures can be found.We then consider, in Section X, the case of classical correla-tions in classical harmonic systems with arbitrary interactionstructure. Interestingly, this case is related to the quantum casefor squared interactions in the sense of Section VIII. Finally,we summarize what has been achieved in the present paper,and present a number of open questions in this context.

II. MAIN RESULTS

Throughout the whole paper we will consider har-monic subsystems on aD-dimensional cubic latticeC =[1, . . . , n]×D. The system thus hasnD canonical degrees of

FIG. 1: A two-dimensional lattice systemC = [1, ..., n]×2 with adistinguished (shaded) regionI , consisting ofv(I) degrees of free-dom . Oscillators representing the exteriorO = C \ I are shownas , whereas pairs belonging to the surfaces(I) of the region aremarked by lines ( ).

freedom. The central question of this work will be: how doesthe degree of entanglement of a distinguished regionI ⊂ Cwith the restO = C \ I scale with the size and shape ofI?

We define the volumev(I) and surfaces(I) of the distin-guished regionI as

v(I) =∑

i∈I

1, s(I) =∑

i∈O

∑

j∈Id(i,j)=1

1,

where

d(i, j) :=D∑

δ=1

|iδ − jδ|

defines the1-norm distance for vectorsi = (i1, . . . , iD) ∈ Cthat specify the position of oscillators on theD-dimensionallattice. More specifically,s(I) is the number of contact points,that is, the number of pairs of sites ofI andO that are immedi-ately adjacent. Note that the distinguished regionI may havearbitrary shape and does not have to be contiguous.

For the pure ground state, we will study the entropy ofentanglement ofI ⊂ C given by

ESC,I = S(I),

whereS = − tr[ρ log ρ] is the von Neumann entropy of astateρ and I = trO[] denotes the reduced state associ-ated with the degrees of freedom of the interiorI. For thepure ground state, this entropy of entanglement is identical toboth the distillable entanglement and the entanglement cost.For pure states, it is indeed the unique asymptotic measure of

3

entanglement. For Gibbs states, in turn, we have to study amixed-state entanglement measure, as the entropy of a sub-system no longer meaningfully quantifies the degree of entan-glement. We will bound the rate at which maximally entan-gled pairs can be distilled, i.e., the distillable entanglementED

C,I(T ). Clearly,EDC,I(0) = ES

C,I for zero temperature.We will subsequently suppress the indexC for notational

clarity. We derive the following properties ofESI andED

I (T ):

(I) For D-dimensional harmonic lattice systems, we derivegeneral upper and lower bounds toES

I for pure groundstates and toED

I (T ) for Gibbs states with respect toa temperatureT > 0. These bounds are expressedentirely in terms of the potential matrix and stated inEqs. (8), (9), and (10).

A necessary condition for the following results to hold is thatthe spectral condition numberκ = λmax(V )/λmin(V ) of thecoupling matrixV satisfiesκ < c < ∞ for somec > 0independent ofI andO.

(II) For nearest-neighbor interactions and the ground state,the entropy of entanglementES

I scales as the surfacearea s(I) of I. More specifically, there exist numbersc1, c2 > 0 independent ofO andI such that

c1 · s(I) < ESI < c2 · s(I).

This is stated in Eqs. (12) and (15). Note that the spe-cific case of cubic regionsI = [1, . . . ,m]×D has al-ready been proven in the shorter paper Ref. [9].

(III) For general finite-ranged harmonic interactions –meaning arbitrary interactions which are strictly zeroafter a finite distance – the entropy of entanglementES

I

of the ground state scales at most linearly with the sur-face area ofI: There exists ac > 0 independent fromO andI such that

ESI < c · s(I).

This is expressed in Eq. (12).

(IV) For Gibbs states with respect to a temperatureT > 0and general finite-ranged interactions the distillable en-tanglementED

I (T ) scales at most linearly with the sur-face area ofI, so

EDI (T ) < c(T ) · s(I),

with c(T ) > 0 being independent ofO andI, see againEq. (12), which applies for any temperature. We alsodetermine temperatures above which the entanglementis strictly zero, see Eq. (13).

(V) We construct a class of Hamiltonians the ground stateof which exhibit an infinite two-point correlation length,for which the entropy of entanglementES

I scales prov-ably at most linearly in the boundary area. This isstated in Eq. (19).

(VI) For interactions that are specified by potential matricesV (see Eq. (1)) that can be written asV = W 2 withWcorresponding to a finite-ranged interaction, we deter-mine the entropy of entanglement. This is made specificin particular forD = 1, see Eq. (18).

(VII) For classical systems of harmonic oscillators withnearest-neighbor interactions prepared in a thermalstate, the mutual information

II(T ) = SI(T ) + SO(T ) − SC(T )

measuring classical correlations scales linearly withthe surface area of the region. Here,SI , SO, andSC

are the discrete classical entropies in phase space ofthe interior, the exterior, and the entire system withrespect to an arbitrary coarse graining as defined inEq. (22), for a classical Gibbs state at temperatureT > 0. The mathematical formulation of the problemis intimately related to case (VI).

The remainder of the paper is now concerned with the de-tailed discussion and rigorous proofs of these statements.

III. HARMONIC LATTICE SYSTEMS – PRELIMINARIES

We consider quantum systems onD-dimensional lattices,where each site is associated with a physical system. Thestarting point is the following Hamiltonian which is quadraticin the canonical coordinates,

H =1

2

(

∑

i

p2i +

∑

i,j

xiVi,jxj

)

. (1)

The matrixV is the potential matrix.Vi,i denotes the potentialenergy contained in the degree of freedom labeled withi. Fori 6= j the elementVi,j describes the coupling between the twooscillators ati andj. We assume thatV is a real symmetricpositive matrix. In this paper, we will consider finite-rangedand nearest-neighbor interactions. Furthermore, interactionsfor which the correlation length diverges are considered.

We will subsequently study properties of both the groundstate(0) as well as of Gibbs states

(T ) =e−H/T

tr[e−H/T ]

corresponding to some nonzero temperatureT . Note that wehave set the Boltzmann constantk = 1. The questions we askare essentially those of the geometric entropy (T = 0), andthose on the distillable entanglement (T > 0) with respect toa distinguished regionI of the latticeC. The specific caseof nearest-neighbor interactions, considered in Refs. [5,9, 20,33] corresponds to the one of a discrete version of free Klein-Gordon fields in flat space-time. Here, we consider regions ofarbitrary shape and allow for more general interactions.

4

A. Phase space, covariance matrices, and ground states

The system we consider embodiesnD canonical degrees offreedom, associated with a phase space(R×2 nD

, σ), wherethe(2nD) × (2nD)-matrixσ,

σ :=

[

0nD 1nD

−1nD 0nD

]

,

specifies the symplectic scalar product, reflecting the canon-ical commutation relations between the2nD canonical coor-dinateso = (x1, . . . , xnD , p1, . . . , pnD ) of position and mo-mentum. Instead of considering states we will refer to theirmoments, considering that both the ground state as well as theGibbs states are quasi-free (Gaussian) states. The second mo-ments can be collected in the(2nD)× (2nD) real symmetriccovariance matrixγ (see, e.g., Ref. [10]), defined as

γj,k := 2ℜ(tr[ojok]) = 2ℜ〈ojok〉.

Here, it is assumed that the first moments vanish, which doesnot restrict generality as they can be made to vanish by lo-cal unitary displacements that do not affect the entanglementproperties.

Following Ref. [5], we find from symplectic diagonaliza-tion that for the ground state at zero temperature

γ0 = V −1/2 ⊕ V 1/2,

i.e., there is no mutual correlation between position and mo-mentum, and the two-point vacuum correlation functions aregiven by the entries ofV −1/2 respectivelyV −1/2,

Gi,j := 〈gs|xixj |gs〉 = [V −1/2]i,j , (2)

Hi,j := 〈gs|pipj |gs〉 = [V 1/2]i,j . (3)

Here,|gs〉 denotes the state vector of the ground state. For thethermal Gibbs state at finite temperature the covariance matrixtakes the form

γ(T ) =(

V −1/2W (T ))

⊕(

V 1/2W (T ))

,

where we define

W (T ) := 1+ 2(exp(V 1/2/T )− 1)−1. (4)

Note that the additional termW (T ) is the same for the posi-tion as well as for the momentum canonical coordinates. As inthe zero temperature case position and momentum are not mu-tually correlated andV −1/2W (T ) (V 1/2W (T )) are the two-point correlation functions of position (momentum), now withrespect to the Gibbs state.

B. Measures of entanglement

The entropy of entanglement can be expressed in terms ofthe symplectic eigenvalues of the covariance matrix corre-sponding to a reduction. Let the(2v(I)) × (2v(I))-matrix

γ0

∣

∣

Idenote the covariance matrix associated with the interior

I; this is the principle submatrix ofγ0 associated with the de-grees of freedom of the interior. The symplectic spectrum ofγ0

∣

∣

Iis then defined as the spectrumµ(BI) of the matrix

BI :=(

γ0

∣

∣

I

)1/2

(i σ|I)(

γ0

∣

∣

I

)1/2

.

For the situation at hand we find that

γ0

∣

∣

I= V −1/2

∣

∣

∣

I⊕ V 1/2

∣

∣

∣

I,

where we denote byV ±1/2|I the principle submatrix ofV ±1/2 associated with the interior and we index the entries ofthe submatrix as[V −1/2]i,j with vectorsi, j ∈ I. Given thedirect-sum structure ofγ0|I and counting doubly degenerateeigenvalues only once, we find that the entropy of entangle-ment can be evaluated as

ESI =

v(I)∑

i=1

[

f(µi − 1

2

)

− f(µi + 1

2

)]

, (5)

wheref(x) = −x log x, and theµi = µi(AI) are the squareroots of the regular eigenvaluesλi = λi(AI) of the v(I) ×v(I) matrixAI ,

µi =√

λi(AI), AI = V −1/2∣

∣

∣

IV 1/2

∣

∣

∣

I.

This reflects the fact that the entropy of a harmonic systemis the sum of the entropies of uncoupled degrees of freedomafter symplectic diagonalization.Using vectorsi, j ∈ I to label the entries ofAI , we find dueto V −1/2V 1/2 = 1,

[

AI

]

i,j=

[

V −1/2∣

∣

∣

IV 1/2

∣

∣

∣

I

]

i,j

= δi,j −∑

k∈O

[

V −1/2]

i,k

[

V 1/2]

k,j.

(6)

This form ofAI = 1 − R hints at an area theorem in thefollowing way (see Fig. 2, where we depict the entries ofR):if the entries ofV ±1/2 decay fast enough away from the maindiagonal, i.e., if the correlation functions decay sufficientlyfast, the main contribution to the entropy comes from oscilla-tors inside a layer around the surface ofI as for all the othersthe product[V −1/2]i,k[V 1/2]k,j will be very small (here vec-torsi, j are insideI, k outside the regionI). Thus, the matrixR has an effective rank proportional tos(I), and the numberof symplectic eigenvalues contributing to the sum in Eq. (5)is approximately proportional tos(I). Much of the remainderof the paper aims at putting this intuition on rigorous grounds.

For mixed states, such as thermal Gibbs states, the von Neu-mann entropy no longer represents a meaningful measure ofthe present quantum correlations, and has to be replaced byconcepts such as the distillable entanglement. The distillableentanglement is the rate at which one can asymptotically dis-till maximally entangled pairs, using only local quantum oper-ations assisted with classical communication. For pure states,

5

0

30

3565

70 30

3565

70

ij

FIG. 2: Entries ofR in one dimension,D = 1, for a finite-ranged coupling matrixV , C = [1, ..., 100] andI = [30, ..., 35] ∪[65, ..., 70], yielding s(I) = 4. Bars showRi,j , color-encoded sur-face depictslog(Ri,j). All units are arbitrary. Note that the entriesdecay exponentially away from the boundary ofI .

it coincides with the entropy of entanglement – the entropy ofa reduction – giving an operational interpretation to this quan-tity [10, 11, 12].

From entanglement theory we know that an upper boundfor the distillable entanglement is provided by the logarithmicnegativity [37]. Note that this is the case even in this infinite-dimensional context for Gaussian states can be immediatelyverified on the level of second moments and a single-modedescription. Moreover, the entropy of entanglement indeedstill has the interpretation of a distillable entanglement, albeitthe fact that distillation protocols leave the Gaussian setting.This is true as long as one includes an appropriate constraintto the mean energy in the distillation protocol [38].

The logarithmic negativity [37] is defined as

ENI (T ) = ‖(T )Γ‖1,

where‖.‖1 denotes the trace norm, and(T )Γ is the partialtranspose of (T ) with respect to the splitI andO = C \ I.Again following Ref. [5], we find after a number of steps

ENI (T ) =

∑

i∈C

log2

[

max{

1, λi

(

Q)

}

]

, (7)

whereλi(Q) labels thenD eigenvalues ofQ,

Q := Pω−Pω+,

and matricesω±(T ) are defined as

ω±(T ) := W (T )−1V ±1/2,

which becomeω± = V ±1/2 for zero temperature following

Eq. (4). The diagonal matrixP , defined as

Pi,j = δij

(

δi∈O − δj∈I

)

,

δi∈S =

{

1 if i ∈ S

0 otherwise,

is the matrix that implements time reversal in the subsystemcorresponding to the inner partI, reflecting partial transposi-tion Γ on the level of states.

IV. UPPER AND LOWER BOUNDS

In this section we will derive upper and lower bounds forthe entropy of entanglement and the distillable entanglementof the distinguished regionI with respect to the rest of thelattice. These bounds only depend on the geometry of theproblem, i.e., the regionI and on properties ofV , namely itsminimal and maximal eigenvalue, which we define asa :=λmin(V ) andb := λmax(V ), respectively, its condition number

κ :=b

a=λmax(V )

λmin(V ),

and the entries ofV ±1/2 respectively the two-point correlationfunctionsG andH as in Eqs. (2,3). In later sections thesebounds will be made specific for a wide range of interactionmatricesV .

A. Upper bound

An upper bound for the entropy of entanglement and thedistillable entanglement is provided by the logarithmic nega-tivity as in Eq. (7). Utilizing this fact, we derive upper boundsusing l1-norms [39] in this section. For brevity and clarity,we will present the case forT = 0 and finite temperature in asingle argument.

A direct calculation shows that the matrixQ is given by

Q = ω−ω+ − 2Xω+

= W (T )−2 − 2Xω+,

introducing the matrixX with entries

Xi,j := ω−i,j

(

δi∈Iδj∈O + δi∈Oδj∈I

)

.

Therefore, we can bound the eigenvalues ofQ according to

λi(Q) ≤ λmin(W (T ))−2 + λi(−2Xω+)

≤ λmin(W (T ))−2 + 2∣

∣

∣λi(Xω+)∣

∣

∣ ,

where we denote byλmin(W (T )) the smallest eigenvalue ofW (T ) which is given by

λmin(W (T ))−1 = λmax

(

eV 1/2/T − 1eV 1/2/T + 1) =

e√

b/T − 1

e√

b/T + 1.

6

Hence, we can write

ENI (T ) ≤

∑

i∈C

log2

[

max

{

1, λmin(W (T ))−2 + 2∣

∣

∣λi(Xω+)∣

∣

∣

}

]

≤ 1

ln(2)

∑

i∈C

max

{

0, λmin(W (T ))−2 − 1 + 2∣

∣

∣λi(Xω+)∣

∣

∣

}

,

i.e., for λmin(W (T ))−2 + 2 maxi |λi(Xω+)| < 1, there is

no longer any bi-partite entanglement in the system. We willlater see that there is a temperatureTc above which this hap-pens. But for now, we use the fact thatλmin(W (T ))−2 ≤ 1 tobound the logarithmic negativity, and relate it to thel1 norm‖.‖l1 [39] of X . We haveλmax(ω+) =

√b/λmin(W (T )) and

therefore

ENI (T ) ≤ 1

ln(2)

∑

i∈C

2|λi(Xω+)| =

2

ln(2)||Xω+||1

≤ 2√b

λmin(W (T )) ln(2)||X ||1 ≤ 2

√b

λmin(W (T )) ln(2)||X ||l1 .

Thel1 norm is defined as the sum of the absolute values of allmatrix entries. Inserting the definition ofX , we find

λmin(W (T )) ln(2)

2√b

EDI ≤ ||X ||l1 =

∑

i,j∈C

∣

∣

∣Xi,j

∣

∣

∣

= 2∑

j∈Ii∈O

∣

∣

∣ω−i,j

∣

∣

∣

(8)

for finite temperature, and

ESI ≤ 4

√b

ln(2)

∑

j∈Ii∈O

∣

∣

∣[V −1/2]i,j

∣

∣

∣ (9)

for zero temperature. These constitute upper bounds on theentropy of entanglement and the distillable entanglement.Both only depend in the distinguished regionI, the maximumeigenvalue ofV , and the entries ofV −1/2, i.e., the two-pointcorrelation function with entriesGi,j . This will be the start-ing point to derive explicit upper bounds for special types ofinteraction matricesV .

B. Lower bound

To achieve lower bounds for the zero temperature case,we consider the entropy of entanglement directly. Startingfrom the general expression Eq. (5), we use the fact that thesymplectic eigenvalues are never smaller than1, i.e., that theeigenvalues ofAI are contained in the interval[1, αI ], withαI being the maximal eigenvalue ofAI . Using the pinch-ing inequality [39], we findαI ≤ (λmax(V )/λmin(V ))1/2 =(b/a)1/2 = κ1/2. Thus we can bound the entropy of entan-

glement as follows,

ESI ≥

v(I)∑

i=1

log[µi] =

v(I)∑

i=1

log[λi]

2≥ log[αI ]

2(αI − 1)

v(I)∑

i=1

(

λi − 1)

≥ log[√κ]

2(√κ− 1)

tr[

AI − 1] .Using vectorsi, j ∈ I to label the entries of thev(I) × v(I)matrixAI , we finally arrive at the lower bound

ESI ≥ − log[

√κ]

2(√κ− 1)

∑

i∈Ij∈O

[

V −1/2]

i,j

[

V 1/2]

j,i. (10)

This lower bound depends only of the geometry ofI, the spec-tral condition numberκ of V and the entries ofV ±1/2, i.e., thetwo-point correlation functions.

Eq. (10) is difficult to evaluate in general but for specialcases of the interaction matrixV it can nevertheless be madespecific. In Section VII, for example, we give an explicit ex-pression for the important case of nearest-neighbor interac-tions. We expect Eq. (10) to be a convenient starting point toderive such lower bounds also for more general cases ofV .

A lower bound for the finite temperature case is difficult toobtain. Generally speaking, a lower bound to the distillableentanglement (two-way distillable entanglement in our case)is given by the hashing inequality [40],

EDI (T ) ≥ max{S(I) − S(), S(O) − S(), 0},

whereI = trO[] andO = trI [ρ]. Yet, naively applied, thisinequality will vanish, as generallyS(I) − S() < 0 andS(O) − S() < 0: this can be made intuitively clear fromthe following argument. The above analysis demonstrates thatboth the interiorI and the exteriorO can be approximatelydisentangled with local unitaries up to a layer of the thick-ness of the two-point correlation length. That is, degrees offreedom associated withi ∈ O for which d(i, j) is suffi-ciently large for everyj ∈ s(I), can to a very good approxi-mation be decoupled and unitarily transformed into a thermalGibbs state. Each such degree of freedom will therefore con-tribute a constant number toS(), such thatS(I)−S() < 0for sufficiently largen. Similarly, one can argue to arrive atS(O) − S() < 0. In order to establish a lower bound tothe distillable entanglement, however, we may start with anyprotocol involving only local quantum operations and classi-cal communication, and apply the hashing inequality on theresulting quantum states. This first step can include in partic-ular local filterings. A bound linear in the boundary area isexpected to become feasible if one first applies an appropriateunitary both inI andO, and then performs a local filtering in-volving degrees of freedom associated withi ∈ I andj ∈ Ofor whichd(i, j) = 1. This option will be explored elsewhere.

V. FINITE-RANGED INTERACTIONS

In this section we will make the upper bound on the entropyof entanglement and the distillable entanglement explicitfor

7

symmetric finite-ranged interaction matricesV , i.e., matricesfor which

Vi,j = 0, for d(i, j) > k/2,

whered(i, j) denotes again the1-norm distance. Denotingas before the maximum and minimum eigenvalue ofV asa = λmin(V ) andb = λmax(V ) respectively, we require thatthe spectral condition numberκ = b/a be strictly less thaninfinity independent ofC, i.e., independent ofn. Note that wedo not require any further assumptions on the matrixV .

A. General upper bounds for finite-ranged interactions

We will make use of a result of Ref. [41] concerning theexponential decay of entries of matrix functions. After gener-alizing this result to matricesV with the properties specifiedabove it enables us to bound the entries ofω− as follows (seeAppendix A)

∣

∣

∣ω−i,j

∣

∣

∣ ≤ Ka,b qd(i,j)κ , qκ =

(

κ− 1

κ+ 1

)2/k

, (11)

and

Ka,b =eη/T − 1

eη/T + 1

κ+ 1

η,

where

η :=

(

aκ

κ+ 1

)1/2

.

For zero temperature we then find

Ka,b =κ+ 1√a

(

κ+ 1

κ

)1/2

.

This shows that off-diagonal terms ofV −1/2 decay exponen-tially, cf. Fig. 3. Substituting Eqs. (11) into the general resultin (8), we find

λmin(W (T )) ln(2)

4√bKa,b

ENI (T ) ≤

∑

i∈Ij∈O

qd(i,j)κ =

∞∑

l=1

qlκNl,

where

Nl =∑

j∈O

∑

i∈Id(i,j)=l

1.

Note thatN1 = s(I) coincides with the definition of thesurface area ofI. We findNl ≤ 2(2l)2D−1s(I) (see Ap-pendix C), i.e., we have

∑

i∈Ij∈O

qd(i,j)κ ≤ 22Ds(I)

∞∑

l=1

qlκl

2D−1 =: KD,κs(I).

0

00 3 6 9

x1

x2

x10

0

3

6

90

3

6

9

FIG. 3: The entries[V −1/2]i,j for a finite-ranged coupling matrixVin two dimensions,D = 2, j = i + (x1, x2). Bars show[V −1/2]i,j

and the color-encoded surface showslog([V −1/2]i,j ). The inset de-picts the same forx1 = x2. All units are arbitrary. Note the expo-nential decay away from the main diagonali = j.

Thus we finally arrive at the desired upper bound linear in thesurface area ofI for both the entropy of entanglement and thedistillable entanglement,

ENI (T ) ≤ 4

√bKa,bKD,κ

λmin(W (T )) ln(2)s(I), (12)

which becomes

ENI ≤ 4(κ+ 1)

32KD,κ

ln(2)s(I)

for zero temperature, i.e., these upper bounds depend solelyon the maximal and minimal eigenvalue of the interaction ma-trix V and the surface area of the regionI. This result demon-strates that indeed, an area-bound of the degree of entangle-ment holds in generality for bosonic harmonic lattice systems.This shows that the previously expressed intuition can indeedbe made rigorous in form of an analytical argument.

B. Disordered systems

Notably, the derived results hold also for systems in whichthe coupling coefficients are not identical, but independent re-alizations of random variables. If the coefficientsVi,j of thereal symmetric matrixV are taken from a distribution with acarrier [x0, x1], such that the interaction is (i) finite-ranged,and (ii) the carrier is chosen such that

κ =λmax(V )

λmin(V )<∞,

then the same result holds true. This follows immediatelyfrom the considerations in Appendix A, where it is not as-sumed that the Hamiltonian exhibits translational symmetry.Eq. (12) is thus valid also for disordered harmonic lattice sys-tems.

8

VI. TEMPERATURES ABOVE WHICH THERE IS NOMORE ENTANGLEMENT LEFT

In Section IV A we found that forλmin(W (T ))−2 +2 maxi |λi(Xω

+)| < 1 there is no entanglement between theregionsI andO. For finite-range interactionsV we found inSection V

maxi

|λi(Xω+)| ≤ λmin(W (T ))−1

√b||X ||l1

≤ 2λmin(W (T ))−1√bKa,bKD,κs(I),

i.e., we have

λmin(W (T ))−2 + 2 maxi

|λi(Xω+)| ≤

λmin(W (T ))−2 + 4λmin(W (T ))−1√bKa,bKD,κs(I).

On the right-hand side onlyKa,b andλmin(W (T ))−2 dependon the temperature, both are decreasing inT and go to zero asT goes to infinity, i.e.,

λmin(W (T ))−2 +4√bKa,bKD,κs(I)

λmin(W (T ))= 1 (13)

gives an implicit equation for the temperatureTc above whichthere is no bi-partite entanglement left in the system.

VII. NEAREST-NEIGHBOR INTERACTIONS

In this section we consider nearest-neighbor interactionsand periodic boundary conditions inD spatial dimensions,i.e., block-circulant matricesV . Forn → ∞ this is a specialcase of the matrices considered in Section V as boundary con-ditions become irrelevant in this limit, i.e., the upper boundcoincides with the one derived for finite-ranged interactions.For a tighter bound see Appendix B. We will now make useof the circulant structure ofV to show that for matrices of thiskind it is possible to also derive a lower bound on the entropyof entanglement that it is proportional to the surface area ofI. We writeM = circ(M) for the circulant matrixM whosefirst block column is specified by the tupel of matricesM . Wecan then recursively defineV = VD via

Vδ+1 = circ(Vδ,−c1nδ , 0nδ , . . . , 0nδ ,−c1nδ),

whereV1 = circ(1,−c, 0, . . . , 0,−c). We choose an energyscale in whichVi,i = 1 and we demandc > 0 [42]. Thiscirculant structure leads to the following properties ofV : theeigenvalues ofV are given by

λi(V ) = 1 − 2cD∑

δ=1

cos(

2πiδ/n)

, i ∈ C,

in particular the maximum eigenvalue is given byλmax(V ) ≤1 + 2cD, where equality holds forn even. The minimumeigenvalue readsλmin(V ) = 1 − 2cD, i.e., positivity ofV de-mandsc < 1/2D. Note that the assumption thatc < 1/2D

independent ofI andO is essential for the argument. If weallow for ann-dependence ofc as it arises, e.g., in the field-limit wherec = 1/(1/n2+2D), then in one spatial dimensionone will encounter an area law up to a logarithmically diver-gent correction. This behavior – resembling the behavior ofcritical spin chains and quadratic fermionic models – will bestudied in more detail elsewhere.

Furthermore the circulant nature ofV yields the followingexplicit expressions for the entries ofV ±1/2

[

V ±1/2]

i,j=

1

nD

∑

k∈C

e2πik(i−j)/n(

λk(V ))±1/2

. (14)

This reasoning leads to the following properties fori 6= j thatare crucial for the present proof (see Appendix B),

− [V −1/2]i,j[V1/2]j,i =

∣

∣

∣[V −1/2]i,j[V1/2]j,i

∣

∣

∣

≥(

c

2

)2d(i,j)(d(i, j)!

(2d(i, j) − 1)∏D

δ=1 |iδ − jδ|!

)2

.

Substituting these results into the general lower bound (10)and keeping only terms withd(i, j) = 1 immediately yields

ESI ≥ log[

√κ]c2

8(√κ− 1)

s(I), (15)

where

κ =1 + 2cD

1 − 2cD

in this case. This generalizes the result of Ref. [9] to regionsI of arbitrary shape. We expect that these lower bounds canbe generalized to other interactionsV and numerical resultssuggest that these bounds hold quite generally.

VIII. SQUARED INTERACTIONS

A simple special case is related to a certain kind of inter-action: this the one where the interaction matrixV can bewritten as

V = M2

with a real symmetric matrixM corresponding to a finite-range interaction. Then, the covariance matrix associatedwiththe interior is nothing but

γ0

∣

∣

I= M−1

∣

∣

∣

I⊕ M

∣

∣

I. (16)

In this case, the symplectic spectrum can be determined in afairly straightforward manner. We mention this case also asit appears to be an appropriate toy model as a starting pointfor studies aiming at assessing the symplectic spectrum of thereduction itself and therefore the spectrum of the reduced den-sity matrix [26].

9

We are looking for the spectrum of thev(I) × v(I) matrixAI as in Eq. (6) which now takes the form (recall that we labelthe entries ofAI by vectorsi, j ∈ I)

[

AI

]

i,j= δi,j −

∑

k∈O

[M−1]i,kMk,j = δi,j −Ri,j .

It is now the central observation that the number of rows ofthe matrixR that are nonzero – and therefore also the numberof eigenvalues that are nonzero – is proportional to the surfacearea ofI, asM is a banded matrix. That is,

Mi,j = 0

for d(i, j) > k/2, meaning thatRi,j = 0 if d(k, j) > k/2for all k ∈ O. As the eigenvalues ofAI are bounded throughthe pinching inequality by1 andλmax(W )/λmin(W ), an up-per bound linear in the surface area of the interior follows im-mediately from the fact that the number of eigenvalues enter-ing the sum in Eq. (5) is proportional tos(I). This argumentdemonstrates that in this simple case, one immediately arrivesat bounds that are linear in the number of contact points.

The task of finding the eigenvalues ofAI explicitly is nowreduced to finding the eigenvalues of the sparse matrixR,

µi(AI) =√

1 − λi(R).

This case is particularly transparent in the one-dimensionalcase,D = 1, and forM being a circulant matrix with firstrow (1,−c, 0, . . . ,−c) for 0 < c < 1/2. The potential ma-trix V corresponds then to nearest-neighbor interactions, to-gether with next-to-nearest-neighbor interactions. In this one-dimensional setting, we setI = [1, . . . ,m]. The matrixRthen takes the simple form (cf. Fig. 4)

Ri,j =n∑

k=m+1

[M−1]i,kMk,j

= −c(

[M−1]i,m+1δj,m + [M−1]i,nδj,1

)

.

To find its eigenvalues, we calculatedet(R − λ1), which isstraightforward for a matrix of this form.

det(R − λ1) = −(−λ)m−2 det

(

R11 − λ R1m

Rm1 Rmm − λ

)

= (−λ)m−2(R1mRm1 − (R11 − λ)(Rmm − λ)),

reflecting the fact thatm − 2 eigenvalues are zero, i.e, thenumber of nonzero eigenvalues iss(I) = s([1, . . . ,m]) = 2.We find for the non-vanishing eigenvalues

2λ± = R11 +Rmm

±(

(R11 −Rmm)2 + 4R1mRm1

)1/2

.

Symmetry ofM−1 yields

λ± = R11 ±R1m = −c(

[M−1]1,n ± [M−1]m,n

)

.

0

0

0

1

4

7

11

14

1

4

7

11

14

1 4 7 11 14i

i

j

FIG. 4: Matrix entries ofR for the caseD = 1, C = [1, ..., 100],I = [1, ..., 15], V = M2, whereM is a nearest-neighbor circulantmatrix. Note that for this particular form ofV the entriesRi,j areexactly zero forj 6= 1, m. This is in contrast to the general form ofR depicted in Fig. 2. The inset shows the entriesRi,1 as bars andlog(Ri,1) in blue (cf. Eq. (17)).

From the circulant structure ofM we have

[M−1]i,j =1

n

n∑

k=1

e2πik(i−j)/n

1 − 2c cos(2πk/n)

=1

n

n∑

k=1

cos(2πk(i− j)/n)

1 − 2c cos(2πk/n),

i.e., for largen we arrive at

[M−1]1,n =1

2c√

1 − 4c2− 1

2c,

[M−1]m,n =1√

1 − 4c2

(

1 −√

1 − 4c2

2c

)m

. (17)

Note that these expressions are asymptotically independent ofm as shown in the inset of Fig. 4. In this limit we finally arriveat

λ± =1

2− 1

2√

1 − 4c2. (18)

This expression specifies the symplectic spectrum of the re-duction in a closed form.

IX. DIVERGING CORRELATION LENGTH ANDAREA-LAW OF THE ENTROPY

In this section we will analytically demonstrate that thereexist Hamiltonians for which the ground state two-point cor-relation functions diverge whereas the geometric entropy is

10

still bounded by an area-law. In spin systems with a long-range Ising interaction such a behavior has been observed inthe one-dimensional case and sketched for higher dimensions[23]. Here, we present a class of examples of this type validin arbitrary dimensionD, for which one can prove the validityof an upper bound linear in the boundary area. Moreover, thisset of examples is not restricted to cubic regions. The interac-tion is here a suitable harmonic long-range interaction. Thisanalysis shows that a divergent two-point correlation functionalone is no criterion for a saturating block entropy in the one-dimensional case, and for an area-dependence in higher di-mensions.

Consider the matrixM with entries

Mi,j =1

d(i, j)α,

Mi,i = 1 +∑

j∈Cj 6=i

1

d(j, i)α

for someα > 0. Now setV = M−2. This choice implies thatthe correlation functionGi,j = [V −1/2]i,j decays only alge-braically. We will show that despite this fact one still has anupper bound linear in the boundary area ofI for appropriatevalues ofα.

Firstly, we have to make sure that the maximum eigenvalueof V can be bounded from above independent ofn. FromGershgorin’s theorem (see, e.g., Ref. [39]) we know that forevery eigenvalueλ(M) there exists ani such that

|λ(M) −Mi,i| ≤∑

j∈Cj 6=i

Mi,j,

i.e.,

λmin(M) ≥ mini

[

Mi,i −∑

j∈Cj 6=i

Mi,j

]

= 1,

and therefore

b = λmax(V ) = λmax(M−2) =

1

λ2min(W )

≤ 1.

Substituting this and the specific form ofM into the generalexpression for the entropy, we obtain

ln(2)

4ES

I ≤∑

j∈Ii∈O

1[

d(i, j)]α =

∞∑

l=1

1

lαNl

≤ 22Ds(I)∞∑

l=1

l2D−α−1,

which converges forα > 2D to

ESI ≤ 22D+2

ln(2)ζ(α− 2D + 1)s(I), (19)

whereζ(x) is the Riemann zeta function. Note that thesebounds are not necessarily tight in the sense that even for

smaller values ofα, such a behavior can be expected. Stepstowards tightening these bounds seem particularly feasible incase of cubic regions, where we conjecture that forα > Dwe arrive at an area-dependence. This analysis shows that forlong-range interactions, an area-law in the degree of entangle-ment can be concomitant with divergent two-point correlationfunctions.

X. AN AREA LAW FOR CLASSICAL CORRELATIONS

In previous sections we have considered the entanglementbetween some region and the rest of a lattice of interactingquantum harmonic oscillators with Hamiltonians that are atmost quadratic in position and momentum. We have demon-strated that both for the ground state and the thermal state ofthe entire lattice the quantum correlations, i.e., the amount ofentanglement, between the region and the remainder of thelattice is bounded by quantities proportional to their boundarysurface area.

This suggests similar questions concerning the classicalcorrelations between a region and the rest of the lattice in thecorresponding classical systems when the lattice system asawhole is prepared in a thermal state. In this section we willdemonstrate that indeed analogous area laws hold. We willnote furthermore that there is a quite striking intimate rela-tion between theclassical systemwith potential matrixVc andthequantum systemwith a potential matrixVq = V 2

c in thiscontext.

A. Hamiltonian, entropy, and mutual information

For the following considerations we use the classical equiv-alent of the quantum mechanical Hamiltonian (1), namelyagain

H =1

2

(

∑

i

p2i +

∑

i,j

xiVi,jxj

)

, (20)

where nowx = (x1, . . . , xnD ) andp = (p1, . . . , pnD) arevectors of classical position and momentum variables, respec-tively, andV denotes again the potential matrix. The state ofthe classical system is characterized by a phase space den-sity = (ξ), a classical probability distribution, whereξ = (x1, . . . , xnD , p1, . . . , pnD ) denotes all the canonical co-ordinates in phase space. For nonzero temperaturesT > 0this phase space distribution is given by the Boltzmann distri-bution

(ξ) =1

Ze−βH(ξ) (21)

whereβ = 1/T (as before,k = 1), and

Z :=∫

dξe−βH(ξ).

Given this density in phase space we will encounter a famil-iar ambiguity when defining the entropy of the system. Using

11

the discrete classical entropy [43], we split the phase spaceinto cubic cells each with a volumeh2N , whereN = kD andh > 0 being an arbitrary constant. From the phase space den-sity we obtain the probability associated with each of thesecells which in turn can be used to determine the entropy func-tion of this probability distribution. We will now make use ofthe multiple indicesi = (i1, . . . , iN) andj = (j1, . . . , jN ),assuming that the cell corresponding to(i, j) is centeredaroundx = (hi1, . . . , hiN) andp = (hj1, . . . , hjN ). Thatis, for each degree of thenD degrees of freedom, the phasespace is discretized. The contribution of each cell to the dis-cretized probability density is then given by

p(i, j) =

∫

Celldξ(ξ) .

As usual, the discrete classical entropy is then defined as thecorresponding Shannon entropy

SC(h) := −∑

i,j

p(i, j) log p(i, j) (22)

We will denote the discrete classical entropy with respect tothe degrees of freedom of the interior withSI(h), the entropyof the exterior withSO(h).

The value for the entropy will depend on the choice ofhand in the limith→ 0 this entropy definition will diverge dueto a term proportional to− log(h2N ). In classical statisticalmechanics this problem is avoided with the help of the thirdlaw of thermodynamics. The entropy itself is however notthe quantity that we wish to compute but rather the mutualinformationII between the interiorI and the rest of a lattice,denoted as before byO. This classical mutual informationmeaningfully quantifies the classical correlations between theinner and the outer. In that case we find that the limith → 0exists and that the mutual informationI can be defined as

II := limh→0

(

SI(h) + SO(h) − SC(h))

.

Following these preparations we are now in a position to de-termine the mutual information between a region and the restof the lattice explicitly when the lattice as a whole is in a ther-mal state.

B. Evaluation of the mutual information

For the evaluation of the mutual information we need todetermine the entropy of the total latticeC, as well as theentropy determined by the reduced densities describing thetwo regionsI andO. To this end we carry out the partialsummation over all degrees of freedom of regionO in orderto find the reduced phase space densityI describing regionI only. Employing the Schur complement we find that thereduced densityI is described by the Boltzmann distributioncorresponding to the same temperature and the Hamiltonian

HI =1

2

∑

i,j∈I

xi[(V∣

∣

I)−1]i,jxj +

∑

i∈I

p2i

(23)

An analogous result holds for the reduced phase space densityof regionO.

For a thermal phase space distribution Eq. (21) correspond-ing to a classical Hamiltonian function of the form Eq. (20)we can compute the entropy straightforwardly, to find

SA = −1

2log det

(

V∣

∣

A

)−1

+ v(A) log2π

β+ v(A)

A ∈ {I,O,C}, which increases with temperature as ex-pected. For the mutual information we find

II =1

2log

det V∣

∣

Idet V

∣

∣

O

detV,

which is, perhaps surprisingly, independent of temperature.Using Jacobi’s determinant identity

det V∣

∣

OdetV −1 = det V −1

∣

∣

∣

I

this expression can be rewritten as

II =1

2log det

[

V∣

∣

IV −1

∣

∣

∣

I

]

=1

2log det

[1−R]

.

It is now advantageous to notice the close connection of thisexpression, in particular of the matrixR

Ri,j =∑

k∈O

[V −1]i,kVk,j

with those that arise in the quantum mechanical problem thatwe have treated previously in Section VIII. Indeed, the clas-sical problem for a system with potential matrixVc is relatedto the quantum mechanical system with the squared poten-tial matrixVq = V 2

c . This formal similarity arises because inSection VIII we have shown that for a lattice of quantum har-monic oscillators with potential matrixVq = V 2

c in its groundstate the symplectic eigenvalues of the covariance matrix de-scribing region I alone are exactly the standard eigenvalues ofthe matrix(1 − R)1/2. The properties of these eigenvalueshave already been discussed in detail in Section VIII. Thisallows us now straightforwardly to establish the area theoremfor the mutual information in a classical system employing theresult for the corresponding quantum system.

This establishes in particular that the classical correlationsas measured by the mutual informationIC,I between the dis-tinguished regionI and the rest of the latticeO = C\I satisfy

c′1 · s(I) ≤ II(T ) ≤ c′2 · s(I)

for largev(I) and appropriate constantsc′1,2 > 0 independentof O and I. (Compare also the assessment of the thermo-dynamical entropy of parts of classical fluids in Ref. [44].)In summary we have seen that the area-dependence of corre-lations is not restricted to quantum systems, as long as onereplaces the notion of entanglement – representing quantumcorrelations – by the notion of classical correlations in a clas-sical system.

12

XI. SUMMARY AND OUTLOOK

In this paper, we have considered the question of the area-dependence of the geometric entropy and the distillable entan-glement in general bosonic harmonic lattice systems of arbi-trary dimension. The question was the general scaling behav-ior of these measures of entanglement with the size of a dis-tinguished region of a lattice. Such an analysis generalizes as-sessments of block entropies in the one-dimensional case. Us-ing methods from entanglement theory, we established boundsthat allow for a conclusion that may be expressed in a nutshellas: in surprising generality, we find that the degree of entan-glement scales at most linearly in the boundary area of thedistinguished region.This analysis shows that the intuitionthat both the interior and the exterior can be approximatelydisentangled up to a layer of the thickness of the two-pointcorrelation length by appropriate local unitaries carriesquitefar indeed.

For cubic regionsI = [1, ...,m]×D the area law can beformulated as

EDI = Θ(mD−1), (24)

whereΘ is the Landau theta.Such area-laws are expected to have an immediate implica-

tion on the accuracy to which ground states can be approxi-mated with matrix-product states and higher dimensional ana-logues in classical simulations of the ground states of quantummany-body systems [18]. After all, the failure of DMRG algo-rithms close to critical points can be related to the logarithmicdivergence of the block entropy in the one-dimensional case.

The findings of the present paper raise a number of interest-ing questions. Notably, in general quantum many-body sys-tems on a lattice (fermionic or bosonic), what are necessaryand sufficient conditions for an area-law in the above senseto hold? Clearly, as we have seen above, the divergence oftwo-point correlation functions alone is not in one-to-onecor-respondence with an area law. It would be interesting to con-sider and possibly decide the conjecture that a one-to-one re-lationship between a system being critical and not satisfyinga law of the form as in Eq. (24) holds if one (i) restricts atten-tion to systems in arbitrary dimension with nearest-neighborinteractions, and (ii) grasps criticality in terms of two-pointcorrelation functions with algebraic decay, concomitant witha vanishing energy gap. Note that the latter two criteria of crit-icality have not been rigorously related to each other yet andmay indeed not be simultaneously satisfied in lattice systems.The general relationship is still awaiting rigorous clarification.

As steps towards such an understanding of a relationshipbetween criticality and properties of ground state entangle-ment in more than one-dimensional fermionic and bosonicsystems, it seems very interesting to study models differentfrom the ones considered in this paper. For example, comple-menting our bosonic analysis, area-laws in fermionic criticalsystems have been addressed [24, 25], where logarithmic cor-rections have been found also in higher-dimensional settings.Other settings in the bosonic case, corresponding to field the-ories beyond this quasi-free setting, are also still not clarified.In particular, the scaling behavior of the geometric entropy

in general bosonic theories in higher dimensions is far fromclear. Then, the case of finite-size effects in harmonic latticesystems where the correlation length is larger than the fullsys-tem, resembling the critical case, will be presented elsewhere.In less generality, it seems also feasible to identify the prefac-tors of the leading and next-to-leading order terms in an area-law of the geometric entropy. It is the hope that the presentwork can contribute to an understanding of genuine quantumcorrelations in quantum many-body systems and inspire suchfurther considerations.

XII. ACKNOWLEDGMENTS

We would like to thank C. H. Bennett, H. J. Briegel,J. I. Cirac, W. Dur, M. Fleischhauer, B. Reznik, T. Rudolph,N. Schuch, R. G. Unanyan, R. F. Werner, and M. M. Wolffor discussions. This work has been supported by the EP-SRC QIP-IRC (GR/S82176/0), the EU Thematic NetworkQUPRODIS (IST-2002-38877), the DFG (SPP 1078 and SPP1116), The Leverhulme Trust (F/07 058/U) and the EuropeanResearch Councils (EURYI).

APPENDIX A: EXPONENTIAL DECAY OF ENTRIES OFMATRIX FUNCTIONS

The result concerning the exponential decay of entries ofmatrix functions of Ref. [41] relies on the fact that thep-thpowerAp of ak-banded matrixA = (Ai,j) is pk-banded, i.e.,[Ap]i,j = 0 for |i − j| > pk/2 for a matrixA with Ai,j = 0for |i − j| > k/2. For the purposes of the present paper, wewill need a generalization of the result of Ref. [41] to blockbanded matrices. We refer toV = (Vi,j) as beingk-banded,if Vi,j = 0 for d(i, j) > k/2. It can be proven by inductionover p that thep-th power ofV is pk banded in this sense.This enables us to formulate the general form of Ref. [41] asfollows.

Let V = (Vi,j) be a k-banded symmetric matrix, i.e.,Vi,j = 0 for

∑

δ |iδ − jδ| > k/2. Definea = λmin(V ),b = λmax(V ), κ = b/a,

ψ : C→ C, ψ(z) =(b− a)z + a+ b

2,

andεχ as an ellipse with foci in−1 and1 and half axesα, β,χ = α+ β.Now letf : C→ C be such thatf◦ψ is analytic in the interiorof the ellipseεχ, χ > 1, and continuous onεχ. Furthermoresuppose(f ◦ψ)(z) ∈ R for z ∈ R. Then there exist constantsK andq, 0 ≤ K, 0 ≤ q ≤ 1 such that

∣

∣

∣[f(V )]i,j

∣

∣

∣ ≤ Kq∑D

δ=1 |iδ−jδ|,

13

where

K = max

{

maxi

∣

∣λi(f(V ))∣

∣ ,2χM(χ)

χ− 1

}

,

q =

(

1

χ

)2/k

, M(χ) = maxz∈εχ

|(f ◦ ψ)(z)|.

To bound the entries ofω−, we apply the above theorem tothe function

ω−(z) =e√

z/T − 1

e√

z/T + 1z−

12 .

For 1 < χ < (√κ + 1)2/(κ − 1), we have thatω− ◦ ψ

is analytic in the interior of the ellipseεχ and continuous onεχ, i.e.,ω− satisfy the assumptions of the above theorem. TomakeK andq specific, we choose

(√κ+ 1)2

κ− 1> χ :=

κ+ 1

κ− 1> 1,

which yields

qκ =

(

κ− 1

κ+ 1

)2/k

, Ka,b =eη/T − 1

eη/T + 1

κ+ 1

η

where

η =

(

aκ

κ+ 1

)1/2

,

and for zero temperatureKa,b reduces to

Ka,b =κ+ 1

η.

These findings explicitly relate the spectral properties oftheHamiltonian to the two-point correlation functions.

APPENDIX B: ENTRIES OF THE CORRELATION MATRIXFOR THE NEAREST-NEIGHBOR CASE

In this appendix, we make the evaluation of the entries ofV 1/2 andV −1/2 of the important case of nearest-neighbor in-teractions specific. The power series expansion of the squareroot is given by

(1 − x)±12 = 1 ∓

∞∑

k=1

a±k xk, a−k =

k∏

l=1

2l − 1

2l,

a−k ≥ a−k2k − 1

= a+k ≥ 1

2k(2k − 1),

which is valid for|x| < 1, i.e., the positivity constraint0 <c < 1/2D allows us to write

(λk(V ))±12 = 1 ∓

∞∑

l=1

a±l

(

2c

D∑

δ=1

cos(2πkδ

n

)

)l

= 1 ∓∞∑

l=1

a±l cl

( D∑

δ=1

e2πikδ

n + e−2πikδ

n

)l

.

Using the multinomial theorem, we have

( D∑

δ=1

e2πikδ

n +e−2πikδ

n

)l

=∑

l!D∏

δ=1

(

e2πikδ

n + e−2πikδ

n

)nδ

nδ!,

where the sum runs over allnδ with∑

δ nδ = l. Now, apply-ing the binomial theorem

(

e2πikδ

n + e−2πikδ

n

)nδ

=

nδ∑

r=o

(

nδ

r

)

e2πikδ(nδ−2r)

n .

Substituting all the above into Eq. (14), we find

[V ±1/2]i,j = δi,j ∓∞∑

l=1

a±l cl∑

l!

D∏

δ=1

f(dδ, nδ),

wheredδ = iδ − jδ and

f(dδ, nδ) =

nδ∑

r=0

1

(nδ − r)!r!

n∑

k=1

e2πik(dδ+nδ−2r)/n

n.

To the sum overr only terms withdδ +nδ−2r = zn for somez ∈ Z contribute. We thus arrive at the following expressionfor the entries ofV ±1/2

[V ±1/2]i,j = δi,j ∓∞∑

l=1

a±l cl∑

∑

δ nδ=l

l!

D∏

δ=1

nδ∑

r=0dδ+nδ−2r=zn

1

(nδ − r)!r!,

i.e., for nearest-neighbor interactions the product−[V −1/2]i,j [V

+1/2]i,j is always positive fori 6= j.This does not hold in general and makes it difficult to obtainexplicit bounds on the entries ofV ±1/2 for more generalinteractionsV . To obtain a lower bound we keep only theterm l = d(i, j) andnδ = |dδ|. The restriction onr is thenfulfilled for r = 0 (r = |dδ|) if dδ < 0 (dδ > 0), yielding

|[V ±1/2]i,j | ≥ a±d(i,j)cd(i,j) d(i, j)!

∏Dδ=1 |iδ − jδ|!

≥(

c

2

)d(i,j)d(i, j)!

(2d(i, j) − 1)∏D

δ=1 |iδ − jδ|!.

It is also possible to obtain an upper bound that is tighter thanthe one derived for finite-ranged interactions: The elements ofV ±1/2 are symmetric underi − j → neδ − (i − j), whereeδ is a unit vector along dimensionδ. Thus, we can demand−n/2 ≤ iδ − jδ ≤ n/2. Then we find thatnδ has to be largeror equal to|iδ − jδ| otherwise the restriction onr can not befulfilled. This in turn means thatl has to be larger or equal tod(i, j). We then obtain an upper bound by summing all terms

14

FIG. 5: Visualization of the enumeration ofNl as in Appendix C. Asbefore, oscillators belonging to the distinguished (shaded) regionI

are marked , the outside ones are shown as. ∂O is shown as theorange shaded area.M4(i) = {j ∈ C|d(i, j) ≤ 4} for a certainoscillator is shaded green, its surface oscillatorsm4(i) = {j ∈C|d(i, j) = 4} are depicted by .

in the sum overr regardless of the given restriction, yielding

|[V ±1/2]i,j | ≤∑

l≥d(i,j)

a±l cl∑

∑

δ nδ=l

l!

D∏

δ=1

2nδ

nδ!

=∑

l≥d(i,j)

a±l (2cD)l

=

∞∑

l=0

a±l+d(i,j)(2cD)l+d(i,j)

≤ a±d(i,j)(2cD)d(i,j)∞∑

l=0

(2cD)l

=a±d(i,j)

1 − 2cD(2cD)d(i,j).

APPENDIX C: ENUMERATING THE RELEVANT TERMSIN THE AREA LAW

We start by identifying the set of oscillators that can con-tribute toNl, l > 1,

Nl =∑

j∈O

∑

i∈Id(i,j)=l

1.

Oscillatorsj ∈ O can only contribute if their distance to theboundary∂O is not larger thanl − 1,

∂O ={

j ∈ O | ∃i ∈ I : d(i, j) = 1}

.Thus, we can restrict the sum overO to the setAl

Al =⋃

i∈∂O

{

j ∈ O | d(i, j) ≤ l − 1}

,

i.e., we can write

Nl =∑

o∈Al

∑

i∈Id(i,o)=l

1 ≤∑

o∈Al

∑

i∈Cd(i,o)=l

1 ≤ |Al|ml,

whereml is the number of surface oscillators of a ball withradius l within the metricd, i.e., ml ≤ 2(2l + 1)D−1 forl ≥ 1. Using the fact that|∂O| ≤ N1 = s(I), |Al| can nowbe bounded from above in the following way

|Al| ≤ |∂O|Ml−1 ≤ s(I)Ml−1,

whereMl is the volume of a ball with radiusl within the met-ric d, i.e.,Ml ≤ (2l+ 1)D. To summarize, we have

Nl ≤ 2(2l− 1)D(2l + 1)D−1s(I) ≤ 2(2l)2D−1s(I).

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