Spatial Entanglement Sudden Death in Spin Chains at All Temperatures

Spatial En tanglemen t Sudden Death in Spin Chains at All T emp eratures Sam uel O. Scalet ∗ Departmen t of Computer Science, Universit y of California, Da vis, CA, 95616, USA Abstract W e pro ve a finite entanglemen t length for the Gibbs state of any lo cal Hamilto- nian on a spin chain at an y finite temp erature: After removing an in terv al of size at least equal to the entanglemen t length, the remaining left and righ t half-c hains are in a separable state. 1 In tro duction En tanglement is one of the most fundamental concepts of quan tum mechanics. While initially p erceiv ed as ”unreal” and therefore necessarily incomplete [ EPR35 ], en tangle- men t has b een demonstrated as a distinguishing and verifiable feature that sets quantum mec hanics apart from classical theories [ Bel64 ]. T o da y , the en tire field of quan tum in- formation theory has b een built on the concept of en tanglemen t, where it facilitates n umerous applications such as quantum telep ortation [ Ben+93 ], encryption [ BB84 ], and ev en quan tum computation [ F ey82 ], to name just a few. Quan tum information theory has also op ened a new p ersp ective on the field of many- b o dy ph ysics. As the underlying interactions in man y-b o dy systems are quantum, it is only natural to take into account entanglemen t in the study of their correlations. This program has led to a plethora of new insigh ts ab out connections betw een en tanglement and correlation measures with phenomenological and computational results, see [ Ami+08 ; Cir+21 ; Alh23 ] for some reviews on this v ast area. Spin chains in one dimension in their thermal state stand as a paradigmatic and also exp erimen tally in teresting example of this line of work. In the translation-inv ariant case, they ha ve been shown to exhibit no phase transitions and to p ossess a unique Gibbs state b y Araki [ Ara69 ; Ara75 ]. Many further results on their entanglemen t structure follow ed including the deca y of other measures of correlations and en tanglement, suc h as m utual and conditional mutual informations [ BCP22 ; Gon+24 ; KB19 ; Kuw24 ; KS22 ], efficien t classical sim ulation [ KS18 ; KAA21 ; FFS23 ; Sca24 ; Ac h+24 ], sampling on quan tum com- puters [ BB10 ], and their fast mixing for op en-system dynamics [ BC25 ], including results that lift the assumption of translation-in v ariance. In fact, the decay of correlations in terms of correlation functions or mutual information is sho wn to be exp onential [ Ara69 ; KK25 ; BC25 ; BCP22 ], which is qualitativ ely optimal as can be easily demonstrated ev en b y a classic al example of a nearest-neighbour spin chain. W e ask the follo wing question: ∗ Email: sscalet@ucda vis.edu 1 What p art of the exp onential ly de c aying c orr elations is nonclassic al? When asking this question, one migh t w onder how to quan tify this en tanglement. T ypical candidates migh t include the en tanglement of formation E F , en tanglement cost E C , the distillable en tanglement E D , or distance measures to the set of separable states, including en tanglement relative entrop y E R , whic h are all motiv ated b y different op erational in ter- pretations. W e omit the detailed definitions of all these measures and refer the reader to [ Hor+09 ]. Somewhat unsatisfactorily , they can display wildly differen t behaviour as demonstrated by the existence of b ound entangled states [ HHH98 ], whic h ha ve nonzero en tanglement but no distillable en tanglemen t. The exp onential deca y of m utual informa- tion b ounds all of these 1 , but is not faithful, so lo wer b ounds remain op en — in particular, it is nonzero for purely classical correlations. W e circumv en t this con undrum b y pro ving the follo wing stronger statemen t: The left and right ends of the chain in its Gibbs state b e c ome unentangle d when sep ar ate d by a finite lengthsc ale. Alb eit generally NP-hard to chec k [ Gur04 ], separabilit y has a clean definition, not suf- fering from the ab o v e dichotom y: A state is separable if it is a conv ex com bination of pro duct states, see Definition 1 . Stated more precisely , the theorem we pro ve is: Theorem 1 (Informal v ersion of Theorem 2 ) . F or a lo c al Hamiltonian on a spin chain, ther e exists a c onstant ℓ ∈ N that only dep ends on the temp er atur e and lo c ality, but not on the system size, such that the fol lowing holds. F or any trip artite interval AB C with | B | ≥ ℓ , and ρ AB C = exp( − β H AB C ) / Z AB C the Gibbs state on the interval, we have that ρ AC = T r B [ ρ AB C ] is sep ar able b etwe en A and C . This implies that all meaningful measures of en tanglemen t, including E F , E D , E C , and E RE are exactly zero on ρ AC , and further, that the state can b e prepared using only lo cal operations and classical comm unication. W e also note that this theorem holds at any fixe d temp erature 1 /β with no phase transition. Ho wev er, as will b e clear from the techniques used, the lengthscale ℓ grows with β . While w e do not spell out the details of this dep endence, the choice of tec hniques based on [ Ara69 ; PP23 ] should result in a dep endence of the form exp(exp( O ( β ))). Since our result holds uniformly in system size, w e can state the following corollary on the thermo dynamic limit. Corollary 1 (Informal v ersion of Corollary 2 ) . F or a uniformly b ounde d finite-r ange inter action, c onsider the infinite system KMS-state ω on the quasilo c al algebr a and the p artial tr ac e tr Λ . Ther e exists ℓ ∈ N such that ω ◦ tr [ − ℓ, 0] is a sep ar able state b etwe en the left and right half-chain. 1 This is immediate for E R and the squashed en tanglement, which in turn b ounds the remaining ones [ Hor+09 ; BCY11 ]. 2 T o a void tec hnical details, w e do not discuss this further here and refer to Section 4 for definitions and details. Before pro ceeding to an outline of our proof, let us describe the notion of the de ath of entanglement and compare it to our result. While not strictly defined, the term was first coined in the context of con tinuous-time noise channels applied to an en tangled state of t wo atoms in ca vities [ YE04 ; YE09 ]. Unav oidably , the correlations b et w een the atoms deca y exp onen tially due to the noise. What is somewhat more striking is that the en tanglement deca ys not just exp onentially . It disapp ears after a finite time t E S D . A simplified explanation of this effect is that there is an op en ball around the maximally mixed state, whic h consists of separable states. As the time-ev olved state approaches the cen ter of this ball, even tually , after a finite time, it has to lie strictly within the ball and b ecomes separable. In a totally different context, [ Bak+24 ] recently show ed that a conceptually similar phenomenon happ ens in high-temp erature Gibbs states. The infinite temperature state is the maximally mixed state, and a small p erturbation lea v es that state separable 2 . A cav eat is that the size of the ball may dep end on the dimension, and it is a highly non trivial insigh t of this work to show that the temp erature at whic h the states b ecome separable does not gro w with system size. Note that the high-temp erature regime in whic h this result holds is the trivial phase, just as one-dimensional systems only possess one phase [ Ara69 ] across all temp eratures, so another natural question answ ered b y our result is: Is ther e an analo gous de ath of entanglement at any temp er atur e for 1D Gibbs states? Clearly , in our setting, w e cannot repro duce a result that prov es separabilit y b etw een arbitrary pairs of spins, since simple examples of suitable interactions will yield entangle- men t b etw een neigh b ouring spins. Instead, w e view our result as another, nov el notion of this sudden death. Rather than with time or temp erature, we sho w the separability at a sufficiently large distance, which one may dub a sp atial sudden de ath of entanglement . Pro of outline W e discuss the main ideas and steps of the pro of of Theorem 2 . A starting p oin t is similar to the idea already used in [ Bak+24 ], that for sufficiently small p erturbations to the maximally mixed state, the state remains separable as long as the strength of the perturbations deca ys exponentially with the size of their support at a sufficien tly large rate. This strategy is inapplicable to our case in several w ays: • W e cannot tune the temp erature up, so we are not close to the maximally mixed state. W e o v ercome this first barrier b y com bining t wo results: the exp onen tial deca y of corre- lations in the form of an appr oximate factorization ρ AC ≈ ρ A ⊗ ρ C (1) and the quantitative faithfulness of marginals, i.e., low er b ounds on their smallest eigen- v alues ρ A/C ⪰ α 1 > 0. In particular, the latter implies that, alb eit far from the maximally 2 Their result holds in an ev en stronger sense, proving full m ultipartite separability . 3 mixed state, the marginals p ossess a p ositive decomposition with a maximally mixed com- p onen t, whic h remains separable when adding the deviation from the pro duct state due to the only approximate Equation ( 1 ). This result forms Prop osition 1 , whic h already implies the separability for constant sizes of the regions A and C . In terestingly , 1D is not a crucial assumption for this statement as long as uniform faithfulness and deca y of correlations can b e pro ven. Ho wev er, the ab ov e argumen t is insufficient to extend to arbitrary system sizes for t wo reasons: • The exponential decay of the appro ximate factorization is optimal, and the rate go es to 0 as β increases. Again, this is true due to classical correlations. A calculation using transfer matrices for a simple classical nearest-neighbour Ising c hain yields the optimal exp onential decay of cor- relations as w ell as m utual information, which lo wer b ounds the error of the appro ximate factorization. Moreo ver: • The exp onen tially small faithfulness α is optimal for a rate log( d ) + Ω( β ). This low er bound is already satisfied for a noninteracting spin c hain. T o ov ercome this issue, w e sho w an additional quasilocal substructure of the deviation from the product state ρ AC − ρ A ⊗ ρ C : It splits in to a term with small v ariable supp ort size k 0 , which is exp onentially deca ying in | B | , and an additional tail of terms with larger supp orts ∆ k , k ≥ k 0 . F or the former part, w e also find a decomp osition in to a separable and a maximally mixed comp onen t Γ + γ 1 . Only for the additional tail, w e are then able to sho w an even stronger, sup erexp onential decay in their supp ort size and | B | . F or an y giv en temp erature, b y increasing the supp ort of the first part, this deca y rate of ∆ k can no w exceed the deca y rate of γ . Viewing the tail as a p erturbation of the maximally mixed component, their sum γ 1 + P k ∆ k remains separable. By com bining all these separable contributions, the theorem follows. Related w ork The question of the deca y of bipartite en tanglemen t has b een studied in [ KS22 ; Kuw24 ]. In particular, the latter sho ws the deca y of the squashed en tanglement in one and higher dimensions, though the higher-dimensional case is restricted to constan t- sized subsystems. In fact, due to the lifted translation-in v ariance assumption in the decay of correlations [ BC25 ] in one dimension, this is by no w subsumed b y the deca y of mutual information [ BCP22 ], see Lemma 5 . Ho w ever, while addressing meaningful measures of en tanglement, all of these prior b ounds pro v e only decay and not the stronger sudden death of en tanglement. While preparing this man uscript, w e became a ware of [ BCP26 ], whic h pro v es a strictly finite Sc hmidt rank for finite-range thermal states in one dimension. While the concept of pro ving a strictly finite prop ert y of the en tanglement, where generically one migh t only exp ect a deca ying sp ectrum, seems related, there is no tec hnical connection b etw een the statemen ts in their work and ours. While the Schmidt rank b ounds the amoun t of en tanglement b etw een neighbouring regions, our result concerns the decay with distance. In fact, the authors men tion the op en question of the en tanglement lengthscale, whic h our work answers. 4 Discussion and outlook In this pap er, w e pro ve the sudden death of en tanglement in one-dimensional Gibbs states. T racing out a finite subsystem from a c hain, the state b ecomes exactly separable b etw een the tw o halv es, despite inevitable quantum correla- tions b etw een close neighbours. T o the b est of our kno wledge this is the first non trivial example of suc h a sp atial death of en tanglement. There are a num b er of interesting op en questions: While in our setting of one- dimensional Gibbs states, classical and quantum correlations are extensiv ely classified, it remains an op en question whether building on our techniques a superexp onen tial deca y of the conditional m utual information can b e sho wn. While its exp onential deca y rate is pro ven in [ Ku w24 ], no lo w er b ound is known and in particular, in the classical setting the stronger Hammersley-Clifford theorem implies exact Mark o vianity b eyond the interaction range. F urther, regarding the death of entanglemen t, it would b e in teresting to prov e analo- gous results in higher-dimensional low-temperature systems or ground states. While this clearly go es b ey ond the scop e of our tec hniques, features lik e the monogamy of entan- glemen t that rule out arbitrarily strong entanglemen t b etw een m ultiple particles migh t giv e a hint in this direction. F or ground states, while their marginals are low-rank, some p erturbations of low-rank separable states can still remain separable. An indication, that the ground state regime is not hop eless is giv en b y the following trivial example: States prepared by constan t-depth quantum circuits exhibit the death of entanglemen t, but maintain short-range en tanglement. Outline In Section 2 , w e in tro duce the notation and prior and elementary results needed for the proof, which follo ws in Section 3 split in to the constan t-size result Prop osition 1 and the main Theorem 2 . In Section 4 , w e discuss the thermo dynamic limit and prov e Corollary 2 . 2 Preliminaries In this section, w e in tro duce the setup and notation as well as some elementary or known results needed for the pro of. F or a finite-dimensional Hilb ert space H , we denote the set of bounded op erators B ( H ). A quan tum state is a normalized p ositiv e semidefinite op erator ρ ∈ B ( H ), ρ ⪰ 0, T r[ ρ ] = 1, where ⪰ denotes the p ositiv e semidefinite order. T r[ · ] denotes the trace, ∥ · ∥ the op erator norm, and ∥ X ∥ 1 = T r h √ X ∗ X i the trace norm. F or tw o p ositive quan tum states, the Umegaki relative entrop y is defined as D ( ρ ∥ σ ) = T r[ ρ (log( ρ ) − log( σ ))]. 2.1 Separable states Definition 1. A p ositive semidefinite matrix X ∈ B ( C d A ) ⊗ B ( C d C ) is c al le d sep ar able if ther e exists a de c omp osition X = r X i =1 X A i ⊗ X C i , wher e X A i ∈ B ( C d A ) , X C i ∈ B ( C d C ) and X A i , X C i ⪰ 0 . The set of sep ar able states in finite dimensions is the set of sep ar able matric es X with T r[ X ] = 1 . A matrix or state that is not sep ar able is c al le d entangle d. 5 A famous criterion for entanglemen t is the PPT criterion: F rom the ab ov e decomp o- sition, it is evident that transp osing one subsystem of a separable state results in a (sepa- rable) state, where the linear transp osition map is ( · ) T B : | i ⟩ ⟨ j | ⊗ | k ⟩ ⟨ l | 7→ | i ⟩ ⟨ j | ⊗ | l ⟩ ⟨ k | . Ho wev er, for some states ρ T B is not p ositive semidefinite, whic h, b y con tradiction, prov es that they are en tangled. The criterion, ho wev er, is not faithful except for dimensions 2 × 2 and 2 × 3 [ W or76 ]. While characterization of bipartite pure state entanglemen t is well understo o d, the landscap e of mixed state en tanglemen t measures is fragmented and dep ends on the sp ecific application as review ed in [ Hor+09 ]. As our pro of of exact separabilit y w orks in an elemen tary w ay with Definition 1 , w e do not go into detail. W e collect a n umber of elemen tary prop erties of separable operators in finite dimen- sions, which we will use rep eatedly . Lemma 1. The set of sep ar able matric es is a close d c one, i.e., it is close d under lin- e ar c ombinations with p ositive c o efficients. The set of sep ar able states is a close d c onic se ction. F urther, if X AC ∈ B ( C d A ) ⊗ B ( C d C ) is sep ar able, so is ( Y A ⊗ Y C ) X ( Y A ⊗ Y C ) † for any Y A ∈ B ( C d A ) , Y C ∈ B ( C d C ) . Pr o of. The cone prop erty is immediate from the definition. T o sho w that it is closed, consider a sequence of separable states X k that con verges to X ∈ B ( C d A ) ⊗ B ( C d C ). By Carath ´ eo dory’s theorem, we can alwa ys choose r = ( d A d B ) 2 in Definition 1 , so w e can write X k = r X i =1 p k,i ( X k ) A i ⊗ ( X k ) C i with T r  ( X k ) A i  = T r  ( X k ) C i  = 1 and p k,i ≥ 0. Since X k con verges, it is b ounded, and so p k,i =   p k,i ( X k ) A i ⊗ ( X k ) C i   1 ≤ ∥ X k ∥ 1 is b ounded to o. No w, all of p k,i , ( X k ) A i , and ( X k ) C i are b ounded sequences in finite dimensions, and hence w e can descend to a subsequence k l , for whic h all of them conv erge, and X = lim k →∞ X k = lim l →∞ r X i =1 p k l ,i ( X k l ) A i ⊗ ( X k l ) C i = r X i =1 p i X A i ⊗ X C i is separable. The set of separable states is the intersection of the separable matrices with the closed affine space defined by T r[ x ] = 1, and therefore a closed conic section. Since X ⪰ 0 implies Y X Y † ⪰ 0 for any matrix Y , the last part follows from the definition of a separable matrix. Since the set of separable matrices and states are con v ex and thereby (path-)connected and closed, they are not open (in the subspace top ology). An explicit example is as follo ws: The sequence of states (1 − 1 /n ) | 00 ⟩ ⟨ 00 | + ( | 00 ⟩ + | 11 ⟩ )( ⟨ 00 | + ⟨ 11 | ) / 2 n con v erges to the separable state | 00 ⟩ ⟨ 00 | but for eac h n do es not satisfy the PPT criterion and is therefore entangled. Ho wev er, there exists an op en ball around the iden tit y in whic h all states are separable as stated in the following Lemma, whic h is central to our main theorem. The optimal dimension dep endent size of suc h balls in v arious norms has b een studied in [ GB02 ] and w e adopt the follo wing result. 6 Lemma 2 ([ GB02 , Theorem 2]) . F or ∆ ∈ B ( C d A ) ⊗ B ( C d C ) with ∥ ∆ ∥ ≤ 1 / √ d A d C 1 AC + ∆ is sep ar able b etwe en A and C . Since the ab ov e statemen t trivially con tin ues to hold when tensoring with the identit y , the dimension can b e restricted to the dimension of the supp ort of the op erator ∆ in a m ultipartite Hilb ert space. Note that in [ Bak+24 ] a version of this statemen t for qubits even prov es multipartite separabilit y . In the con text of our work nearest-neigh b our en tanglement is naturally una v oidable, so we restrict ourselv es to the abov e bipartite statemen t with slightly b etter dimension dep endence. 2.2 1D Gibbs states W e consider a spin system on the chain Z and consider the lo cal algebra A 0 = ∪ n ∈ N A [ − n,n ] , where A X = ⊗ i ∈ X A i , A i ∼ = B ( C d ) and up to the natural embedding, i.e., for Λ ⊂ Λ ′ , X Λ ∈ A Λ is iden tified with X Λ ⊗ 1 Λ ′ \ Λ . Its closure A = A 0 in operator norm is the quasilo cal algebra. 1 denotes the iden tity matrix. W e denote the trace on a region Λ, b y T r Λ : A Λ → C and omit the subscript when clear from the con text. F or X ⊂ Λ, tr X : A Λ → A Λ \ X denotes the (unnormalized) partial trace, i.e., tr X [ 1 Λ ] = d | X | 1 Λ \ X . F or an op erator X ∈ A 0 , its supp ort is the smallest set Λ such that X ∈ A Λ . F or P 0 ( Z ) the set of finite subsets of Z , let Φ : P 0 ( Z ) → A 0 b e a lo cal interaction, i.e., such that Φ( X ) ∈ A X . W e assume that it has a finite-range r such that Φ( X ) = 0 if diam( X ) > r and b ounded strength J J = sup i ∈ Z X X : i ∈ X ∥ Φ( X ) ∥ Throughout the pap er, when writing that a term is constan t, w e mean that it is an explicit function only of J , r , and d , but not any system size. F or any region A , we define H A = X X ⊂ A Φ( X ) W e consider a finite interv al AB C ⊂ Z with A , B , and C interv als with A preceding A B C ∂ k B Figure 1: Graphical represen tation of the regions AB C . and C succeeding B . W e use the following notation for the k -neighbourho o d of B : ∂ k B = { k ∈ Λ | d ( k , B ) ≤ k } , see Figure 1 . 7 W e denote the Gibbs state on a region Λ b y ρ Λ = exp( − H Λ ) / Z Λ , Z Λ = T r Λ [exp( − H Λ )] and its marginals ρ Λ X = tr Λ \ X [ ρ Λ ], where we omit the sup erscript when clear from the con text. Here, w e c ho ose the inv erse temperature β = 1 as it can be absorb ed in to the in teraction strength J . Note that ρ Λ will sometimes be considered as an op erator in A 0 but it is only normalized in A Λ . If not noted otherwise ρ = ρ AB C . W e use the shorthand ρ k = ρ ∂ k B , H k X = H X ∩ ∂ k B , and Z k 0 = Z ∂ k B . W e need the follo wing standard Lemma on the gro wth of partition functions. Lemma 3. F or Hamiltonians H, H ′ ∈ B ( C d ) , T r[exp( − H ′ )] T r[exp( − H )] ≤ exp ( ∥ H − H ′ ∥ ) . (2) In p articular, for an inter action of str ength J and Z Λ as define d ab ove and any r e gion A exp((log( d ) − J ) | A | ) ≤ Z A ≤ exp((log( d ) + J ) | A | ) (3) and for neighb ouring intervals A , B , exp( − r J ) ≤ Z A Z B Z AB ≤ exp( r J ) . (4) Pr o of. W e in tro duce the smo oth function f : [0 , 1] → R f ( s ) = T r h e − H + s ( H − H ′ ) i . By Duhamel’s form ula f ′ ( s ) = T r h ( H − H ′ ) e − H + s ( H − H ′ ) i ≤ ∥ H − H ′ ∥ f ( s ) and so b y Gr¨ on wall’s inequalit y f (1) ≤ f (0) exp( ∥ H − H ′ ∥ ), whic h prov es ( 2 ). Equa- tion ( 3 ) follows b y c ho osing H = H A , H ′ = 0 and vice versa. Equation ( 4 ) follows b y c ho osing H = H A + H B , H ′ = H AB and vice v ersa and noting that the terms in H AB that differ from the ones in H A + H B ha ve supp ort containing one of the first r sites of B , so ∥ H A + H B − H AB ∥ ≤ r J . In [ Ara69 ] the Araki expansionals w ere introduced and their lo cality structure ana- lyzed. The result allo ws to relate an (unnormalized) Gibbs state to a pro duct of Gibbs states on decoupled subin terv als and is based on Lieb-Robinson type b ounds for the com- plex time evolution. W e use the following alternativ e formulation of Araki’s results from a more recen t w ork (see also [ PP23 ]). Lemma 4 ([ BCP22 , Corollary 3.4]) . L et Φ b e a lo c al inter action of r ange r and str ength J , s ∈ C with | s | ≤ 1 , and c onsider E X,Y ( s ) := e − sH X Y e sH X + sH Y ∈ A X ∪ Y wher e X , Y ar e two adjac ent intervals. Then ther e exists a c onstant G > 0 only dep endent on r and J but not on the size of X , Y such that ∥ E X,Y ( s ) ∥ , ∥ E X,Y ( s ) − 1 ∥ ≤ G 8 uniformly in the size of X and Y . F urther, let ˜ X , ˜ Y b e (p ossibly empty) intervals im- me diately pr e c e e ding and suc c e e ding X and Y r esp e ctively, and | X | , | Y | ≥ ℓ . Then for | s | ≤ 1   E X,Y ( s ) − E ˜ X X,Y ˜ Y ( s )   ,   E X,Y ( s ) − 1 − E ˜ X X,Y ˜ Y ( s ) − 1   ≤ G ℓ ( ⌊ ℓ/r ⌋ + 1)! . As another shorthand notation w e define the Araki expansional restricted to the k - neigh b ourho o d of B : E k X,Y ( s ) = E X ∩ ∂ k B ,Y ∩ ∂ k B W e will mak e use of results on the deca y of correlations in Gibbs states. In particu- lar, w e need a form of approximate factorization to pro ve separability of constan t-sized regions. The results in [ BCP22 ] sho w the equiv alence of sev eral notions of the decay of correlations including that of correlation functions as w ell as said appro ximate factoriza- tion. While the final results in there are restricted to the translation-inv arian t case, as the result relies on [ Ara69 ], this restriction can b e readily lifted by applying the equiv alence to the more recent result on decay of correlations in [ BC25 ]. Lemma 5. F or any c onse cutive adjac ent intervals ˜ AAB C ˜ C , and p otential Φ of b ounde d r ange r and str ength J , c onsider the Gibbs state ρ = ρ ˜ AAB C ˜ C . Then, ther e exist c onstants C, α > 0 only dep endent on r , J , and d such that ∥ ρ AC − ρ A ⊗ ρ C ∥ ≤ C exp( − α | B | ) Pr o of. [ BC25 , Corollary I.2] pro ves the exp onen tial deca y of correlations   Corr( X A , X C ) ρ AC   : =   T r AC  ( X A ⊗ X C ) ρ AC  − T r A [ X A ρ A ] T r C [ X C ρ C ]   ≤ C ′ exp( − α ′ | B | ) for constan ts C ′ , α ′ > 0 only dep enden t on the in teraction range, strength, and local dimension. This is equiv alen t to the uniform exp onen tial clustering condition in [ BCP22 , Theorem 8.2], whic h thereb y yields the exp onential deca y of the m utual information I ( A : C ) ≤ C ′′ exp( − α ′′ | B | ) . The Lemma follo ws from Pinsker’s inequality ∥ ρ AC − ρ A ⊗ ρ C ∥ 2 1 ≤ 2 I ( A : C ) and the matrix norm inequalit y ∥ · ∥ ≤ ∥ · ∥ 1 . Remark 1. The e quivalenc e in [ BCP22 ] is inevitably r estricte d to 1D systems making he avy use of the r esults in L emma 4 . However, it turns out that a we aker version of the e quivalenc e b etwe en de c ay of c orr elation functions and mutual information with an exp onen tial o verhead in the sizes of A and C holds mor e gener al ly. It is a str aightforwar d c onse quenc e of the e quivalenc e b etwe en tr ac e norm and r elative entr opy distanc e to gether with the e quivalenc es of finite-dimensional norms up to dimensional pr efactors. It turns out that this exp onential overhe ad p oses no b arrier to our pr o of. The ab ove lemma makes our pr o of cle aner and impr oves c onstants. However, for p otential extensions to higher dimensions, it is inter esting to note that the de c ay of c orr elations is sufficient to ensur e this we aker form of appr oximate factorization. The following Lemma pro ves that the deca y of smallest eigen v alues of marginals is no faster than exp onential. Due to a slightly differen t setup, we presen t the pro of whic h is analogous to [ FFS23 , Lemma 4.2]. 9 Lemma 6. F or an inter action Φ of r ange r and str ength J , ther e ar e c onstants C , α > 0 , only dep endent on r , J , and d , such that for any trip artite interval AB C , and the Gibbs state ρ AB C its mar ginal is lower b ounde d as ∥ ρ − 1 B ∥ ≤ C exp( α | B | ) uniformly in the size of A and C . Pr o of. Consider E AB ,C ( − 1 / 2) E A,B ( − 1 / 2) E † A,B ( − 1 / 2) E † AB ,C ( − 1 / 2) = e H AB C / 2 e − ( H A + H B + H C ) e H AB C / 2 . By Lemma 4 , we hav e    E AB ,C ( − 1 / 2) E A,B ( − 1 / 2) E † A,B ( − 1 / 2) E † AB ,C ( − 1 / 2)    ≤ G 4 or equiv alently G 4 1 ⪰ e H AB C / 2 e − ( H A + H B + H C ) e H AB C / 2 . W e rearrange G 4 e − H AB C ⪰ e − ( H A + H B + H C ) , and using that partial traces preserve the positive semidefinite order Z AB C G 4 ρ B ⪰ Z A Z C exp( − H B ) ⪰ Z A Z C exp( −∥ H B ∥ ) 1 . Since, by Lemma 3 Z A Z C Z AB C = Z AB Z C Z AB C Z A Z B Z AB 1 Z B ≥ exp( − ( J + log( d )) | B | − 2 rJ ) , and ∥ H B ∥ ≤ J | B | , the pro of follows with C = G 4 e 2 rJ and α = 2 J + log( d ). Remark 2. A simple example of a non-inter acting, tr anslation-invariant on-site p otential shows that the ab ove L emma is optimal with α ≥ log ( d ) + J . 3 Pro of of Theorem 2 In this section, w e presen t the pro of of the main theorem. W e will make use of the follo wing tec hnical lemma. Lemma 7. F or any r e gions A , B , and a normalize d state ρ B ∈ A B , the map A AB → A A , X 7→ tr B [ ρ B X ] is c ontr active in the op er ator norm. Pr o of. Let ρ B = P i p ( i ) | i ⟩ ⟨ i | B where | i ⟩ B is an orthonormal basis diagonalizing ρ B . Then, expanding the partial trace in this basis, we hav e ∥ tr B [ ρ B X ] ∥ = sup | ψ ⟩ A : ⟨ ψ | ψ ⟩ A =1 X i p ( i ) ⟨ ψ | A ⟨ i | B X | ψ ⟩ A | i ⟩ B ≤ X i p ( i ) ∥ X ∥ = ∥ X ∥ . 10 Next, we prov e a separable structure of the marginals AC , where the en tanglemen t lengthscale | B | still depends on the size of A and C . W e will b o otstrap the general, system-size indep enden t en tanglement lengthscale from this proposition. Prop osition 1. L et Φ b e an inter action of r ange r and str ength J , and fix the lo c al dimension d . Ther e exists a function ℓ 1 ( k ) (also dep endent on r , J , and d ) such that for | B | ≥ ℓ 1 ( k ) , ther e exists a de c omp osition e H k AC / 2 ρ k AC e H k AC / 2 = γ ( k ) 1 + Γ( k ) , wher e γ ( k ) = exp( −O ( k )) 1 and Γ( k ) is sep ar able. F urthermor e, we c an cho ose ℓ 1 ( k ) = O ( k ) . Pr o of. W e assume | B | ≥ r , suc h that H k AC = H k A + H k C . Define ∆ b y ˜ ρ k AC := e H k AC / 2 ρ k AC e H k AC / 2 = ˜ ρ k A ⊗ ˜ ρ k C + ∆ , where ˜ ρ k A := e H k A / 2 ρ k A e H k A / 2 and ˜ ρ k C := e H k C / 2 ρ k C e H k C / 2 . By Lemma 5 , there exist constants C, α > 0 suc h that ∥ ∆ ∥ = ∥ e H k AC / 2 ρ k AC e H k AC / 2 − ˜ ρ k A ⊗ ˜ ρ k C ∥ ≤    e H k AC      ρ k AC − ρ k A ⊗ ρ k C   ≤ C exp(2 J k − α | B | ) , where we use   exp  H k AC    ≤ exp(2 J k ). F urthermore, using Lemma 6 , we hav e that ˜ ρ k A = e H k A / 2 ρ k A e H k A / 2 ⪰ e H k A   ( ρ k A ) − 1   − 1 ⪰    e − H k A    − 1   ( ρ k A ) − 1   − 1 ⪰ C ′ exp( − α ′ k ) 1 , where w e com bined the contribution from the b ound ∥ exp  − H k A  ∥ ≤ exp( J k ) and the constan t from the Lemma into a new constant α ′ . The same bound holds for ˜ ρ k C . W e c ho ose γ ( k ) = C ′ 2 exp( − 2 α ′ k ) / 2 and decomp ose ˜ ρ k AC = ˜ ρ k A ⊗ ˜ ρ k C + ∆ =  ˜ ρ k A − p 2 γ ( k ) 1 A  ⊗  ˜ ρ k C − p 2 γ ( k ) 1 C  + p 2 γ ( k ) 1 A ⊗  ˜ ρ k C − p 2 γ ( k ) 1 C  +  ˜ ρ k A − p 2 γ ( k ) 1 A  ⊗ p 2 γ ( k ) 1 C + 2 γ ( k ) 1 AC + ∆ . Consequen tly , Γ( k ) :=  ˜ ρ k A − p 2 γ ( k ) 1 A  ⊗  ˜ ρ k C − p 2 γ ( k ) 1 C  + p 2 γ ( k ) 1 A ⊗  ˜ ρ k C − p 2 γ ( k ) 1 C  +  ˜ ρ k A − p 2 γ ( k )  ⊗ p 2 γ ( k ) + γ ( k ) 1 AC + ∆ and due to our c hoice of γ , the first three terms are separable as ˜ ρ k A − p 2 γ ( k ) 1 A , ˜ ρ k C − p 2 γ ( k ) ⪰ 0 are p ositiv e op erators. Finally , b y Lemma 2 the remaining term γ ( k ) 1 AC + ∆ 11 is separable if ∥ ∆ ∥ γ ( k ) = C exp(2 J k − α | B | ) C ′ 2 exp( − 2 α ′ k ) / 2 ≤ d − k , whic h is achiev ed b y the c hoice | B | ≥ ℓ 1 = max { r , (log(2 C /C ′ 2 ) + k (2 α ′ + log ( d ) + 2 J )) /α } = O ( k ). Remark 3. In fact, the ab ove the or em alr e ady pr oves sep ar ability of constan t-sized r e- gions (the factors e H k AC / 2 in the statement ar e sup erfluous by L emma 1 and include d only for c onvenienc e later). F urthermor e, it only r elies on two ingr e dients: Uniform faithful- ness of the state, i.e., a uniform lower b ound on al l c onstant-size d mar ginals, and de c ay of c orr elations. F or the latter, even a we aker notion than the appr oximate factorization suffic es sinc e for c onstant dimension they c an b e pr oven e quivalent, se e R emark 1 . This p aves the way for higher-dimensional extensions of the ar gument, though it would only b e inter esting for de c ay of c orr elations b eyond the high-temp er atur e r e gime as otherwise, the r esult of [ Bak+24 ] alr e ady applies. Let us no w pro ceed to the proof of our main theorem. Theorem 2. L et Φ b e a p otential on a one-dimensional spin-chain of r ange r and in- ter action str ength J . Ther e is a function ℓ ( d, J, r ) , such that for any interval A ∪ B ∪ C with disjoint c ontiguous subintervals A , B , and C in or der, wher e | B | ≥ ℓ ( d, J, r ) and the Gibbs state ρ = ρ AB C = exp( − H AB C ) / Z AB C , its mar ginal ρ AC is sep ar able b etwe en A and C . In p articular, ℓ ( d, J, r ) do es not dep end on | A | or | C | . Remark 4. While state d for adjac ent intervals AB C , which c over the entir e system, it fol lows imme diately that sep ar ability holds for any subr e gions at distanc e ℓ + 1 as the set of sep ar able states is invariant under LOCC pr oto c ols including p artial tr ac es. Pr o of. W e assume again that | B | ≥ r suc h that H AC = H A + H C . By Lemma 1 , to show separabilit y of ρ AC , we can equiv alently sho w that the following op erator is separable. Z AB C Z B exp( H AC / 2) ρ AC exp( H AC / 2) = 1 Z B e H AC / 2 tr B  e − H AB C  e H AC / 2 (5) = tr B h ρ B e ( H A + H B ) / 2 e − H AB / 2 e H AB / 2 e H C / 2 e − H AB C (6) × e H C / 2 e H AB / 2 e − H AB / 2 e ( H A + H B ) / 2 i (7) = tr B h ρ B E † A,B (1 / 2) E † AB ,C (1 / 2) E AB ,C (1 / 2) E A,B (1 / 2) i (8) = tr B h ρ B E k 0 † A,B (1 / 2) E k 0 † AB ,C (1 / 2) E k 0 AB ,C (1 / 2) E k 0 A,B (1 / 2) i (9) + ∞ X k = k 0  tr B h ρ B E k +1 † A,B (1 / 2) E k +1 † AB ,C (1 / 2) E k +1 AB ,C (1 / 2) E k +1 A,B (1 / 2) i (10) − tr B h ρ B E k † A,B (1 / 2) E k † AB ,C (1 / 2) E k AB ,C (1 / 2) E k A,B (1 / 2) i (11) Note that the ab ov e summation is in fact finite as the terms are zero whenev er k ≥ max {| A | , | C |} . Let us first consider the terms in the sum separately . ∆ k := tr B h ρ B E k +1 † A,B (1 / 2) E k +1 † AB ,C (1 / 2) E k +1 AB ,C (1 / 2) E k +1 A,B (1 / 2) i − tr B h ρ B E k † A,B (1 / 2) E k † AB ,C (1 / 2) E k AB ,C (1 / 2) E k A,B (1 / 2) i (12) 12 A B C 1 1 1 1 1 1 1 1 Γ(3) separable 1 1 1 1 1 1 ∆ 3 1 1 1 1 ∆ 4 1 1 ∆ 5 en tangled, deca ying Figure 2: Illustration of the decomp osition used in our pro of. In addition to the separable con tribution Γ( k 0 ), the superexp onentially decaying remainder terms ∆ k are added to the iden tity contribution γ ( k ) 1 AC (not depicted), lea ving it separable. W e note that ∆ k is supp orted on ∂ k +1 B ∩ ( A ∪ C ), whic h is of size 2 k + 2. F urther we can b ound the op erator norm of ∆ k : ∥ ∆ k ∥ ≤    E k +1 † A,B (1 / 2) E k +1 † AB ,C (1 / 2) E k +1 AB ,C (1 / 2) E k +1 A,B (1 / 2) − E k † A,B (1 / 2) E k † AB ,C (1 / 2) E k AB ,C (1 / 2) E k A,B (1 / 2)    ≤    E k +1 † A,B (1 / 2) − E k † A,B (1 / 2)       E k +1 † AB ,C (1 / 2) E k +1 AB ,C (1 / 2) E k +1 A,B (1 / 2)    +    E k † A,B (1 / 2)       E k +1 † AB ,C (1 / 2) − E k † AB ,C (1 / 2)      E k +1 AB ,C (1 / 2) E k +1 A,B (1 / 2)   +    E k † A,B (1 / 2) E k † AB ,C (1 / 2)      E k +1 AB ,C (1 / 2) − E k AB ,C (1 / 2)     E k +1 A,B (1 / 2)   +    E k † A,B (1 / 2) E k † AB ,C (1 / 2) E k AB ,C (1 / 2)      E k +1 A,B (1 / 2) − E k A,B (1 / 2)   ≤ 4 G 3 G k ( ⌊ k /r ⌋ + 1)! , where the first inequality follo ws from Lemma 7 , the second from the triangle inequal- it y and submultiplicativit y of the op erator norm, and the last line is an application of Lemma 4 . This how ever do es not quite suffice to apply Lemma 2 y et, as ∆ k is a difference of op erators and ma y not b e bounded a wa y from zero. Let us therefore, consider the first expression in Line ( 9 ), whic h is equal to Z k 0 Z B exp  H k 0 AC / 2  ρ k 0 AC exp  H k 0 AC / 2  = Z k 0 Z B Γ( k 0 ) + Z k 0 Z B γ ( k 0 ) 1 AC (13) b y Prop osition 1 as long as | B | ≥ ℓ 1 ( k 0 ). While b oth terms are already separable, we will view the ∆ k as a p erturbation to the second term that remains separable. Noting again that by Lemma 3 , we ha ve Z k 0 Z B ≥ exp  −   H k 0 AB C − H B   + 2 k 0 log( d )  ≥ exp( − 2 k 0 J + 2 k 0 log( d )) (14) 13 Hence, we can write Z k 0 Z B γ ( k 0 ) ≥ C exp( − αk 0 ) (15) for some constants C , α > 0 that com bine the ones from Equation ( 14 ) with the ones lo wer b ounding γ ( k ) in Prop osition 1 . W e no w decomp ose C exp( − αk 0 ) 1 AC + ∞ X k = k 0 ∆ k = ∞ X k = k 0 C e − αk 0 − log(2)( k − k 0 +1) 1 AC + ∆ k . F or this to b e separable by Lemma 2 , w e need to ensure that ∥ ∆ k ∥ ≤ d − k − 1 1 C e − αk 0 − log(2)( k − k 0 +1) , i.e., G 3 G k ( ⌊ k /r ⌋ + 1)! ≤ d − k − 1 1 C e − αk 0 − log(2)( k − k 0 +1) , for all k ≥ k 0 . This condition can b e summarized as C ′ exp( α ′ k + α 0 k 0 ) ≤ ( ⌊ k /r ⌋ + 1)! for some new constants C ′ ≥ 1, α ′ > 0, α 0 and is required ∀ k ≥ k 0 . By Robbin-Stirling’s form ula n ! ≥ ( n/e ) n , this is satisfied if we enforce the (rather lo ose) b ound 3 k 0 := r e exp( r (log ( C ′ ) + α 0 + α ′ )) and so we set our entanglemen t lengthscale ℓ ( J, d, r ) = ℓ 1 ( k 0 ), where the dep endency on the parameters en ters through the choices of constants and the implicit dep endence of the function ℓ 1 from Prop osition 1 . Let us no w summarize the decomp osition to sho w that this closes the pro of: W e com bine ( 13 ) and ( 12 ) into ( 5 )-( 11 ) Z AB C Z B e H AC / 2 ρ AC e H AC / 2 = Z k 0 Z B (Γ( k 0 ) + γ ( k 0 ) 1 AC ) + X k = k 0 ∆ k = Z k 0 Z B Γ( k 0 ) +  Z k 0 Z B γ ( k 0 ) − C e − αk 0  1 AC + ∞ X k = k 0 C e − αk 0 − log(2)( k − k 0 +1) 1 AC + ∆ k , where the first term was already c hosen separable b y Prop osition 1 , the second term is p ositiv e (and thereb y separable) b y Equation ( 15 ), and separabilit y of the last term was just shown for the given en tanglement lengthscale. 3 Cho osing k 0 ≥ r exp( rα ′ + 1), the righ t-hand side of C ′ exp( α ′ k + α 0 k 0 ) ≤ exp( k (log( k /r ) − 1) /r ) gro ws faster than the left-hand side in k so it suffices to enforce it for k = k 0 . Replacing C ′ b y C ′ k 0 strengthens the inequalit y and makes it solv able. 14 4 The thermo dynamic limit In this section, w e discuss the implications of our result for the thermo dynamic limit b y formulating it in terms of states on the quasilo cal algebra of the infinite systems. This is a straightforw ard corollary as our statement already holds uniformly in s ystem size, but requires some additional setup and definitions. W e recall the quasilo cal algebra A = ∪ n A [ − n,n ] and in addition define the left and righ t algebras A L = ∪ n A [ − n, 0] , A R = ∪ n A [1 ,n ] , where all closures are with resp ect to the op erator norm. Note that the quasilo cal algebra is not the algebraic tensor pro duct of the left and righ t algebras. Ho wev er, eviden tly we can still em b ed the left and righ t algebra in the quasilo cal algebra A L , A R ⊂ A and A L ∩ A R = C 1 . The following definition is taken from [ Key+06 , Definition 3.2]. Definition 2. A state ω on A is c al le d a pr o duct state b etwe en A L and A R if for al l A ∈ A L , B ∈ A R it satisfies ω ( AB ) = ω ( A ) ω ( B ) . A state is c al le d sep ar able if it lies in the we ak*-closur e of the c onvex hul l of pr o duct states. F urther, recall that our mo del of b ounded finite-range interaction defines a time evo- lution on A as the automorphism group α t = lim n →∞ exp  it [ H [ − n,n ] , · ]  and that a state ω is a KMS state ω (at β = 1) if ω ( Aα i ( B )) = ω ( B A ) for all A , B in a norm-dense, α t -in v ariant *-subalgebra of A [ BR87 , Definition 5.3.1], but w e will not w ork directly with the definition. W e still denote b y tr [ − n, 0] : A L → A L the partial trace extended by con tin uity to the closure of the algebra. Note that according to the usual em b edding w e identify tr [ − n, 0] [ A ] = tr [ − n, 0] [ A ] ⊗ 1 [ − n, 0] . Corollary 2. L et Φ b e an inter action of r ange r and inter action str ength J and let ω b e the KMS state. Ther e exists an ℓ ∈ N only dep endent on r , J , and d , such that ω ◦ tr [ − ℓ, 0] is sep ar able. Pr o of. Consider the densit y matrices ρ [ − m,m ] , whic h define states on A [ − m,m ] = A [ − m, 0] ⊗ A [1 ,m ] . The t ypical construction of thermal states extends these states using the Hahn- Banac h theorem to the quasilo cal algebra and then pro ceeds to pro ve that the weak-* limit in Λ is a KMS state. Ho w ever, since the first part of this construction, the extension b y Hahn-Banach, may in tro duce additional correlations/entanglemen t b etw een the left and righ t subsystems, we choose a more explicit construction as follows: Define the states ω m on A Λ for any Λ ⊃ [ − m, m ] b y ω m ( A ) = T r Λ  ρ [ − m,m ] ⊗ 1 Λ \ [ − m,m ] d 2 m +1  A  Since these are consistent with the embedding in to larger algebras this defines a state on A 0 , which we also denote b y ω m . F or A ∈ A , w e c ho ose A n ∈ A 0 with A n → A and define ω m ( A ) = lim n →∞ ω m ( A n ), which is unique since for A 0 ∋ B n → A lim n →∞ | ω m ( A n ) − ω m ( B n ) | ≤ lim n →∞ ∥ A n − B n ∥ = 0 . 15 First note that the sequence ω m has a weak-* conv ergent subnet ω α , and that every limit p oint is a KMS state [ BR87 , Prop osition 6.2.15], whic h by [ Ara75 ] is unique in our setup of one-dimensional finite-range interactions, so all limit p oints coincide. W e denote the KMS state by ω and observe that ω ◦ tr [ − ℓ, 0] =  lim α ω α  ◦ tr [ − ℓ, 0] = lim α  ω α ◦ tr [ − ℓ, 0]  as the limit is tak en in the w eak-* sense and tr [ − ℓ, 0] ( A ) ⊂ A . Due to the closedness of the set of separable op erators in the weak-* top ology w e are left to sho w that eac h ω α ◦ tr [ − ℓ, 0] (or each ω m ◦ tr [ − ℓ, 0] ) is separable. By Theorem 2 , c ho osing a sufficien tly large ℓ , w e can decomp ose ω m ◦ tr [ − ℓ, 0] = X k p k ω m,k L ⊗ ω m,k R := X k p k ω m,k (16) on A 0 , where the sum is finite for eac h m b y Carath´ eo dory’s theorem. Again we define ω m,k on A using limits of approximating sequences, whic h are unique due to b oundedness as b efore. Since Equation ( 16 ) holds on a dense subset of A , it remains true for their extensions to A . Finally , for eac h ω m,k and for A ∈ A L , B ∈ A R , w e can find an appro ximating sequence A n → A , B n → B (recall the limit is indep enden t of the c hoice), where A n ∈ A 0 ∩ A L , B n ∈ A 0 ∩ A R and ω m,k ( AB ) = lim n →∞ ω m,k ( A n B n ) = lim n →∞ ω m,k ( A n ) ω m,k ( B n ) = lim n →∞ ω k,m ( A n ) ω m,k ( B n ) = ω m,k ( A ) ω m,k ( B ) . The first equality uses con tin uity of tr [ − ℓ, 0] , the second equalit y the fact that ω m,k is pro duct in A 0 , and the rest is definition. 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