Separability Gap and Large Deviation Entanglement Criterion

Separability gap and large deviation entanglement criterion

Jakub Czartowski,1 Konrad Szymanski,´ 1 Bartłomiej Gardas,1, 2 Yan V. Fyodorov,3 and Karol Zyczkowski˙ 1, 4 1Jagiellonian University, Marian Smoluchowski Institute of Physics, Łojasiewicza 11, 30-348 Kraków, Poland 2Institute of Physics, University of Silesia, ul. Bankowa 12, 40-007 Katowice, Poland 3Department of Mathematics, King’s College London, London WC2R 2LS, UK 4Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland (Dated: June 2, 2020)

For a given Hamiltonian H on a multipartite quantum system, one is interested in finding the energy E0 of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value λmin⊗ of H among all product states. For several concrete model Hamiltonians, we investigate the difference λmin⊗ E0, called separability gap, which vanishes if the ground state has a product structure. In the generic case− of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled.

I. INTRODUCTION. depends clearly on the analyzed Hamiltonian H. The aim of the present work is to investigate to what ex- Describing complex many-body physical systems one often tent this issue poses a fundamental limitation to the near- postulates a suitable Hamiltonian H and tries to find its ground term quantum annealing technology. In particular, we identify Hamiltonians for which the separability gap (1) becomes state energy E0. From the mathematical perspective, one thus faces an optimization problem when searching for the mini- significant. As for those Hamiltonians there exists a systematic upper bound for the precision of the separable state ap- mal expectation value among all normalized pure states ψ . | i proximation commonly used by noisy intermediate scale de- That is to say, E0(H) = minψ ψ H ψ . In principle, if a hermitian Hamiltonian matrix H ish provided,| | i one can diagonalize vice [13], quantum annealers in particular. it, find its spectrum and thus easily identify the smallest eigen- Since the latter devices are far from being perfect in many aspects [14], the measurement process they perform has not value E0. Nevertheless, if the system in question consists of L interacting particles (e.g. spins), the dimension N of the been put under theoretical scrutiny. However, as the quan- matrix grows exponentially, N = 2L, rendering this simplistic tum technology improves, this problem becomes more and approach ineffective for L 1. more relevant for practical applications [15]. In this paper, we show that for a generic Hamiltonian the separable state Although heuristic algorithms for large systems exist [1,2], approximation leads to a significant and systematic error of they are most likely to fail in the high-entanglement limit [3]. the ground state energy. Our findings allow us to formulate In such cases of practical importance one applies various the large deviation entanglement criterion based on a generic, methods based on quantum annealing [4,5] and can depend macroscopic, observable that is applicable for any multipartite on an increasing number of dedicated physical annealing sys- quantum system. The term “generic Hamiltonian” refers to a tems [6–9]. Relying on this approach, however, one faces a typical realization of a random Hermitian matrix pertaining to variety of difficulties and challenges [10]. There is one par- the Gaussian orthogonal ensemble of a fixed dimension. ticular drawback that is not readily evident. Namely, at the We emphasize that it is the measurement process performed end of a quantum annealing, one measures the orientation of by current (and most likely also by near-term [16]) quantum individual spins forming the system and obtains an approx- annealers that serves as the main motivation behind our work. imation to the ground state energy related to a product state, As far as we know, with these machines one can only mea-

arXiv:1812.09251v3 [quant-ph] 1 Jun 2020 (H) = H λmin⊗ minψsep ψsep ψsep , where the minimum is taken sure individual spins in the computational basis. A primary over all product states,h |ψ| =i φ φ φ [11]. | sepi | 1i ⊗ | 2i ⊗ ··· ⊗ | Li example is the D-Wave 2000Q machine where all spins are Such separable states, admitting the simplest tensor network measured in the z-basis to reconstruct the final (classical) en- structure with bond dimension being one [12], are physically ergy. Here we simply pin point far reaching consequences of associated with the mean field like approximations. this fact, indicating the very limit for the underlying present- 3 Although for a system composed out of L 10 spins se- day technology. lecting the optimal configuration of signs out∼ of 2L possibili- ties is already a great achievement, in this way one cannot obtain any approximation for the ground state energy better than II. EXTREME SEPARABLE VALUES AND PRODUCT the minimal product value λmin⊗ (H). The size of the separa- NUMERICAL RANGE bility gap ∆sep(H), defined by the difference of both minima, To tackle the aforementioned issue we begin with basic no- tions and definitions concerning spectrum of quantum sys- ∆ (H) = λ ⊗ (H) E (H), (1) tems. The set of possible expectation values of an operator sep min − 0 0.35 L = 8 b) L = 16 0.25 L = 32 L

/ Neel´ state

sep √ 0.15 (h 2 2)

∆ ≈ 2

H among all normalized states, W(H) = z : z = ψ H ψ , 0.7 0.35 − ) L = 8 { h | | i} λmin⊗ a b) is called numerical range [17]. For any hermitian matrix, E L = 16 † 0 = 0.25 L = 32 H H of order N, this set forms an interval along the real 0.9 axis between the extreme eigenvalues, W(H) = [E ,E ], − L L 0 N 1 0.05 / Neel´ state − /

sep √ where the eigenvalues (possibly degenerated) are ordered, E 0.15 (h 2 2)

∆ ≈ L 0 E0 E1 ..., EN 1. 1.1 / − − → Assume≤ ≤ now≤ that (i) N = MJ so that the Hilbert space has sep J ∆ 0.05 a tensor structure, HN = HM⊗ , and (ii) the product states → 0 1.3 ψ are defined. By analogy, the set of expectation val- − 2 4 6 8 10 0 2 4 6 | sepi 2 0 2 2 2 2 2 4 6 ues of H among normalized product states, W ⊗(H) = z : system size L magnetic field h z = ψ H ψ , is called product numerical range [18{ ]. h sep| | sepi} By definition it is a subset of W(H) and for a Hermitian H it FIG. 1. A numerical solutionmagnetic obtained for the 1D Heisenberg field modelh forms an interval between extreme product values, W (H) = ⊗ in Eq. (3). Panel a) shows the ground state energy, E0, together with [λ ⊗ ,λ ⊗ ]. Product numerical range found several applica- min max the minimal reachable energy, λmin⊗ , as a function of the system size tions in the theory of quantum information [19]. For instance, L. Analytic calculations yield λ ⊗ = 1/L 1 and the best fit re- min − if the minimal product value of a hermitian matrix H of size sults in E0/L = 0.63/L 1.27. Panel b) shows the separability gap, 2 − d is non-negative, then H represents an entanglement witness ∆ /L := (λ ⊗ E )/L versus the magnetic field h. The apparent sep min − 0 or a positive map useful for entanglement detection [20]. local minimum at h 2√2 corresponds to a Néel product state. ≈

A. Linear chain of interacting qubits. nearest neighbour coupling – the 1D Heisenberg model in the transverse magnetic field [24], The model we are going to discuss first is motivated by the L 1 L idea of finding the ground state of a physical system (con- − z z x x z H = ∑ σî σî+1 + σî σî+1 h ∑ σî . (3) sisting of interaction qubits) with spin-glass quantum anneal- − i=1 − i=1 ers [6]. After the annealing cycle has been completed, just before the final measurement, the system Hamiltonian reads [5] Although for a general Hamiltonian it is hardly possible to evaluate the minimal product value analytically, it is doable in the case of vanishing magnetic field, h = 0. z z z H = ∑ Ji jσî σˆ j ∑ hiσî . (2) In order to simplify the matter we assume spherical coor- − i, j E − i V h i∈ ∈ dinates (θ 0,φ 0) on a Bloch sphere, rotated such that the main axis lies along the y axis of the standard Cartesian coordinates. ˆ z Here, σi is the z-th component of the spin-1/2 operator (act- Under such assumption it can be shown that expectation value ing on a local Hilbert space H2) associated with i-th qubit. NL on a separable state Ψsep0 = i=1 ψ(θi0, φi0) yields Input parameters Ji j, hi are defined on a graph G = (E ,V ), | i | i specified by its edges and vertices. They encode the initial L 1 − problem to be solved [6]. Clearly, this Hamiltonian is classi- Ψsep0 H Ψsep0 = ∑ sinθi sinθi+1 cos(φi φi+1), (4) cal in a sense that all its terms commute. Thus, the final mea- h | | i i=1 − surement can be carried out on individual qubits, in any order, thus the minimal product value reads λ ⊗ = 1 L. without disturbing the system [21]. After that, the ground state min − energy is easily reconstructed from the eigenvalues that were A numerical simulation (cf. Fig.1a) shows that the sepa- measured. This is of great practical importance. However, rability gap ∆sep plays a crucial role for any system size. For to become general purpose computing machines [13] near- a large number of qubits the gap grows linearly with the system size, ∆ CL with C 0.27. In the asymptotic limit, term annealers will need to include interactions between the sep ≈ ≈ remaining components of the spin operator, σ x, σ y [22]. L ∞, the ground state energy of (3) was derived analytically, i i E→/L = 4/π, for the same system with periodic boundary General purpose computing machines are those that realize 0 − the gate model of quantum computation to which adiabatic conditions [25, 26]. As in this limit E0 does not depend on the quantum computing is equivalent (with possible polynomial boundary conditions, we arrive at the explicit result for the overhead); cf. Ref. [22]. Although one cannot establish this asymptotic separability gap, equivalence with only ZZ interactions, it is sufficient to add 4 only XX or ZX type of interactions to the annealer Hamilto- ∆sep(H) = λ ⊗ E0 ( 1)L. (5) min − −−−→L ∞ π − nian to demonstrate universality [23]. → For the sake of argument, assume that the final measure- This implies a systematic error if the ground state energy is ment can be accomplished faithfully. Also, let the system be approximated by reconstructing the ground state by an opti- shielded from its environment for as long as it is necessary mal product state. To put it differently, in this case the true to perform computation. Even then, there exists a fundamen- minimal energy of the system can never be reached by any tal limitation on how much information can be extracted from annealing procedure. the system by measuring it in the computational basis. We The separability gap is maximal at h = 0 and vanishes in the demonstrate this feature studying a chain of L spins with a case of very strong fields, h 1, for which the interaction | | 3

L L part of H can be neglected. Interestingly, this dependence is form, HL = σ+⊗ + σ ⊗ , where σ = σx iσy. The only non- not monotonic, as the separability gap ∆ exhibits its mini- zero eigenvalues are− 1 and thus±E = ±1. To calculate the sep ± 0 − mum at h 2√2. At this value of the ﬁeld the gap tends to minimum value over the product states λmin⊗ we again resort zero, since≈ the ground state of the system becomes separable to the polar coordinates on the Bloch ball and deﬁne state NL Néel product state [27]. Ψsep = i=1 ψ(θi,φi) . Calculating the expectation value on| suchi state yields| i

L ! L ! B. Toy model with interaction between all subsystems 1 L Ψsep HL Ψsep = 2 − ∏sin(θi) cos ∑ φ j , (9) h | | i i j=1 Consider an arbitrary Hamiltonian H describing a system of L qudits and acting on the space of dimension dL. If the n which is easily minimized with θi = 0 and ∑ j=1 φ j = π. The eigenstate ψ0 corresponding to the eigenvalue E0 is separa- resulting minimal separable expectation value, λ ⊗ (HL) = ble, the separability| i gap vanishes by definition. However, the min 21 L, tends to zero as L ∞ (recall that E = 1). Simi- reverse implication does not hold, as the gap ∆ can be arbi- − 0 sep lar conclusions can be drawn→ by analyzing a family− of real trary small even if two eigenstates with the smallest energies, symmetric and antidiagonal Hamiltonians with no more than E and E are strongly entangled. 0 1 2L non-zero entries, To investigate this problem consider a model Hamiltonian matrix representing a two-qubit system  a k = 0,...,L 1,  k   − 0 0 0 1 (HL0 )i, j = (i, j) = (1 + k,N k) (i, j) = (N k,1 + k) 0 0 a 0  − ∨ − H =   =: A(1,a,a,1), (6) 0 otherwise. 2 0 a 0 0 1 0 0 0 In particular, setting a1 = 1 one obtains E0 = 1, thus the support of the spectrum is [ 1,1]. On the other hand,− one can where A(x ,...,x ) denotes a matrix with the vector x at the 1 N show using analogous method− as before that antidiagonal and zero entries elsewhere. Then the Hamilto- nian can be written as [25] L 1 L 2 2 λmin⊗ (HL0 ) = 2 − ∑ ak . (10) H = (2 + 2a)σ ⊗ + (2a 2)σ ⊗ . (7) | | 2 x − y k=1 We shall assume that a [0,1], so the ordered spectrum of Hence, the above model extends the family of Hamiltonians ( , , , ) ∈ = H2 reads 1 a a 1 and E0 1. In the non-degenerate for which λ ⊗ tends to zero in the case of a large number of case, a (−0,1)−, all the eigenvectors− of H are maximally en- min ∈ 2 qubits, despite the support of HL being fixed. tangled and they form the Bell basis [28]. Due to the special As we will shortly see, this non–intuitive property is char- form of H2 it is possible to perform optimization over product acteristic for generic Hamiltonians. This is an important result states analytically. By assuming angular parametrisation on especially since λ ⊗ (H) can not be calculated analytically in the Bloch sphere ψ(θ ,φ ) = (cosθ /2,eiφi sinθ /2) for each min | i i i i i general [29] and furthermore all known numerical methods qubit, we arrive at the expectation value of H2 on a product are restricted to small system sizes (cf. Appendix I). state ψ (θ ,θ ,φ ,φ ) ψ | sep 1 2 1 2 i ≡ | sepi 1 ψsep H2 ψsep = sinθ1 sinθ2 III. GENERIC HAMILTONIANS OF L-QUBIT SYSTEMS h | | i 2 (8) [cos(φ1 + φ2) + acos(φ1 φ2)], × − The situation in which separable states do not approximate which is to be minimized. By setting θ1 = θ2 = π/2, φ1 + well the ground state is in some sense generic (or typical). φ = π and φ φ = π we arrive at the minimal value To substantiate this statement let us consider random hermi- 2 1 − 2 λmin⊗ (H2) = (1+a)/2. Note that the separability gap ∆sep = tian matrices drawn from the Gaussian Orthogonal Ensemble (1 a)/2 is− the largest for a = 0 and it vanishes for a = 1. (GOE) of size N = 2L, which describe Hamiltonians acting Analyzing− dimension of a subspace which contains at least on L qubits. For each sample matrix H we wish to deter- a single separable state one can show [18] that for a hermitian mine minimal eigenvalue E0 and estimate minimal separable matrix of order N = 4 the minimal product value is not larger expectation value λmin⊗ . Due to the concentration of measure than the energy of the first excited state, E0 λmin⊗ E1, so in the limit of a large system size these quantities become in this case the separability gap is bounded, ∆≤ ∆≤= E self-averaging, so that for a typical realization their values are sep ≤ 1 1 − E0. Hence in the limit a 1 the spectrum of H2 becomes close to the ensemble averages [30]. degenerated and thus the separability→ gap vanishes. Generically no product states are found in subspaces with Let us now generalize the above model for L qubits by con- dimension comparable to N. In the case of L qubits a sub- sidering a symmetric, antidiagonal real matrix of size N = 2L space of dimension 2L L 1 almost surely (a.s.) contains no − − such that (HL)1,N = (HL)N,1 = 1 and all other entries equal product state [31]. It is therefore reasonable to expect that the to zero. This Hamiltonian captures an all-to-all type of in- range of expectation values of a GOE Hamiltonian over prod- teractions between qubits and can be written in a compact uct states shrinks with increasing system size: product states 4

computational basis scribed in Appendix I or a standard optimization algorithm * any separable basis ( ). Results obtained confer to the bounds (11). The lower ∗ average E0 bound corresponds to a measurement of the energy in an opti- * mized separable basis, while the upper one to a measurement carried out in the ﬁxed separable basis. * ) Proposition 1 implies that for a typical random matrix H L (H) ⊗ min acting on an –qubit system λmin⊗ 0 with probability

λ →

( one, although E0(H) 2. This observation implies that for P a large system described→ − by a generic Hamiltonian the separability gap is constant, ∆sep 2, so it is not possible to obtain any accurate estimation of the→ ground state energy, if the measurement is performed in any separable basis. It is worth emphasizing that the above observation has key consequences for the theory of multipartite entanglement in -2 -1 0 large quantum systems: Measuring any generic observable H ψ A of a composed system of total dimension N in a separa- h i| i ble state yields outcome close to the average of eigenvalues A¯ = TrA/N. This above statement can be connected with ear- FIG. 2. Collection of six distributions P(λmin⊗ ) of minimal separable expectation values for generic GOE Hamiltonians of dimension lier results of Wiesniak´ et al. [34], who proposed to consider N = 2L for L = 3,...,8. Red squares (blue dots) denote asymptotic macroscopic quantities, like magnetic susceptibility, as entan- lower (upper) bounds for λmin⊗ obtained in Eq. (11) and (B15) (in Ap- glement witness. In fact our observation can be formulated in pendix) and with fixed M = 4, green triangles represent the average a similar spirit. J ground state energy E0. Dashed lines are plotted to guide the eye. Any generic hermitian observable A of order N = M allows one to construct two dual entanglement witnesses, corresponding to both wings of the semicircular spectrum, are superpositions of almost all eigenstates of the Hamilto- W (A) := I c A, such that any negative expectation value, nian. This behavior holds true as it is a consequence of the Tr±ρW < 0,± implies± entanglement of the state ρ. The actual ± following two results: value of the parameter, c = N/(J√TrA2 TrA), as a func- Proposition 1. Consider a generic Hamiltonian represented tion of the total system± size N, number of∓ parties J, mean by a GOE matrix H of size N = MJ, with M,J 1 normal- 2 value and the variance of A, follows from the bound (11), since ized as TrH = N, so that the minimal energy asymptoti- it implies that the matrix W is positive among all states sepa- h i ± cally reads E0 2. Then the minimal value λmin⊗ among all J → − rable with respect to the partition HN = HM⊗ . The above re- product states of the J–partite system satisfies the following sult can be reformulated into the following simple, yet a very estimates with probability one (almost surely), general large deviation entanglement criterion. Namely, if an expectation value of a typical observable A r of order N = MJ in the state ρ is sufficiently distant from the 2J 4lnN ¯ λmin⊗ . (11) barycenter of the spectrum, A = TrA/N, that is when −√N a≤.s. a≤.s. − N TrAρ A¯ > 2J√TrA2/N2, (12) Proposition 2.. The above estimates work also for the par- | − | tition of total space into L qubits. Let us assume that M = 2K so that N = 2L with L = K + J, and any state separable with then the state ρ is almost surely entangled with respect to the L J partition into J subsystems with M levels each. respect to partition H2⊗ is separable for splitting HM⊗ as well. Hence this criterion belongs to the class of double–sided To derive the upper estimate note that the diagonal entries entanglement witnesses 2.0 recently analyzed in [35]. Note of H correspond to expectation values among product states that the reasoning holds in one direction only as there exist i i ...i . For any random GOE matrix of size N its diago- also entangled states for which the expectation value is close 1 2 J ¯ nal,| D =idiagH, is a sequence of N numbers independently to the mean A. However, numerical computations confirm a p drawn from the normal distribution N (0, 1/N). Therefore, natural conjecture that the larger the absolute value of the deviation, δ = φ A φ A¯ , the larger average entanglement the typical minimal entry on diagonal min D GOE behaves p h i of the analyzed|h state| | i − (cf.| numerical results presented in as 4lnN/N [32] and leads to the right inequality in (11). φ Fig.3). To quantify entanglement| i of pure states of an L-qubit The− reasoning leading to the lower estimate relies partly upon system we used the family of measures introduced by Meyer the use of the so-called “replica trick” and saddle point ap- and Wallach [36], which are based on the linear entropy of re- proximation (cf. [33] and Appendix II for a more detailed duced states averaged over all possible reductions consisting analysis). of k subsystems, Fig.2 presents histograms of the smallest separable ex- 3 pectation value λ ⊗ obtained for a sample of 10 random k 1 min 2 L − Hamiltonians from the Gaussian Orthogonal Ensemble of size Qk( ψ ) = Slin(ρX ), (13) L k ∑ N = 2 . Numerical data are obtained by the algorithm de- | i 2 1 k X: X =k − | | 5

1 that it is non-zero for several model Hamiltonians acting on multipartite quantum systems. Moreover, we studied Hamil- tonians constructed by random matrices from the Gaussian orthogonal ensemble and demonstrated that for such a generic Hamiltonian involving L qubits the minimal value of energy

) λ ⊗ among all product states is signiﬁcantly larger than the i min

ψ 1 | ground state energy E0. Thus making use of near-term quan- ( 2 2 tum annealers, in which the ﬁnal result is obtained by inde- Q pendent measurements of each of L qubits and corresponds to a product state, can not provide a reliable approximation for the ground state energy of a typical problem. Furthermore, we formulated an entanglement criterion based on the expectation value of a generic observable A among an arbitrary state ρ of 0 -2 -1 0 1 2 a composed quantum system and showed that TrρA provides direct information concerning the degree of entanglement of A ψ h i| i the investigated state ρ.

FIG. 3. Range of allowed values for pure states of the system consisting of L = 7 qubits in the plane spanned by the expectation ACKNOWLEDGEMENTS L value A ψ of a GOE observable A = H of size N = 2 , and the h i| i Meyer-Wallach measure of entanglement deﬁned in Eq. (13). Black It is a pleasure to thank Jacek Dziarmaga, Bálint Koc- crosses denote eigenstates of H, red region – numerically determined ˙ range attained by pure states, shaded blue region denotes the bound zor, Tomasz Maci ˛azek, Andy Mason, Zbigniew Puchała, Marek Rams, Ilya Spitkovsky and Jakub Zakrzewski for in- TrAρ A¯ 2L√TrA2/N2 implied by Eq. (12), beyond which the |states are− entangled.| ≤ In addition, yellow circles represent a sample of spiring discussions. Financial support by Narodowe Cen- 10 random pure states, green squares – 10 random product states. trum Nauki under the grant number 2015/18/A/ST2/00274 (KZ)˙ and 2016/20/S/ST2/00152 (BG) is gratefully acknowl- edged. The research at KCL was supported by EPSRC grant 2 where Slin(ρ) = 1 Trρ being the linear entropy of a state EP/N009436/1. This research was supported in part by PL- ρ of dimension 2k−. This function captures the mean entan- Grid Infrastructure. glement of k-qubit subsystems with the rest of the system. Although Fig.3 depicts data obtained for Q2, similar results were also analyzed for other measures of entanglement, in- V. APPENDIX cluding quantities Qk with k = 1,...,L. All these results support the statement that the deviation of the expectation value A. Numerical technique to estimate λmin⊗ (H) A beyond the bounds (12) can be used to quantify the de- h iψ gree of entanglement of the analyzed state ψ . We brieﬂy sketch here the approach employed in this work | i For comparison Fig.3 contains also data for random sep- to calculate the separability gap for a random Hamiltonian arable states and generic random states, which are known to pertaining to Gaussian Orthogonal Ensemble (GOE), in the be highly entangled [37, 38]. The set of separable pure states case of a small system size (up to N = 28). The ground has a lower dimension and carries zero measure in the entire state energy E0 can be obtained easily in this case. The al- set of all pure states, so its projection W ⊗(A) onto an axis de- gorithm used for calculation of minimal separable expecta- termined by the observable A is typically much smaller than tion – λmin⊗ , on the other hand, utilizes the divide and conquer the entire range W(A). Asymptotically, in the limit of large strategy [39]. To begin with, let us consider a general case dimension N of the Hilbert space, the ratio of the volumes of of minimizing expectation value of α β H α β , where both sets tends to zero. α H is a qubit state and β belongsh ⊗ to| a|d-dimensional⊗ i | i ∈ 2 | i space Hd. The expectation value can be rewritten as

IV. DISCUSSION AND OUTLOOK α β H α β = α H β α , (A1) h ⊗ | | ⊗ i h | | i | i where H = Tr [H(1 )] is a matrix of size 2. If In this work we have investigated to what extent the near- β B β β β is fixed,| furtheri optimization⊗ | ih over| α is trivial: the result| isi term quantum annealing technology may become fundamen- | i minimal eigenvalue of 2 2 hermitian matrix H β : tally limited by its intrinsic measurement process allowing to × | i ask only yes or no questions to individual qubits. This type of s 2 “polling” on a quantum system is probably the most natural TrH β H β min α H β α = | i | i detH β . (A2) one and definitely the easiest to realize experimentally. Un- α h | | i | i 2 − 2 − | i fortunately, as we have argued, it does not allow to extract all | i relevant information from the system in question. The above expression can be written in a more succinct form. In particular, we analyzed the separability gap and showed Let H = Tr [H(σ 1)] (with σ = 1 ). Then, the above ex- i A i ⊗ 0 2 6 pression becomes with w := w1 ... wJ [40]. To this end, we assume a hermitian⊗ Hamiltonian⊗ H⊗drawn from GOE with scale parameter q 2 H0 1 2 2 2 a such that TrH = aN. h i H1 + H2 + H3 , (A3) 2 − 2 h i h i h i To beginh with, wei introduce the partition function [41] where all averages are taken over the d-dimensional vector Z β . The minimization of α β H α β can be now inter- Zβ = exp( β w H w )dw , (B3) − h ⊗| | ⊗i ⊗ preted| i as minimization of hconvex⊗ |function| ⊗ overi 4-dimensional convex set of simultaneous expectation values called numeri- where β plays a role of the inverse temperature. Here, dwi de- cal range: notes the integration measure over a single qubit space. Then, the typical separable expectation value, Eq. (B1), can be found W(H0,H1,H2,H3) = conv ( H0 , H1 , H2 , H3 ) β : { h i h i h i h i | i as the zero-temperature limit of the associated free energy: β H . | i ∈ d} (A4) lnZβ λmin⊗ = lim . (B4) h i − β ∞ β This problem is easily numerically solved with an arbitrary → high accuracy. To calculate the latter limit, consider the following function Solution of the H2 Hd case can be leveraged to the more defined for positive integer n, ( k) ⊗ general H ⊗ Hd, where k N: using the procedure de- ! 2 ⊗ ∈ Z n n scribed above, it is possible to determine arbitrarily close ap- n (i) (i) (i) Zβ = exp β ∑ w H w ∏dw , (B5) proximation of the set − i, j=1 h ⊗ | | ⊗ i i=1 ⊗

W ⊗(H0,H1,H2,H3) = conv ( H0 , H1 , H2 , H3 ) γ which is the n-th power of Z . Then we can formally write { h i h i h i h i | i β : γ H2 H , γ = α β . | i ∈ ⊗ d | i | ⊗ i} D E (A5) d n lnZβ = Zβ , (B6) h i dn n=0 This (convex) set W ⊗ can then be used in place of W in cal- which is interpreted as a derivative of an analytic continuation culation of λmin⊗ – the result is minimal energy over sepa- D nE rable states in tripartite case. This result can be alike used of Zβ . This average can be further simplified using the further – the recursive structure provides natural extensions. equality which holds for any matrix X Complexity of the algorithm is exponential, owing to the NP- completeness of the problem, but it is possible to determine exp( β TrHX) = exp[aβ 2 TrX2 /2] (B7) h − i H certified lower and upper bounds of λmin⊗ this way in deter- ministic time and linear space complexity. with a being the scaling parameter of the GOE. Here, XH denotes a hermitian part of a matrix X. Therefore, ! Z n 2 n B. Lower estimate in Eq. (11) of main text n 1 2 (i) ( j) (i) Zβ = exp aβ ∑ w w ∏dw . (B8) h i 2 i, j=1 h ⊗ | ⊗ i i=1 ⊗ In this section we provide a reasoning, which leads to the left hand side of Eq. (11) in the main text for any generic (i, j) (i) ( j) By introduction of a collection of matrices Q = u u , Hamiltonian H of order N = MJ, where M denotes a high k k k so that h | i (M 1), but otherwise arbitrary dimension of each subsys- (M n 1)/2 tem, while J stands for their number. The method used relies n k=1...J J ! − − upon the use of the so-called “replica trick”, which is a pow- (i) J (i, j) ∏du = C(n,M) ∏ dQk ∏ detQk , erful but not fully rigorous method of theoretical physics. i=1 ⊗ i, j=1,...n k=1 We wish to show that the minimal separable expectation (B9) ( , ) value, λmin⊗ (H), vanishes in the large system limit, N ∞. where the number C n M does not depend on J [42], the Here we analyze separability with respect to partition of→ the above integral can be written in a form suitable for the sad- system into J subsystems of size M each. Due to effect of con- dle point approximation [33, 43]. The domain of integration centration of measure the above quantity is "self-averaging", over matrices Q goes over positive-definite matrices of size which means that its distribution becomes strongly localized n with diagonal entries fixed to be unity. Making use of this around the expectation value. Therefore it is sufficient to approximation one arrives at an expression, study the average value and demonstrate that  Φ     z }| { λ ⊗ 2J/√N, (B1) Z M ( , ) ( , )2  h mini ∼ − Z n = exp β 2 Q i j ...Q i j + lndetQ h β i 2 ∑ 1 J ∑ k where the brackets denote the ensemble average, and    

λ ⊗ = min w H w , (B2) (i, j) (n+1)/2 min J − w h ⊗| | ⊗i C(n,M) dQk detQk . (B10) | ⊗i × ∏ ∏ 7

1/J where we chose a nonstandard GOE scaling, a = M = N . Finally, taking the limit β ∞ in Eq. (B4) we obtain λ ⊗ → h mini ∼ In the limit of large M, this integral is dominated by the MJ, with M = N1/J, where N is the total system size, J is maximum of the exponent argument. Furthermore, C(0,M) = the− number of partitions and M is their local dimension. Since 1 and we have worked with the scaling a = M, the ensemble average, λmin⊗ , needs to be compared to the average minimal MΦ(Qoptim) h i Z n lim . (B11) eigenvalue, E0. Then we arrive at the desired expression h β i ∼ n 0 n → 1/J λ JN 1 min⊗ (1 J− )/2 Henceforward, we assume that Q1 = QJ = Q, where Q is h i = − 1 = JN− − . (B15) ··· E0 N(1+J− )/2 parametrized with a single parameter q: − We have assumed that M 1 such that the saddle point 1 q ... q method can be used. Let us now consider the case of N ∞; q 1 ... q   M 1 is kept constant and L ∞. In this limit the following→ Q(q) = . . . .. (B12) → . . .. . holds q q ... 1 λ ⊗ log N h mini = M . (B16) Such an ansatz is compatible with properties of maximal E0 √N Φ[Q(q)] at low temperature (high β). This demonstrates that the estimate (B1) holds for J 1, what By calculating maximum of Φ[Q(q)] one can easily deter- completes the reasoning concerning Proposition 1. mine that q = 1 1/β in the limit β ∞. Thus, − → Therefore, when dimension N = MJ increases, the minimal J2 separable expectation value λmin⊗ of generic Hamiltonian of Φ[Q(q)] = ∑ Q◦ i, j + ∑lndetQk J size N with respect to partition HM⊗ approaches 0. n(n 1) Let us now proceed to Proposition 2. Formally, to conduct = − q2J (B13) 2 the proof we require that M = N1/J 1. Let us now assume K + J (n 1)ln[1 q] + ln[1 + q(n 1) , that the local dimension forms a power of two, M = 2 , so the { − − − } total dimension reads N = MJ = 2L with L = K +J. Any state where denotes the Hadamard (i.e. elementwise) product of ψ entangled with respect to the partition of the entire system | i matrices.◦ Therefore the following estimate holds into J subsystems of size M is also entangled with respect to the ﬁner partition into L qubits. Therefore the estimate (B16) 2J L n M Jq q holds also for the physically motivated partition HN = H2⊗ Zβ + J ln[1 q] h i ∼ 2 1 q − 2 − and implies that the ratio λmin⊗ /E0 tends to zero in the limit − (B14) N ∞. h i (1 β 1)2J → = J(β 1) + − − J lnβ . − 2 −

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