Should Entanglement Measures Be Monogamous Or Faithful?

Should Entanglement Measures be Monogamous or Faithful?

Cecilia´ Lancien,1, 2 Sara Di Martino,1 Marcus Huber,3, 1 Marco Piani,4 Gerardo Adesso,5 and Andreas Winter1, 6 1F´ısicaTeòrica:Informaciói FenòmensQuàntics,Universitat Autònomade Barcelona, ES-08193 Bellaterra (Barcelona), Spain 2Institut Camille Jordan, UniversitéClaude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France 3Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland 4SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom 5School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 6ICREA – InstitucióCatalana de Recerca i Estudis Avan¸cats,Pg. Lluis Companys 23, ES-08010 Barcelona, Spain (Dated: 27 June 2016) “Is entanglement monogamous?” asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett’s legacy on quantum information theory. While the answer is affirmative in the quali- tative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.

Introduction. Entanglement is a quintessential manifesta- number of parties attracted increasing interest. The concept tion of quantum mechanics [1, 2]. The study of entanglement of monogamy became synonymous with the validity of an in- and its distribution reveals fundamental insights into the na- equality due to Coffman, Kundu and Wootters (CKW) [25]. ture of quantum correlations [3], on the properties of many- Given any tripartite state ρABC, and choosing a bipartite en- body systems [4, 5], and on possibilities and limitations for tanglement measure E, the CKW inequality reads [26] quantum-enhanced technologies [6]. A particularly interest- ing feature of entanglement is known as monogamy [7], that EA:BC(ρABC) > EA:B(ρAB) + EA:C(ρAC) , (1) is, the impossibility of sharing entanglement unconditionally ρ ρ ρ ρ across many subsystems of a composite quantum system. with AB = TrC[ ABC] and AC = TrB[ ABC]. Intuitively, Eq. (1) means that the sum of the individual pairwise en- In the clearest manifestation of monogamy, if two parties tanglements between A and each of the other parties B or C A and B with the same (finite) Hilbert space dimension are cannot exceed the entanglement between A and the remaining maximally entangled, then their state is a pure state |ΦiAB parties grouped together. Eq. (1) was originally proven for [8], and neither of them can share any correlation—let alone arbitrary states of three qubits, adopting the squared concur- entanglement—with a third party C, as the only physically rence as entanglement measure [25]. Variations of the CKW allowed pure states of the tripartite system ABC are product inequality and generalizations to n parties have been estab- states |ΦiAB ⊗ |ΨiC. Consider now the more realistic case of lished for a number of entanglement measures in discrete as A and B being in a mixed, partially entangled state ρAB. It is well as continuous variable systems [20, 27–39]. In particu- then conceivable for more parties to get a share of such entan- lar, the squashed entanglement [40] and the one-way distill- glement. Namely, a state ρAB on a Hilbert space HA ⊗ HB is able entanglement fulfill Eq. (1) in composite systems of ar- termed “n-shareable” with respect to subsystem B if it admits bitrary dimension [28]. Hybrid CKW-like inequalities involv- a symmetric n-extension, i.e. a state ρ0 on H ⊗ H ⊗n in- AB1...Bn A B ing entanglement and other forms of correlations have also variant under permutations of the subsystems B1,..., Bn and been proven [28, 41], while measures of quantum correlations such that the marginal state of A and any B j amounts to ρAB. weaker than entanglement generally violate the CKW inequal- While even an entangled state can be shareable up to some ity [42]. To some extent, therefore, Eq. (1) does capture the number of extensions, a seminal result is that a state ρAB is spirit of monogamy as a distinctive property of entanglement. n-shareable for all n > 2 if and only if it is separable, that is, The main problem with CKW inequalities, however, is that no entangled state can be infinitely-shareable [7, 9–12]. This their validity is not universal, but rather depends on the spe- statement formalizes exactly the monogamy of entanglement cific choice of E. Perhaps counterintuitively, several promi- (in an asymptotic setting), and has many important implica- nent entanglement monotones, such as the entanglement of tions, including the equivalence between asymptotic quantum formation or the distillable entanglement [3, 43, 44], do not cloning and state estimation [13, 14], the emergence of objec- obey the constraint formalized by Eq. (1), unless one intro- tivity in the quantum-to-classical transition [15], the security duces ad hoc rescalings (see e.g. [45]). Since entanglement as of quantum key distribution [16–19], and the study of frustra- a concept is monogamous in the n-shareability sense [7], one tion and topological phases in many-body systems [20–24]. is led to raise the following key question: Should any valid Over the last two decades, the goal to formalize monogamy entanglement measure be monogamous in a CKW-like sense? of entanglement in precise quantitative terms and for a finite In this Letter we address the question in general terms. 2 EA:C

EA:BC FIG. 1: For any tripartite state ρABC , an entanglement measure E obeys monogamy if, given the global entanglement EA:BC , the pairwise terms EA:B and EA:C are non-trivially constrained. We formalize these constraints via a function f (EA:B, EA:C ) in Eq. (2). Choosing f (x, y) = x+y, one gets the CKW inequality (1), which limits the pairwise terms to the triangular darker region with dashed p boundary. The other depicted shaded regions correspond to f (x, y) = x2 + y2 (dotted boundary) and f (x, y) = max(x+cy4, y+cx4) with c a constant (dot-dashed boundary). Any measure E is termed monogamous if, for all tripartite states, the ensuing entanglement distribution can be conﬁned to a region strictly smaller than the white square with solid boundary. The latter denotes the trivial choice EA:B 0 EA:BC f (x, y) = max(x, y), which is satisﬁed a priori by any entanglement monotone E.

Given an entanglement measure E, we shall say that it is cally unfaithful) ones, like the squashed entanglement [28, monogamous if there exists a non-trivial function f : R>0 × 40], and geometrically faithful (yet non-monogamous) ones, R>0 → R>0 such that the generalized monogamy relation like EF , ER, and their regularizations. Finally, we show that this dilemma can be resolved if one relaxes the definition (2) EA:BC(ρABC) > f EA:B(ρAB), EA:C(ρAC) , (2) to introduce monogamy relations for any fixed dimension of ∞ HA ⊗ HB ⊗ HC. Explicitly, we prove that EF and ER are re- is satisfied for any state ρABC on any tripartite Hilbert space trievable as monogamous for any finite dimension, by provid- HA ⊗HB ⊗HC. Recalling that E is an entanglement monotone ing dimension-dependent choices of f in Eq. (2), which only (which in turn implies that it is nonincreasing under partial reduce to the trivial one in the limit of infinite dimension. traces), the function f in Eq. (2) is without loss of generality Result (1) Generic non-monogamy for entanglement such that f (x, y) > max(x, y). Thus, to give rise to a non-trivial of formation and relative entropy of entanglement. We constraint, we need f (x, y) > max(x, y) for at least some range begin by defining the measures employed in our analysis of values of x and y. One might further impose that f (x, y) is [3, 43, 44, 46]. The entanglement of formation EF is the con- monotonic in both its arguments, but we will not require this vex roof extension of the entropy of entanglement, EF (ρA:B) = here. We will however require that f is continuous in general. P inf i piS (TrB[|ψiihψi|AB]), where S (ρ) = − Tr[ρ log ρ] While the CKW form (1) of a monogamy relation (which is {pi,|ψiiAB} recovered for the particular choice f (x, y) = x + y) implicitly is the von Neumann entropy, and log ≡ log2. On the presumes some kind of additivity of the entanglement mea- other hand, the relative entropy of entanglement ER quanti- sure in question, our general form (2) transcends this and can fies the distance from the set of separable states, ER(ρA:B) = be applied to recognize any entanglement measure E as de inf S (ρAB||σAB), where the minimization is over all separable σAB facto monogamous, based on the intuition that it should obey σAB, and S (ρ||σ) = Tr[ρ log ρ−ρ log σ] is the relative entropy. some trade-off between the values of EA:B and EA:C for a given The fact that both EF and ER violate the CKW inequal- EA:BC, see Fig. 1. Oppositely, if the only possible choice in ity (1) may be traced to their subadditivity, meaning that Eq. (2) were f (x, y) = max(x, y), then the measure E would there exist states ρAB and σA0C such that E(ρA:B ⊗ σA0:C) < fail monogamy in the most drastic fashion: given a state ρABC, E(ρA:B) + E(σA0:C), with E denoting either EF [53] or ER [54]. having EA:BC > 0 a priori would not imply that EA:B and EA:C We now show that these measures fail monogamy even in the have to constrain each other in the interval [0, EA:BC]. general sense of Eq. (2). These results are based on random Quite remarkably, we rigorously show in the following that induced states, defined as follows. Given n, s ∈ N, a random n s the entanglement of formation EF [43] and the relative en- mixed state ρ on C is induced by C if ρ = TrCs |ψihψ| for |ψi tropy of entanglement ER [46], which are two of the most a uniformly distributed random pure state on Cn ⊗ Cs. Note important entanglement monotones for mixed states [3, 44], that if s 6 n, this is equivalent to ρ being uniformly distributed cannot satisfy a non-trivial monogamy relation in the sense of on the set of mixed states of rank at most s on Cn. Here we Eq. (2), with violations becoming generic [47] with increas- focus on the (balanced) bipartite Hilbert space Cd ⊗ Cd, and ing Hilbert space dimension. We further show that a whole aim to determine, given d, s ∈ N, what is the typical value d d class of additive entanglement measures, including the entan- of EF (ρA:B) and ER(ρA:B) for ρAB a random state on C ⊗ C ∞ s glement cost EF [43, 48] and the regularized relative entropy induced by an environment C . The answer is as follows, of entanglement E∞ [49, 50], also fail monogamy as captured R h i 2 2 2 P − − 1 − −cd t / log d by Eq. (2). The latter result is proven by a constructive argu- EF (ρA:B) log d 2 ln 2 6 t > 1 e , (3) ment which exploits the peculiar properties of the maximally h 2 i 2 antisymmetric state on Cn ⊗ Cn, which is (n − 1)-shareable yet P − d 1 s − −cst ER(ρA:B) log s + 2 ln 2 d2 6 t > 1 e , (4) far from separable [51], and has hence been dubbed the ‘universal counterexample’ in quantum information theory [52]. where s 6 Cd2t2/ log2(1/t) for any fixed t > 0 in Eq. (3), and Specifically, any additive entanglement measure which is ge- Cd log(1/t)/t2 6 s 6 d2 for any fixed 0 < t < 1 in Eq. (4), ometrically faithful in the sense of being lower-bounded by while c, C > 0 denote universal constants in both equations. a quantity with a sub-polynomial dimensional dependence on While Eq. (3) was established in [47] (Theorem V.1), although the antisymmetric state, cannot be monogamous in general. with a looser dependence on the parameters, Eq. (4) is an en- Our analysis then reveals that entanglement measures di- tirely original result of independent interest. Both proofs are vide into two main categories: monogamous (yet geometri- mathematically quite involved, and are relegated to [55]. 3

Importantly, failure of monogamy is then retrieved as a 퐴0 퐴1 퐴2 … … 퐴 푘 퐴 푘 퐴 푘 … … 퐴 푘+1 generic trait of entanglement quantified by these measures. 2 2 +1 2 +2 2 Namely, the main result of this section is that there exist states 퐴 퐵 퐶 (x) (x) (x) (x) ρABC on Hilbert spaces HA ⊗HB ⊗HC such that, as x → ∞, FIG. 2: Schematic of the antisymmetric state αA2k+1+1 partitioned in (x) (x) (x) three subsystems, A ≡ A0, B ≡ A1 ... A2k , and C ≡ A2k+1 ... A2k+1 . EA:BC ρABC 6 x, while EA:B ρAB ∼ EA:C ρAC ∼ x, (5) for E denoting either EF or ER. To sketch the proof [55], set x (x) (x) (x) d (x) monogamy relation in the sense of Eq. (2). We first establish d = b2 c and HA ≡ HB ≡ HC ≡ C . Next, consider ρABC a n d the following result. Let n ∈ N, d = 2 + 1, and set HA j ≡ C (x) (x) (x) (x) n random state on HA ⊗ HB ⊗ HC , induced by some HE ≡ for each 0 6 j 6 2 . Assume next that E satisfies conditions s (x) (x) (a) and (b). Then, there exists 0 k n − 1 such that C , with s ∼ log d. In that way, ρAB and ρAC are random states 6 6 on Cd ⊗ Cd, induced by some Cs ⊗ Cd, with sd satisfying both EA :A ...A α 2k+1 = EA :A ...A α 2k+1 the conditions for Eqs. (3) and (4) to apply. We then have: 0 1 2k A 0 2k+1 2k+1 A (x) (x) (x) ln(nt+1/c) E{F,R} ρ 6 log d 6 x, while EF ρ and EF ρ are k+1 A:BC A:B A:C > 1 − EA :A ...A k+1 α 2 +1 (6). both equal to log d −O(1) ∼ log d ∼ x with probability greater n 0 1 2 A −cd2/ log2 d (x) (x) than 1−2e , and ER ρA:B and ER ρA:C are both equal To prove Eq. (6), consider a partition of the antisym- to log d−log(log d)−O(1) ∼ log d ∼ x with probability greater metric state α 2k+1+1 as illustrated in Fig. 2, and set g = A k than 1 − 2e−cd log d. EA0:A1...A k αA0...A k for each 0 6 k 6 n. Then, Result (2) Non-monogamy for a whole class of additive 2 2 entanglement measures. We now show that a class of addi- c c ≈ 6 g0 6 g1 6 ··· 6 gn−1 6 gn 6 log d ≈ n. (7) tive entanglement measures also fail monogamy in the sense nt (log d)t of Eq. (2). A key role in this result is played by the antisym- The last inequality is by property (a), because dA0 = d = metric state, defined as follows [48, 51]. Given a subsystem A n 2 + 1. The first inequality is by property (b), because g0 = with (finite-dimensional) Hilbert space HA, the (maximally) t t ⊗n EA0:A1 αA0 A1 > c/(log d) ≈ c/n . And the middle inequal- antisymmetric state αAn on HA is the normalized projector ⊗n ities are by monotonicity of E under discarding of subsys- onto the antisymmetric subspace of HA . A crucial property tems, because for each 0 6 k 6 n − 1, αA ...A is the re- of αAn is that its reduced state on any group of k subsystems, 0 2k ⊗k duced state of αA ...A . Now, Eq. (7) implies that there exists for any 0 6 k 6 n, is α k , i.e. the antisymmetric state on H . 0 2k+1 A A ¯ t+1 0 6 k 6 n−1 such that gk¯ /gk¯+1 > 1−ln(n /c)/n. Indeed, oth- We now focus on entanglement monotones E satisfying the t+1 n g0 Qn−1 gk ln(n /c) c ρ erwise we would have = < 1 − 6 t+1 , following conditions: (a) Normalization: For any state AB gn k=0 gk+1 n n on H ⊗ H , E (ρ ) 6 min(log d , log d ); (b) Lower- which contradicts Eq. (7). Then, as we have on one hand A B A:B AB A B ¯ boundedness on the bipartite antisymmetric state: Denoting gk¯+1 = EA0:A1...A ¯ α 2k+1+1 , and on the other hand gk¯ = 2k+1 A by α 0 the antisymmetric state on H ⊗ H 0 (with d = d 0 ), ¯ ¯ AA A A A A EA :A ...A ¯ α 2k+1 = EA :A ¯ ...A ¯ α 2k+1 , Eq. (6) is proven. t 0 1 2k A 0 2k+1 2k+1 A EA:A0 (αAA0 ) > c/(log dA) , where c, t > 0 are universal con- The main result of this section then follows immediately. stants; (c) Additivity on product states: For any state ρAB on Namely, once again, there exist states ρ(x) on Hilbert spaces ⊗m ABC H ⊗ H , E m m (ρ ) = m E (ρ ); (d) Linearity on mix- x x x A B A :B AB A:B AB H ( ) ⊗H( ) ⊗H( ) such that Eq. (5) holds, for E now denoting tures of locally orthogonal states: For any 0 λ 1, and any A B C 6 6 any entanglement measure satisfying conditions (a)–(d). states ρ , σ on H ⊗H such that Tr[ρ σ ] = Tr[ρ σ ] = AB AB A B A A B B The proof goes as follows. As E satisfies (a) and (b), 0, E (λ ρ + (1 − λ)σ ) = λ E (ρ ) + (1 − λ)E (σ ). A:B AB AB A:B AB A:B AB we know by Eq. (6) that, for any d ∈ N, there exists a Important examples of entanglement measures fulfilling the d state ρABC on HA ⊗ HB ⊗ HC, where HA ≡ C and HB ≡ above requirements are the regularized versions of the entan- k H ≡ (Cd)⊗2 for some 0 k blog dc (see Fig. 2), such glement of formation (aka entanglement cost) E∞ [43, 48] and C 6 6 F that E (ρ ) and E (ρ ) are both lower bounded by of the relative entropy of entanglement E∞ [49, 50]. Indeed, A:B AB A:C AC R log(logt+1 d/c) condition (c) holds by construction for any regularized entan- 1 − log d EA:BC (ρABC) = (1 − o(1))EA:BC (ρABC). Now, ∞ 1 ⊗n glement measure, defined as E (ρAB) = lim EAn:Bn (ρ ). ⊗m A:B n→∞ n AB by property (c), for any m ∈ N, considering ρABC instead of ∞ ∞ ⊗m Furthermore, in the case of EF and ER , conditions (a) and (d) ρABC (and relabelling A into A etc.) will multiply all values are inherited as they hold for EF and ER. Finally, condition of E by a factor m. By property (d), for any 0 6 λ 6 1 and any (b) can be seen as some kind of faithfulness (or geometry- separable state σABC (across the cut A : BC) which is locally preserving) property: given that the antisymmetric state has orthogonal to ρABC, considering λ ρABC + (1 − λ)σABC instead constant trace distance from the set of separable states, one of ρABC will multiply all values of E by a factor λ. Conse- may wish for an entanglement measure to stay bounded away quently, any value x > 0 for EA:BC (ρABC) is indeed attainable, (x) (x) (x) from 0 on the antisymmetric state, dimension-independently on some suitably large Hilbert space HA ⊗ HB ⊗ HC . as well (or with a sub-polynomial dependence). For E being Recapitulating, we demonstrated that entanglement mea- ∞ ∞ EF or ER , a condition stronger than (b) in fact holds, namely sures which faithfully capture the properties of the antisym- EA:A0 (αAA0 ) > c, where c > 0 is a universal constant [51]. metric state cannot be monogamous in general. Conversely, What we show here is that any entanglement measure there exist relevant entanglement measures for which the de- E, obeying properties (a)–(d), cannot satisfy a non-trivial sirable condition (b) does not hold—such as the squashed 4 entanglement, which scales as o(1/dA) on the antisymmetric entangled, as both measures vanish only on separable states. state αAA0 —yet monogamy holds instead, even in the original Conclusions. We addressed on general grounds the ques- CKW form (1) [28, 40]. This is the origin of the “monogamy tion of whether entanglement measures should be monoga- vs faithfulness” dilemma discussed in the introduction. mous in the sense of obeying a quantitative constraint akin Result (3) Recovering monogamy: dimension-dependent to Eq. (1) introduced in [25]. We showed that paradigmatic relations. In the previous two sections, we proved that several measures such as the entanglement of formation and the rela- important entanglement measures cannot obey a monogamy tive entropy of entanglement, as well as their regularizations, relation of the form (2) with f a universal function. Neverthe- cannot be monogamous in general, as they cannot satisfy any less, it may become possible to establish such an inequality non-trivial general relation of the form (2) limiting the distri- if we allow the function f to be dimension-dependent. For bution of bipartite entanglement in arbitrary tripartite states. instance, the squared entanglement of formation obeys the Monogamy can nonetheless be recovered if the constraints are CKW inequality for arbitrary three-qubit states [36], which p made dependent on the (finite) dimension of the system. means that choosing f (x, y) = x2 + y2 in Eq. (2), as de- The present study substantially advances our understand- picted in Fig. 1 (dotted boundary), makes EF monogamous ing of entanglement and the complex laws governing its dis- when restricted to Hilbert spaces with dA = dB = dC = 2. tribution in systems of multiple parties, and paves the way The third main result of this Letter is to show that non- to more practical developments in quantum communication trivial dimension-dependent monogamy relations can be es- and computation. The concept of monogamy as studied here ∞ tablished for EF and ER in any finite dimension. Concretely, is of particular physical relevance, as the structure of Eq. (2) for any state ρABC on a Hilbert space HA ⊗ HB ⊗ HC, it holds lends itself to be applied repeatedly to establish limitations in a many-body scenario [5], allowing one to compare the dis- c 8 EF (ρA:BC) > max EF (ρA:B) + 8 EF (ρA:C) , tribution of entanglement on equal footing across the various dAdC log dA,C c 8 parts of a composite system, unlike e.g. the case of hybrid EF (ρA:C) + 8 EF (ρA:B) , (8a) dAdB log dA,B monogamy relations involving different quantifiers [28]. ∞ ∞ c0 ∞ 4 It will be worth investigating further links between the phe- ER (ρA:BC) > max ER (ρA:B) + 4 ER (ρA:C) , dAdC log dA,C nomenon of monogamy and the so-called quantum marginal 0 E∞ (ρ ) + c E∞ (ρ ) 4 , (8b) problem [59] as well as the fact that information cannot be R A:C d d log4 d R A:B A B A,B arbitrarily distributed in multipartite quantum states [60–63]. 0 where c, c > 0 are universal constants, and we set dA,B = Implications of our study for progress in other fields like con- min(dA, dB), dA,C = min(dA, dC). An instance of Eq. (8b) is densed matter [5] and cosmology [64], where monogamy of qualitatively illustrated in Fig. 1 (dot-dashed boundary). entanglement takes centre stage, also deserve further study. The proof of Eqs. (8) makes use of results from Refs. [50, Acknowledgments. We acknowledge financial support 56–58], and is provided in [55]. While Eqs. (8) may not be from the European Union under the European Research Coun- tight [55], they do establish that the involved entanglement cil (StG GQCOP No. 637352, AdG IRQUAT No. 267386), measures can be effectively regarded as monogamous accord- the European Commission (STREP RAQUEL No. FP7- ing to Eq. (2) in any finite dimension, even though the con- ICT-2013-C-323970) and the Marie Skłodowska-Curie Grant straints become trivial in the limit of infinite dimension, in No. 661338, the Spanish MINECO (Project No. FIS2013- agreement with results (1) and (2). Notice further that Eqs. (8) 40627-P), the Generalitat de Catalunya CIRIT (Project encapsulate strict monogamy, as the constraining functions No. 2014-SGR-966), the Swiss National Science Founda- satisfy f (x, y) > max(x, y) for all positive x and y. This im- tion (AMBIZIONE PZ00P2 161351), and the French CNRS plies that if, say, E(ρA:B) = E(ρA:BC), then E(ρA:C) = 0 for E (ANR projects OSQPI 11-BS01-0008 and Stoq 14-CE25- ∞ being either EF or ER , which means that A and C must be un- 0033). G.A. thanks B. Regula for fruitful discussions.

[1] E. Schrodinger,¨ Proc. Camb. Phil. Soc. 31, 553 (1935). 255 (1988). [2] E. Schrodinger,¨ Proc. Camb. Phil. Soc. 32, 446 (1936). [10] R. F. Werner, Lett. Math. Phys. 17, 359 (1989). [3] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, [11] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A Rev Mod Phys 81, 865 (2009). 69, 022308 (2004). [4] J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, [12] D. Yang, Phys. Lett. A 360, 249 (2006). 277 (2010). [13] J. Bae and A. Ac´ın, Phys. Rev. Lett. 97, 030402 (2006). [5] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. [14] G. Chiribella and G. M. D’Ariano, Phys. Rev. Lett. 97, 250503 Phys. 80, 517 (2008). (2006). [6] J. P. Dowling and G. J. Milburn, Phil. Trans. Roy. Soc. A 361, [15] F. G. Brandao,˜ M. Piani, and P. Horodecki, Nature Commun. 6, 1655 (2003). 7908 (2015). [7] B. Terhal, IBM J. Res. Dev. 48, 71 (2004). [16] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [8] D. Cavalcanti, F. G. S. L. Brandao,˜ and M. O. Terra Cunha, [17] I. Devetak and A. Winter, Proc. Roy. Soc. A 461, 207 (2005). Phys. Rev. A 72, 040303 (2005). [18] M. Pawłowski, Phys. Rev. A 82, 032313 (2010). [9] M. Fannes, J. T. Lewis, and A. Verbeure, Lett. Math. Phys. 15, [19] U. Vazirani and T. Vidick, Phys. Rev. Lett. 113, 140501 (2014). 5

[20] T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2014). (2006). [46] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. [21] A. Ferraro, A. Garc´ıa-Saez, and A. Ac´ın, Phys. Rev. A 76, Rev. Lett. 78, 2275 (1997). 052321 (2007). [47] P. Hayden, D. W. Leung, and A. Winter, Commun. Math. Phys. [22] S. M. Giampaolo, G. Gualdi, A. Monras, and F. Illuminati, 265, 95 (2006). Phys. Rev. Lett. 107, 260602 (2011). [48] G. Vidal, W. Dur,¨ and J. I. Cirac, Phys. Rev. Lett. 89, 027901 [23] S. M. Giampaolo, B. C. Hiesmayr, and F. Illuminati, Phys. Rev. (2002). B 92, 144406 (2015). [49] F. G. S. L. Brandao˜ and M. B. Plenio, Nature Phys. 4, 873 [24] K. Meichanetzidis, J. Eisert, M. Cirio, V. Lahtinen, and J. K. (2008). Pachos, Phys. Rev. Lett. 116, 130501 (2016). [50] M. Piani, Phys. Rev. Lett. 103, 160504 (2009). [25] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, [51] M. Christandl, N. Schuch, and A. Winter, Commun. Math. 052306 (2000). Phys. 311, 397 (2012). [26] We will denote bipartite entanglement either by EA:B(ρAB) or [52] S. Aaronson, S. Beigi, A. Drucker, B. Fefferman, and P. Shor, E(ρA:B) throughout the manuscript, depending on convenience. in CCC’08. 23rd Annual IEEE Conference on Computational [27] K. A. Dennison and W. K. Wootters, Phys. Rev. A 65, 010301 Complexity (IEEE, 2008), p. 223. (2001). [53] M. B. Hastings, Nature Phys. 5, 255 (2009). [28] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004). [54] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 [29] G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006). (2001). [30] G. Adesso and F. Illuminati, Phys. Rev. Lett. 99, 150501 (2007). [55] See Supplemental Material at [EPAPS] for technical proofs. [31] G. Adesso, D. Girolami, and A. Serafini, Phys. Rev. Lett. 109, [56] F. G. S. L. Brandao,˜ M. Christandl, and J. T. Yard, Commun. 190502 (2012). Math. Phys. 306, 805 (2011). [32] J. S. Kim, A. Das, and B. C. Sanders, Phys. Rev. A 79, 012329 [57] W. Matthews, S. Wehner, and A. Winter, Commun. Math. Phys. (2009). 291, 813 (2009). [33] J. S. Kim and B. C. Sanders, J. Phys. A: Math. Theor. 43, [58] A. Winter, Communications in Mathematical Physics p. 1 445305 (2010). (2016), ISSN 1432-0916. [34] F. F. Fanchini, M. C. de Oliveira, L. K. Castelano, and M. F. [59] A. Coleman and V. Yukalov, in Coulson’s Challenge (Springer, Cornelio, Phys. Rev. A 87, 032317 (2013). Berlin, 2000), pp. XII, 282. [35] T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini, Phys. Rev. [60] E. H. Lieb and M. B. Ruskai, J. Math. Phys. 14, 1938 (1973). A 89, 034303 (2014). [61] N. Linden and A. Winter, Commun. Math. Phys. 259, 129 [36] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang, Phys. Rev. Lett. 113, (2005), ISSN 1432-0916. 100503 (2014). [62] J. Cadney, M. Huber, N. Linden, and A. Winter, Linear Algebra [37] Y. Luo and Y. Li, Ann. Phys. 362, 511 (2015). and its Applications 452, 153 (2014), ISSN 0024-3795. [38] M. F. Cornelio, Phys. Rev. A 87, 032330 (2013). [63] C. Eltschka and J. Siewert, Phys. Rev. Lett. 114, 140402 (2015). [39] B. Regula, S. Di Martino, S. Lee, and G. Adesso, Phys. Rev. [64] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, JHEP 2013, Lett. 113, 110501 (2014). 1 (2013). [40] M. Christandl and A. Winter, J. Math. Phys. 45, 829 (2004). [65] H. Sommers and K. Zyczkowski,˙ J. Phys. A. 34, 7111 (2001). [41] M. Horodecki and M. Piani, J. Phys. A.: Math. Theor. 45, [66] P. Levy,´ Problèmesconcrets d’analyse fonctionnelle (Gauthier- 105306 (2012). Villars, Paris, 1951). [42] A. Streltsov, G. Adesso, M. Piani, and D. Bruß, Phys. Rev. Lett. [67] V. Milman, Funct. Annal. Appl. 5, 28 (1971). 109, 050503 (2012). [68] G. Aubrun, S. Szarek, and E. Werner, Commun. Math. Phys. [43] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot- 305, 85 (2011). ters, Phys. Rev. A 54, 3824 (1996). [69] G. Pisier, The Volume of Convex Bodies and Banach Spaces Ge- [44] M. B. Plenio and S. Virmani, Quantum Info. Comput. 7, 1 ometry, Cambridge Tracts in Mathematics, vol. 94 (Cambridge (2007). University Press, Cambridge, 1989). [45] K. Salini, R. Prabhu, A. S. De, and U. Sen, Ann. Phys. 348, 297 1

Supplemental Material

Should Entanglement Measures be Monogamous or Faithful?

APPENDIX A: ENTANGLEMENT OF FORMATION AND RELATIVE ENTROPY OF ENTANGLEMENT OF RANDOM STATES

Let us first fix a few definitions and notations. For any n ∈ N and 1 6 p 6 ∞, we denote by k · kp the Schatten p-norm on the n set of Hermitian operators on C . Particular instances of interest are the trace-class norm k · k1, the Hilbert–Schmidt norm k · k2 and the operator norm k · k∞. n s Given n, s ∈ N, we say that a random mixed state ρ on C is induced by C , which we denote by ρ ∼ µn,s, if ρ = TrCs |ψihψ| for |ψi a uniformly distributed random pure state on the global “system + ancilla” space Cn ⊗ Cs. It is known that in the case s 6 n, ρ ∼ µn,s is equivalent to ρ being uniformly distributed (for the measure induced by the Hilbert-Schmidt distance) on the set of mixed states of rank at most s on Cn (see e.g. [65]). In the sequel, we shall use the following shorthand notation: for any n s n pure state ψ on C ⊗ C , ρψ stands for the corresponding marginal state on C , i.e. ρψ = TrCs |ψihψ|. In the following we will always work on the bipartite Hilbert space Cd ⊗ Cd, where we denote by D(Cd ⊗ Cd) the set of all states and by S(Cd : Cd) the set of states which are separable across the cut Cd : Cd. For any ρ ∈ D(Cd ⊗ Cd), its entanglement of formation is defined as   X X X  E (ρ) = inf  p S (ρ ): ρ = p |ϕ ihϕ |, p > 0, p = 1, |ϕ i ∈ Cd ⊗ Cd , F  i ϕi i i i i i i   i i i  and its relative entropy of entanglement as

n d d o n d d o ER(ρ) = inf S (ρkσ): σ ∈ S(C : C ) = inf Tr ρ log ρ − log σ : σ ∈ S(C : C ) .

We are interested in determining, given d, s ∈ N, what is the typical value of EF (ρ) and ER(ρ) for ρ ∼ µd2,s. The answer is stated in Theorems 1 and 2 below. They both rely on technical lemmas appearing in Section A 1. Theorem 1 is then obtained as a direct corollary of Proposition 8. While Theorem 2 follows from the upper bound of Proposition 10 and the lower bound of Proposition 13. Let us also emphasize that the statement about the typical value of EF , appearing in Theorem 1, was already more or less established in [47, Thm. V.1], but only the lower bound appears there and with a not as tight dependence in the parameters, which is why we brieﬂy repeat the whole argument here. Oppositely, the statement about the typical value of ER, appearing in Theorem 2, is new.

Theorem 1. Fix t > 0. Let ρ be a random state on Cd ⊗ Cd induced by some environment Cs, with s 6 Cd2t2/ log2 d for some universal constant C > 0. Then, " # ! 1 −cd2t2/ log2 d P EF (ρ) − log d − 6 t > 1 − e , (A1) 2 ln 2 where c > 0 is a universal constant.

Theorem 2. Fix 0 < t < 1. Let ρ be a random state on Cd ⊗ Cd induced by some environment Cs, with Cd log(1/t)/t2 6 s 6 d2 for some universal constant C > 0. Then, " # ! 1 s −cst2 P ER(ρ) − 2 log d − log s + 6 t > 1 − e , (A2) 2 ln 2 d2 where c > 0 is a universal constant.

1. A few technical lemmas

The two crucial tools that we will use in order to obtain Theorems 1 and 2 are Levy’s Lemma and Dvoretzky’s Theorem (the former being at the heart of the derivation of the latter). These guarantee that, in high dimension, regular enough functions typically do not deviate much from their average behaviour. The precise version of this general paradigm that we will need are quoted in the following Lemmas 3 and 4. 2

Lemma 3 (Levy’s Lemma for Lipschitz functions on the sphere [66]). Let n ∈ N. For any L-Lipschitz function f : S Cn → R and any t > 0, if ψ is uniformly distributed on S Cn , then

2 2 P(| f (ψ) − E f | > t) 6 e−cnt /L , where c > 0 is a universal constant.

Lemma 4 (Dvoretzky’s Theorem for Lipschitz functions on the sphere [67, 68]). Let n ∈ N. For any circled L-Lipschitz function 2 2 n f : S Cn → R and any t > 0, if H is a uniformly distributed Cnt /L -dimensional subspace of C , with C > 0 a universal constant, then

−cnt2/L2 P (∃ ψ ∈ H ∩ S Cn : | f (ψ) − E f | > t) 6 e , where c > 0 is a universal constant.

For the sake of completeness, we now re-derive (more or less well-known) concentration results for two functions that will pop up later on while establishing Theorems 1 and 2.

Lemma 5. Fix n, s ∈ N with s 6 n and t > 0. If ψ is uniformly distributed on S Cn⊗Cs , then ! 1 s −cnst2/ log2 n P S (ρψ) − log s + > t 6 e , 2 ln 2 n where c > 0 is a universal constant. Proof. Deﬁne the function f : ψ ∈ S Cn⊗Cs 7→ S ρψ ∈ R. We know from [47, Lemma III.2], that f is 3 log n-Lipschitz. Besides, we also know from [47, Lemma II.4], that f has average (w.r.t. the uniform probability measure over S Cn⊗Cs ) 1 s s E f (ψ) = log s − + o . ψ 2 ln 2 n n Having at hand these two estimates, Lemma 5 is a direct consequence of Levy’s Lemma 3.

Lemma 6. Fix d ∈ N and t > 0. If H is a uniformly distributed Cd2t2/ log2 d-dimensional subspace of Cd ⊗ Cd, with C > 0 a universal constant, then ! 1 −cd2t2/ log2 d P ∃ ψ ∈ H ∩ S Cd⊗Cd : S (ρψ) − log d + > t 6 e . 2 ln 2 Proof. Deﬁne the function f : ψ ∈ S Cd⊗Cd 7→ S ρψ ∈ R. We know from [47, Lemma III.2], that f is 3 log d-Lipschitz. Besides, we also know from [47, Lemma II.4], that f has average (w.r.t. the uniform probability measure over S Cd⊗Cd ) 1 E f (ψ) = log d − + o (1) . ψ 2 ln 2 Having at hand these two estimates, Lemma 6 is a direct consequence of Dvoretzky’s Theorem 4.

n s Lemma 7. Fix n, s ∈ N with s 6 n, as well as x ∈ S Cn , and let ρ be a random state on C induced by an environment C . Then, ! p 1 t −cst2 ∀ t > 0, P hx|ρ|xi − √ > √ 6 e , n n where c > 0 is a universal constant. p Proof. Deﬁne the function g : ψ ∈ S Cn⊗Cs 7→ hx|ρψ|xi. We know from [68, App. B], that g is 1-Lipschitz. Besides, we also know from [68], Appendix B, that g has average (w.r.t. the uniform probability measure over S Cn⊗Cs ) ! 1 1 Eψg(ψ) = √ + o √ . n n

Having at hand these two estimates, the conclusion of Lemma 7 is a direct consequence of Levy’s Lemma 3. 3

2. Typical value of the entanglement of formation of random induced states

Proposition 8. Fix t > 0. Let ρ be a random state on Cd ⊗ Cd, induced by some environment Cs, with s 6 Cd2t2 log2 d for some universal constant C > 0. Then, ! 1 −cd2t2 log2 d P ∃ ϕ ∈ supp(ρ) ∩ S Cd⊗Cd : S (ρϕ) − log d + > t 6 e , ln 2 where c > 0 is a universal constant. Proof. One simply has to observe that supp(ρ) is a uniformly distributed s-dimensional subspace of Cd ⊗ Cd, and apply Lemma 6.

3. Upper and lower bounds on the typical relative entropy of entanglement of random induced states

Observe the following alternative form of the relative entropy of entanglement: For a state ρ on Cd ⊗ Cd,

n d d o ER(ρ) = min − Tr(ρ log σ): σ ∈ S(C : C ) − S (ρ). (A3)

Lemma 9. For any state ρ on Cd ⊗ Cd, we have

ER(ρ) 6 2 log d − S (ρ). Proof. It is clear from equation (A3) that we have, in particular, ! Id E (ρ) 6 − Tr ρ log − S (ρ) = 2 log d − S (ρ). R d2 This concludes the proof of Lemma 9. Proposition 10. Fix 0 < t < 1. Let ρ be a random state on Cd ⊗ Cd induced by some environment Cs, with s 6 d2. Then, ! 1 s 2 2 2 P E (ρ) > 2 log d − log s + + t 6 e−cd st / log d, R 2 ln 2 d2 where c > 0 is a universal constant. Remark 11. Note that the bound appearing in Proposition 10 is non-trivial only in the regime d 6 s 6 d2. Indeed, it also holds d d for any state ρ on C ⊗ C that ER(ρ) 6 EF (ρ) 6 log d, and if s < d, we do not learn anything better than that from Proposition 10 for a random state ρ on Cd ⊗ Cd induced by some environment Cs. Hence in words, Proposition 10 actually tells us the 2 2 following: if ρ ∼ µd2,s with d s 6 d , then ER(ρ) is w.h.p. smaller than 2 log d − log s + O(s/d ), as d, s → ∞. Proof. From Lemma 9, it is clear that, for any 0 < t < 1, ! ! 1 s 1 s P E (ρ) > 2 log d − log s + + t 6 P S (ρ) < log s − − t . (A4) R 2 ln 2 d2 2 ln 2 d2

2 Now, we know from Lemma 5 that the probability on the r.h.s. of inequality (A4) is smaller than e−cd2 st2/ log d, which is precisely the announced result. Lemma 12. For any state ρ on Cd ⊗ Cd, we have ER(ρ) > − log max hx ⊗ y|ρ|x ⊗ yi : x, y ∈ S Cd − S (ρ). Proof. We see from equation (A3) that the only thing we have to prove is that

n d d o min − Tr(ρ log σ): σ ∈ S(C : C ) > − log max hx ⊗ y|ρ|x ⊗ yi : x, y ∈ S Cd . (A5)

d d Now, for any σ ∈ S(C : C ), we see by concavity of log that − Tr(ρ log σ) > − log Tr(ρσ). Indeed, denoting by {λ1, . . . , λd2 } the eigenvalues and by {e1,..., ed2 } an eigenbasis of σ, we have

d2  d2  X X  Tr(ρ log σ) = he |ρ|e i log λ 6 log  he |ρ|e iλ  = log Tr(ρσ). i i i  i i i i=1 i=1 4

As a consequence, n o n o n o min − Tr(ρ log σ): σ ∈ S(Cd : Cd) > min − log Tr(ρσ): σ ∈ S(Cd : Cd) = − log max Tr(ρσ): σ ∈ S(Cd : Cd) . It then follows from extremality of pure separable states amongst all separable states that

n d d o max Tr(ρσ): σ ∈ S(C : C ) = max hx ⊗ y|ρ|x ⊗ yi : x, y ∈ S Cd , so that equation (A5) indeed holds. Proposition 13. Fix 0 < t < 1. Let ρ be a random state on Cd ⊗Cd induced by some environment Cs, with Cd log(1/t)/t2 6 s 6 d2 for some universal constant C > 0. Then, ! 1 s 2 P E (ρ) < 2 log d − log s + − t 6 e−cst , R 2 ln 2 d2 where c > 0 is a universal constant. 2 Remark 14. Note that Proposition 13 actually tells us the following: if ρ ∼ µd2,s with d s d , then ER(ρ) is w.h.p. bigger than 2 log d − log s − o(1), as d, s → +∞. And in the case of sufficiently mixed states, the refinement: if ρ ∼ µd2,s with 4/3 2 2 d s 6 d , then ER(ρ) is w.h.p. bigger than 2 log d−log s−O(s/d ), as d, s → ∞. That is, in that regime, the fluctuations from 2 2 log d − log s are of the same order ±O(s/d ), cf. Remark 11. We conjecture that for 1 6 s 6 O(d), w.h.p. ER(ρ) > log d − O(1).

Proof. Set M(ρ) = max {hx ⊗ y|ρ|x ⊗ yi : x, y ∈ S Cd }. From Lemma 12, it is clear that, for any 0 < t < 1, ! ! 1 s t 1 s t P E (ρ) < 2 log d − log s + − t 6 P − log M(ρ) < 2 log d − + P S (ρ) > log s − + . (A6) R 2 ln 2 d2 2 2 ln 2 d2 2

2 00 00 Indeed, deﬁning the events A = “ ER(ρ) < 2 log d − log s + 1/(2 ln 2)s/d + t and B1 = “ − log M(ρ) < 2 log d − t/2 , 2 00 B2 = “ S (ρ) > log s − 1/(2 ln 2)s/d + t/2 , we have ¬B1 ∩ ¬B2 ⇒ ¬A, i.e. equivalently A ⇒ B1 ∪ B2. Hence,

P(A) 6 P(B1 ∪ B2) 6 P(B1) + P(B2). Now on the one hand, we know from Lemma 5 that ! 1 s t 2 2 2 P S (ρ) > log s − + 6 e−cd st / log d. (A7) 2 ln 2 d2 2 And on the other hand, one may observe that ! ! t p et/4 p 1 + t/4 P − log M(ρ) < 2 log d − = P M(ρ) > 6 P M(ρ) > . 2 d d

So ﬁx 0 < δ < 1/8 and consider Nδ a δ-net for k · k within S Cd . By a standard volumetric argument (see e.g. [69, Ch. 4]) we 2d know that we can impose |Nδ| 6 (3/δ) . Set next Mδ(ρ) = max {hx ⊗ y|ρ|x ⊗ yi : x, y ∈ Nδ}. Then, let x, y ∈ S Cd andx ¯, y¯ ∈ Nδ be such that u = x − x¯, v = y − y¯ satisfy kuk, kvk 6 δ, and observe that hx ⊗ y|ρ|x ⊗ yi = hx¯ ⊗ y¯|ρ|x¯ ⊗ y¯i + hx¯ ⊗ y¯|ρ|x¯ ⊗ vi + hx¯ ⊗ v|ρ|x¯ ⊗ yi + hx¯ ⊗ y|ρ|u ⊗ yi + hu ⊗ y|ρ|x ⊗ yi.

Hence, hx ⊗ y|ρ|x ⊗ yi 6 Mδ(ρ) + 4δM(ρ), so that taking supremum over x, y ∈ S Cd yields Mδ(ρ) > (1 − 4δ)M(ρ). Yet, we know from Lemma 7 that, for ﬁxed x, y ∈ S Cd , ! p 1 + t/8 0 2 ∀ t > 0, P hx ⊗ y|ρ|x ⊗ yi > 6 e−c st . d We therefore get by the union bound that ! !4d p 1 + t/8 3 0 2 P M (ρ) > 6 e−c st . δ d δ √ Consequently, we eventually obtain, choosing δ = t/10, in order to have (1 + t/4) 1 − 4δ > 1 + t/4 − 5δ/4 = 1 + t/8, that ! ! !4d p 1 + t/4 p 1 + t/8 30 0 2 P M(ρ) > 6 P M (ρ) > 6 e−c st . (A8) d t/10 d t

00 2 And whenever s > Cd log(1/t)/t2, the r.h.s. of inequality (A8) above is smaller than e−c st . So combining the deviation probabilities (A7) and (A8) with inequality (A6), we get precisely the announced result, just 00 2 2 2 2 2 observing that e−c st + e−cd st / log d 6 e−c0 st . 5

APPENDIX B: DIMENSION-DEPENDENT MONOGAMY RELATIONS FOR THE ENTANGLEMENT OF FORMATION AND THE REGULARIZED RELATIVE ENTROPY OF ENTANGLEMENT

0 Theorem 15. For any state ρABC on HA⊗HB⊗HC, and any η, η > 0, we have, setting dA,B = min(dA, dB) and dA,C = min(dA, dC), ! c c E (ρ ) E (ρ ) E (ρ ) 4+η, E (ρ ) E (ρ ) 4+η , F A:BC > max F A:B + 4+η F A:C F A:C + 4+η F A:B dAdC log dA,C dAdB log dA,B

0 0 ! c 0 c 0 E∞ (ρ ) E∞ (ρ ) E∞ (ρ ) 2+η , E∞ (ρ ) E∞ (ρ ) 2+η , R A:BC > max R A:B + 2+η0 R A:C R A:C + 2+η0 R A:B dAdC log dA,C dAdB log dA,B where c, c0 > 0 are constants depending on η, η0. Remark 16. For concreteness, in the main text we choose η = 4 and η0 = 2. To prove Theorem 15, we will need as a starting point the two monogamy-like relations appearing in Lemma 17 below. In the ← ∞ latter, we denote by ER,LOCC the relative entropy of entanglement ﬁltered by one-way LOCC measurements, and by ER,LOCC← its regularized version. These were introduced and studied in [50], to which the reader is referred for more details.

Lemma 17. For any state ρABC on HA ⊗ HB ⊗ HC, we have

∞ ∞ ∞ ← EF (ρA:BC) > EF (ρA:B) + ER,LOCC (ρA:C) and ER (ρA:BC) > ER (ρA:B) + ER,LOCC← (ρA:C) . (B1) Proof. The second inequality in (B1) was proved in [56, Lemma 2]. So let us turn to proving the ﬁrst inequality in (B1). i P i P i For this, assume that {pi, ψABC}i is such that ρABC = i piψABC and EF (ρA:BC) = i piEF ψA:BC . Now, for a pure state ψABC, denoting by ρAB and ρAC its reduced states on HA ⊗ HB and HA ⊗ HC respectively, we know from [28, Cor. 2], that EF (ψA:BC) = EF (ρA:B) + S LOCC← (ρA:CkρA ⊗ ρC). Hence,   X h i i i i i  X i i  E (ρ ) p E ρ S ← ρ kρ ⊗ ρ E (ρ ) S ← ρ p ρ ⊗ ρ  , F A:BC = i F A:B + LOCC A:C A C > F A:B + LOCC  A:C i A C i i where the inequality follows from the convexity of EF and the joint convexity of S LOCC← , combined with the observation that P i P i P i i piρ = ρAB and piρ = ρAC. Now, the state piρ ⊗ ρ is obviously separable across the cut HA : HC, and therefore i AB i AC i A C ← P i i ← S LOCC ρA:C i piρA ⊗ ρC > ER,LOCC (ρA:C), which yields precisely the advertised result.

Proof of Theorem 15. Since we can exchange the roles played by HB and HC, we just have to show that c E (ρ ) E (ρ ) E (ρ ) 4+η, F A:BC > F A:B + 4+η F A:C (B2) dAdC log min(dA, dC)

0 c 0 E∞ (ρ ) E∞ (ρ ) E∞ (ρ ) 2+η . R A:BC > R A:B + 2+η0 R A:C (B3) dAdC log min(dA, dC)

By Pinsker inequality and the Pinsker-type inequality established in [56, Lemma 3], we know that, for any state ρAC,

1 2 ∞ 1 2 E ← (ρ ) > kρ − S(H : H )k ← and E ← (ρ ) > kρ − S(H : H )k ← . R,LOCC A:C 2 ln 2 AC A C LOCC R,LOCC A:C 8 ln 2 AC A C LOCC

Now, we also know from [57] that, for any states ρAC and σAC,

c0 kρAC − σACkLOCC← > √ kρAC − σACk1. dAdC 0 And ﬁnally, it was shown in [58], Corollaries 4 and 7, that for any states ρAC and σAC, and any η0, η0 > 0

1/2−η0 |EF (ρAC) − EF (σAC)| 6 C log min(dA, dC) kρAC − σACk1 ,

0 ∞ ∞ 0 1−η0 |ER (ρAC) − ER (σAC)| 6 C log min(dA, dC) kρAC − σACk1 . 6

So putting everything together, we get that, for any state ρAC,

c 4+η E ← (ρ ) E (ρ ) , R,LOCC A:C > 4+η F A:C dAdC log min(dA, dC)

0 ∞ c ∞ 2+η0 E ← (ρ ) E (ρ ) . R,LOCC A:C > 2+η0 R A:C dAdC log min(dA, dC)

And combining these two lower-bounds with Lemma 17 yields, as wanted, the two inequalities (B2) and (B3).