Optimized Correlation Measures in Holography

PHYSICAL REVIEW D 101, 066009 (2020)

Optimized correlation measures in holography

Newton Cheng* Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720, USA

(Received 15 October 2019; accepted 21 February 2020; published 12 March 2020)

We consider a class of correlation measures for quantum states called optimized correlation measures defined as a minimization of a linear combination of von Neumann entropies over purifications of a given E E state. Examples include the entanglement of purification P and squashed entanglement sq. We show that when evaluating such measures on “nice" holographic states in the large-N limit, the optimal purification has a semiclassical geometric dual. We then apply this result to confirm several holographic dual proposals, including the n-party squashed entanglement. Moreover, our result suggests two new techniques for determining holographic duals: holographic entropy inequalities and direct optimization of the dual geometry.

DOI: 10.1103/PhysRevD.101.066009

I. INTRODUCTION We will consider a class of correlation measures called One of the most important tools in AdS=CFT is the Ryu- optimized correlation measures, whose bipartite forms n Takayanagi/Huberny-Rangamani-Takayanagi (RT/HRT) pre- were extensively studied in [3]. Given an arbitrary -party ρ scription [1,2] for computing the von Neumann entropy of state A1…An , these measures are defined to be a minimi- fα a boundary region A in the state ρAB: zation of a function : A M Eα A ∶…∶A fα ψ ψ ; S ρ ½ ; ð 1 nÞ¼ inf ðj ih jÞ ð2Þ Að ABÞ¼ ð1Þ jψiA …A A0 …A0 4GN 1 n 1 n where A½M is the area of a homologous extremal surface where the minimization is taken over all purifications A anchored to the boundary of in the dual spacetime. ψ 0 0 of ρA …A ,sothattrA0 …A0 ψ ψ ρA …A . j iA1…AnA1…An 1 n 1 n j ih j¼ 1 n Equation (1) provides an elegant interpretation of the bulk- 2n Denoting the set of all 2 − 1 nonempty combinations of boundary correspondence: An information-theoretic quan- α these regions by R, the function f is a linear combination of tity about the boundary state can be computed as the area of von Neumann entropies on the extended Hilbert space: a “dual” geometric object in the bulk, which is often X simpler to do than working directly in the boundary theory. α f ðjψihψjÞ ¼ αJ SJ ðjψihψjÞ; ð3Þ If ρAB is pure, then SA captures the entire entanglement J ∈R structure of the state. However, the von Neumann entropy is not the only correlation measure for a generic quantum 2n where α ∈ R2 −1 and the sum runs over all elements J of R. state; indeed, it is well known that the von Neumann As in [3], we will impose a non-negativity condition entropy is a poor measure of entanglement once one P αJ ≥ 0, or else choosing part of the extension to be considers mixed or n-party states with n>2. Despite this, J with a few exceptions, correlation measures beyond maximally mixed on an enormous Hilbert space gives Eα → −∞ entropy have received little attention in a gravitational . An example of such a measure that has recently context. We therefore aim to study holographic duals for garnered significant interest is the entanglement of purifi- measures better suited to studying the entanglement struc- cation [4]: ture of holographic mixed states, which are ubiquitous E A∶B S ; when studying, e.g., black holes. Pð Þ¼ min AA0 ð4Þ jψiAA0BB0

* [email protected] for which αAA0 ¼ 1 and all other αJ ¼ 0. For holographic states, EP is conjectured to be dual to a particular minimal Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. surface in the bulk called the entanglement wedge cross Further distribution of this work must maintain attribution to section EW [5–8].TheEP ¼ EW conjecture, as well as the author(s) and the published article’s title, journal citation, related proposals for the dual to EW, has since attracted and DOI. Funded by SCOAP3. significant attention [9–29]. Other examples include the R

2470-0010=2020=101(6)=066009(11) 066009-1 Published by the American Physical Society NEWTON CHENG PHYS. REV. D 101, 066009 (2020)

Q n and correlation introduced in [3] and the squashed To make notation cleaner, we will use U ¼ ⋃i 1Ai to E ¼ entanglement sq [30], which we discuss in the main body refer to the union of all n parties A1; …;An of an n-party U0 ⋃n A0 of the paper. state, and use ¼ i¼1 i to refer to the purifying ρ ρ A generic correlation measure of the form (2) is gen- subsystem, e.g., A1…An ¼ U. We will also draw a dis- erally very difficult to compute due to the enormous tinction between entanglement measures and total corre- parameter space of purifications that needs to be searched. lation measures; entanglement measures are intended to As a result, finding the holographic duals to such measures capture purely quantum correlations, while the more can be similarly difficult, relying on indirect evidence such general total correlation measures are allowed to capture as consistency checks with inequalities, or requiring strong a combination of quantum and classical correlations. assumptions like the surface-state correspondence [31]. Entanglement measures are usually required to satisfy Holographic duals to other optimized correlation measures more stringent properties that are not satisfied by a generic have been proposed [32–34], but their conclusions rely on Eα, as we discuss below. Finally, although we use the term the assumption that the optimal purification in (2) can be “minimal surface,” our arguments do not rely on time taken to be geometric, in the sense that the purification lives symmetry, and we therefore expect no issues to arise in in the boundary CFT Hilbert space and is dual to a considering dynamical setups with extremal surfaces. semiclassical geometry. The question of the assumption’s validity was first posed in [32] in the context of the II. PRELIMINARIES E squashed entanglement sq and has since remained an open problem in holography. Some recent progress toward A. Correlation measures an answer appeared in [17], where it was argued that We first briefly introduce the general, axiomatic geometric minimization was sufficient to compute the approach to correlation measures in quantum information. bipartite EP, and hence, one could employ the bit thread We are given a functional E defined on the set of density formulation of holography [35] on the purifying state to matrices DðHÞ of a Hilbert space H, and in order for E to provide evidence for EP ¼ EW. be a good correlation measure, we require that E satisfy In this work, we argue that, up to reasonable assumptions some given set of axioms that are believed to be reasonable. we will outline below, something even stronger holds: For Arguably the most important axiom for an entanglement any n-party optimized correlation measure evaluated on measure is monotonicity under local operations and “nice” holographic states, minimizing over the subset of classical communication (LOCC): geometric purifications is sufficient to achieve the minimum over all purifications. Using our result, we confirm EðρÞ ≥ EðΛðρÞÞ; ð5Þ the holographic proposals in [32–34], and in particular, we find a dual to the n-party squashed entanglement. for any LOCC operation Λ, as entanglement never Moreover, our result suggests that holographic duals can increases under LOCC. Other axioms that are usually be determined by directly optimizing the purified geometry, enforced for entanglement measures include that they as opposed to relying on indirect evidence such as obeying should vanish on separable states, be continuous in the n → ∞ the same set inequalities. asymptotic regime corresponding to copies of a E Φd Throughout this paper, we will work at leading order in given state, and be normalized in the sense that ð þÞ¼ d Φd d the large-N, large central charge limit. We will also assume log for the maximally mixed state þ of size . One can that a suitable regularization scheme is in place, e.g., a show that any such E that satisfies these axioms and is radial cutoff, such that all the correlation measures of monotonic under LOCC reduces to the von Neumann interest are finite and Hilbert spaces can be tensor factor- entropy on pure states, singling it out as the unique ized. We will moreover assume that the purifying sub- (bipartite) entanglement measure for pure states [36]. system is finite dimensional and, crucially, that the infimum The converse is not true: A correlation measure E that in (2) can turn into a minimum; this is an open question for reduces to the von Neumann entropy on pure states is not generic quantum states, but we argue below that this is not a generically a good entanglement measure. However, major obstacle for holographic states. such measures are still of interest if they are monotonic The paper is organized as follows: In Sec. II, we review under local operations (LO)—these are total correlation the necessary assumptions and properties of holographic measures, which may be sensitive to both quantum and states to prove our claim. In Sec. III, we apply the classical correlations in a given state, rather than just properties of holographic states to prove the sufficiency quantum correlations from entanglement. The dependence of minimizing over states with semiclassical duals. In on classical correlations allows these E to increase under Sec. IV, we discuss applications of the result to holographic classical communication (CC). The most well-known exam- systems and determining duals to various correlation ple of such a measure is the bipartite mutual information: measures. Finally, in Sec. V, we discuss potential avenues I A∶B S S − S ; of future work for understanding holographic states. ð Þ¼ A þ B AB ð6Þ

066009-2 OPTIMIZED CORRELATION MEASURES IN HOLOGRAPHY PHYS. REV. D 101, 066009 (2020) which captures all correlations, quantum and classical, optimized correlation measure and already suffers from this between A and B [37], and reduces to SA for a pure state. issue. The geometric dual generally makes these measures In other words, SA is really the unique correlation measure easier to compute or understand, which can lead to insight for bipartite pure states in general. EP is also an example of a into the fine-grained structure of holographic states. total correlation measure, as it is monotonic under LO (but Conversely, holographic systems are a useful testing not CC) and trivially reduces to SA on pure states. ground for quantum information, precisely because we While the story is very clean for bipartite, pure states, the are able to understand quantities that are usually extremely situation is unclear for any other kind of state. Consider, for complicated for generic quantum states. instance, a generic mixed state ρAB. Then, we will generally find B. Linearity and geometric purifications

SAðρABÞ ≠ SBðρABÞ;SABðρABÞ ≠ 0; ð7Þ To make our argument, we will need to employ two crucial properties of holographic states. The first property so SA is not only capturing the entanglement between A and we will use is that entropies in holography are linear at B, but it can be seen as receiving contributions from the leading order in the 1=N expansion [39]. MoreP precisely, M i entanglement between A and a purifying subsystem. The given a superposition of holographic states ρAB ¼ i piρAB search for good entanglement measures for mixed states with exponentially suppressed overlap (as is the case for remains an active subfield of quantum information; see, states with macroscopically distinct dual geometries), the e.g., [38] for a discussion on various, generally inequiva- entropy of the A subregion is approximately lent, measures. The inequivalence corresponds to differing M operational interpretations. Moreover, SA is a bipartite X S ρ p S ρi S ; correlation measure and cannot capture intrinsically n- Að ABÞ¼ i Að ABÞþ mix þ ð10Þ party entanglement for n>2. The simplest example is the i P 3-qubit Greenberger-Horne-Zeilinger (GHZ) state: S − M p p where mix ¼ i i log i is the entropy of mixing, and 1 the ellipses indicate terms of order Oð1=N2Þ and smaller. jGHZ3iABC ¼ pffiffiffi ðj000iþj111iÞ; ð8Þ 2 The assumptions made in computing (10) break down when M ∼ eOðcÞ, with the most apparent effect being that S with the party labels corresponding to each qubit. The mix can become leading order. reduced density matrix of any two qubits is Now let jψiABA0B0 be a purification of ρAB defined on the entire boundary. The second property we will use is that we 1 can expand jψi as a superposition of states fjϕiig dual to trAjGHZ3ihGHZ3jABC ¼ ðj00ih00jþj11ih11jÞ; ð9Þ 2 semiclassical geometries which is a separable state, and hence, contains no entan- XM glement between qubits BC. The entropy SA ¼ log 2 is jψi¼ cijϕii; ð11Þ only able to capture the bipartite correlations between A i and BC, rather than the intrinsic tripartite correlations. In holography, we are interested in the theory of such that fjϕiig are themselves purifications of ρAB [17]: correlation measures because mixed states and multipartite states are ubiquitous, e.g., a black hole with inverse trA0B0 jϕiihϕijABA0B0 ¼ ρAB ð12Þ temperature β or multiple boundary subregions, and the issues with the von Neumann entropy mean studying the for all jϕii. Because our result relies so crucially on this entanglement structure of such states is difficult. However, result, we reproduce the argument from [17] for the just as the von Neumann entropy has a geometric dual, we existence of such a decomposition here with added detail expect that other information-theoretic quantities defined and attention paid to the necessary assumptions. on the boundary state should have a geometric dual. Given a holographic state ρAB, let jψiABA0B0 be any Understanding the dual objects can be extremely useful, purification on the boundary with a geometric dual. We are because there are situations where the geometric descrip- free to act on the purifying subsystem HA0B0 with any tion is easier to work with than the CFT description. operation that preserves the purity of the overall state; this A simple example is computing the entropy of a single is the standard statement from quantum mechanics that all interval in the vacuum state of a holographic 2D CFT: The purifications of a given state in the same Hilbert space are CFT calculation, while not particularly complicated, is related by a unitary acting on HA0B0 . In particular, we are more work than computing the area of a geodesic in AdS3. free to insert, e.g., pairs of black holes using the gluing α Indeed, E is very difficult to compute for an arbitrary procedure of [40]. Let fjϕiig be the set of purifications quantum state; EP is one of the simplest examples of an generated by this procedure—these states all live in the

066009-3 NEWTON CHENG PHYS. REV. D 101, 066009 (2020) holographic CFT Hilbert space and will generally be a holographic state, it is sufficient to minimize only over proper subset of the set of all geometric purifications. purifications that are geometric, so that the purifying We now make a reasonable assumption: We can fill the subsystem corresponds to a spatial region on the boundary. purifying subsystem with black hole microstates. That is, Suppose we are interested in computing some n-party because we are free to tune the parameters of the black optimized correlation measure Eα for an arbitrary n-party ρ ρ H ⊗H holes in the purifying subregion to fill very small or very CFT state U ¼ A1…An on a Hilbert space ¼ n that large regions, such states should have support over any is a subsystem of the full boundary Hilbert space H . Let H CFT complete basis of A0B0 . Therefore, given an arbitrary basis jψiUU0 be a purification of ρU, and moreover, let us choose β H j ii of A0B0 , we should be able to write jψiUU0 such that it achieves the desired minimum over all X X purifications: σ ϕ ϕ c β β c β β : i ¼ trABj iih ij¼ ijj jih jjþ ijkj jih kj ð13Þ X j j α α E ðA1∶…∶AnÞ¼f ðjψihψjÞ ¼ αJ SJ ðjψihψjÞ: ð15Þ J ∈R We now make another assumption: The black hole microstates we consider are typical in the sense that there is a For our arguments to hold, we will need to assume that basis in which the cijk are exponentially suppressed in N, jψiUU0 exists and lives in the large-N CFT Hilbert space and cij is approximately invertible; for example, we could H CFT. This is not immediately obvious, given that the consider the eigenstate thermalization hypothesis for the purifying subsystem may have unbounded, but finite, σ σ i ≠ j energy eigenbasis. For two i, j with , their bulk dimension. As a heuristic argument for why these assump- H geometries in A0B0 are macroscopically distinct, and tions are reasonable, we note that we are free to tune our σ σ hence, i, j are approximately orthogonal up to exponen- regulator to make H have arbitrarily large, but finite, tial corrections. Then, any superposition of the jϕii will dimension. A caveat is that we are ultimately limited by ρ c also be a purification of AB. The invertibility of ij implies a minimal cutoff scale that vanishes in the infinite-N limit. we can also invert the σi, so that Modulo this caveat, the idea is then to keep tuning H until it X can fit such a jψiUU0 . For a particular choice of α, it may be −1 jβjihβjj ∼ cij σi; ð14Þ possible that there exists a suitably finite bound on the i dimension of the purifying subsystem, in which case this assumption is unnecessary. This is the case for EP. which implies that there are at least dim HA0B0 many linearly Now expand jψiUU0 in the basis of holographic states independent σi. In other words, we are able to construct with semiclassical duals, with the added property that any density matrix in the purifying subsystem as a super- each state in the expansion also be a geometric purification position of geometric states on the subsystem. Again of ρU: invoking the unitary equivalence of purifications, we conclude that, up to exponential corrections, any arbitrary XM purification on the boundary can be expanded as a super- jψiUU0 ¼ cijϕiiUU0 ; ð16Þ position of geometric purifications. i We will assume the requisite assumptions are satisfied to use the above properties: We do not work beyond leading 2 such that trU0 jϕiihϕijUU0 ¼ ρU for all i. Writing pi ¼jcij , 1=N order in , we work with an appropriately dense and we can lower bound Eα as follows: typical subspace of black hole microstates, and the number of terms M that appear in (11) is exponentially suppressed α α E ðA1∶…∶AnÞ¼f ðjψihψjÞ ð17Þ relative to the full Hilbert space dimension. In other words, the results of this paper will only apply to suitably nice X holographic states. Note, however, that these assumptions are ¼ αJ SJ ðjψihψjÞ ð18Þ common in much of the literature, and much of the standard J ∈R lore in holography similarly only holds for such nice states; X X for example, it is known that the error correction properties of ∼ α p S ϕ ϕ S J ½ i J ðj iih ijÞ þ mixð19Þ AdS=CFT, and hence, other properties, can be radically J ∈R i changed with large M, e.g., [41]. Indeed, we do not expect the X results of this work to apply beyond these nice states. ≥ min αJ SJ ðjϕiihϕijÞ: ð20Þ i J ∈R III. MINIMIZATIONS OVER P GEOMETRIC EXTENSIONS α S ≥ 0 Note that ð J ∈R J Þ mix as both terms are non- We now present the main result of this paper: When negative. By the minimality of our choice of jψiUU0 ,we evaluating a correlation measure Eα of the form (2) on a must also have the upper bound

066009-4 OPTIMIZED CORRELATION MEASURES IN HOLOGRAPHY PHYS. REV. D 101, 066009 (2020) X α min αJ SJ ðjϕiihϕijÞ ≥ E ðA1∶…∶AnÞ: ð21Þ 1 i J ∈R ERðA∶BÞ¼ min½SAB þ 2SAE − SABE − SEð28Þ 2 ρABE

We therefore find 1 X ¼ min ½SAB þ 2SAA0 − SABA0 − SA0 : ð29Þ α 2 ψ E ðA1∶…∶AnÞ¼min αJ SJ ðjϕiihϕijÞ ð22Þ j iABA0B0 i J ∈R The interest in these measures stems from the fact that not for some geometric purification jϕiiUU0 of the original state all correlation measures of the form (2) are good correlation α ρU. We have hence found that the minimum over all measures in the sense that E are generically not monotonic purifications can always be achieved, to leading order, under LO. Reference [3] found that enforcing the monot- by only considering geometric purifications. Ponicity condition, along with the non-negativity condition Many correlation measures of interest are usually J ∈R αJ ≥ 0, leads to the intersection of convex cones in extensions defined as optimizations over all of a given α space whose extreme rays include EP, EQ, ER, and the state, rather than purifications. An extension of a state squashed entanglement E , which we discuss in the next ρ ρ sq U is simply any state UE in an extended Hilbert space subsection. H ⊗ H ρ ρ U E such that trE UE ¼ U. Such correlation mea- In [33,34], holographic duals to EQ and ER were sures have the form proposed but with the necessary assumption that the X α minimization over geometric purifications is sufficient to E ðA1∶…∶AnÞ¼inf αJ SJ ðρUEÞ; ð23Þ ρ UE J ∈R achieve a minimum over all purifications. With this assumption, one finds that EQ is dual to the entanglement n−1 where R is the order 2 − 1 set of all nontrivial wedge mutual information EM defined by combinations of A1; …;An;E. However, it is clear that this is entirely equivalent to the definition in (2),as 1 EMðA∶BÞ¼ min½SA þ SB þ SAA0 − SBA0 ; ð30Þ purifying the ρUE to jψiUEE0 leaves all the entropies 2 A0B0 appearing on the rhs unchanged, and hence, the optimization will yield the same result. The difference lies only in which is nothing more than the statement of EQ but with the 0 0 the redundancy of the search. Using EP as an example, we minimization over geometric regions A ;B that purify ρAB could equivalently write its definition as and the entropies interpreted as minimal surfaces. Such an expression is useful precisely because, as described in [33], EPðA∶BÞ¼minSAA0 ðρABA0 Þð24Þ ρABA0 there are cases where the geometric description of the system is particularly simple, e.g., two disjoint regions in S ψ ψ : ¼ min AA0 ðj ih jABA0B0 Þ ð25Þ AdS3, so that EM can be understood intuitively and is even jψiABA0B0 tractable to compute. Similarly, one finds the holographic Therefore, our result also applies to correlation measures dual to ER: defined in this way. For the sake of clarity, with respect to other literature, we will always use the most common 1 ERðA∶BÞ¼ min½SAB þ 2SAA0 − SABA0 − SA0 ð31Þ definitions of correlation measures, even if such definitions 2 A0B0 are not written as optimizations over purifications. ¼ EWðA∶BÞ; ð32Þ IV. APPLICATIONS IN HOLOGRAPHY A. Optimizing geometries which reduces to nothing more than the entanglement wedge cross section, adding ER to a growing list of Our result finds immediate application in the search for E Eα candidate information duals to W. holographic duals to , whereby we can directly optimize Our result confirms the results above, as we have shown the appropriate combination of surfaces dual to the desired that the crucial assumption regarding geometric minimi- linear combination of entropies on a given geometry. Two Q E R zations is also valid. More generally, we can make a recent examples are the correlation Q and correlation proposal for the holographic dual of any Eα as simply ER [3] defined as α α E ðA1∶…∶AnÞ¼ min f ðjψihψjÞ; ð33Þ 1 0 0 A …An EQðA∶BÞ¼ min½SA þ SB þ SAE − SBEð26Þ 1 2 ρABE A0 …A0 1 with the 1 n interpreted as geometric regions that ¼ min ½SA þ SB þ SAA0 − SBA0 ; ð27Þ purify the original state. Again, the utility of such an 2 ψ j iABA0B0 expression is that there likely exist situations where the

066009-5 NEWTON CHENG PHYS. REV. D 101, 066009 (2020) geometric description is relatively simple, making Eα easier because holographic entropy satisfies the monogamy of to understand or compute. mutual information (MMI): We remark here that applying this result to the bipartite EP ¼ EW conjecture gives −I3ðA∶B∶EÞ ≥ 0; ð39Þ

0 EPðA∶BÞ¼minSðAA Þ¼EWðA∶BÞ; ð34Þ where I3ðA∶B∶EÞ¼SA þSB þSE −SAB −SAE −SBE þSABE A0B0 is the tripartite information. It is quite easy to find counter- examples to (39) for generic quantum states, e.g. the so that EW can be interpreted as a single von Neumann entropy in some purified geometry. This is very similar to four-party GHZ state, so (39) is a special property of holographic states. Rewriting E as a minimization over an independent proposal for the information dual of EW sq given in [19], where a simple von Neumann entropy called the tripartite information gives the reflected entropy SR defined on a canonical purification 2E 1 1 of the given state is dual to W. Understanding the E ðA∶BÞ¼ IðA∶BÞþ min½−I3ðA∶B∶EÞ: ð40Þ sq 2 2 ρ relationship between the geometries dual to the purifica- ABE tions in EP, ER, and SR could provide insight into why In general, the state ρABE that achieves the minimum in (40) multiple distinct correlation measures reproduce the same E value in holography. need not be geometric in the sense that may not correspond to a spatial region, and hence, MMI need not apply. Fortunately, per our result, the minimum in (40) can indeed B. Holographic entropy inequalities be achieved by considering only geometric extensions, so As another method of determining holographic duals, the computation is simple: Choosing the empty extension our result permits the use of holographic entropy inequal- E¼0 achieves the lower bound −I3ðA∶B∶EÞ¼0. Note that ities in computing optimized correlation measures. One this corresponds to a purification jψiABEE0 where E is empty E 0 example of note is the squashed entanglement sq defined and E is any purifier for ρAB. We conclude that the squashed by [30,42] entanglement always saturates its upper bound 1 1 E A∶B I A∶B E E A∶B I A∶B sqð Þ¼ min ð j Þð35Þ sqð Þ¼ ð Þð41Þ 2 ρABE 2

1 in holographic systems, giving the holographic dual to the ¼ minðSAE þ SBE − SABE − SEÞ: ð36Þ E 2 ρABE bipartite sq as the appropriate sum of minimal surfaces dual to SA, SB, and SAB. Moreover, taking the squashed entan- The squashed entanglement has been extensively studied in glement to be a genuine measure of purely quantum the context of quantum information [43–50], as it satisfies correlations, we conclude that correlations within nice several abstract properties that make it arguably the most holographic states are dominated at leading order by promising measure of purely quantum correlations between quantum entanglement. parties in mixed states. For example, if ρAB is a pure state, This technique of using a holographic entropy inequality then the only extensions of the state are tensor products to make correlation measures computable is quite general: ρABE ¼ ρAB ⊗ ρE,so As observed in [33], the conditional entanglement of mutual information (CEMI) [51] 1 1 I A∶B E I A∶B S ; 2 ð j Þ¼2 ð Þ¼ A ð37Þ 1 0 0 0 0 EIðA∶BÞ¼ min ½IðAA ∶BB Þ − IðA ∶B Þ ð42Þ 2 ρ E S ABA0B0 and sq ¼ A, reducing to the von Neumann entropy on E pure states. Moreover, it was proven in [44] that sq is can be evaluated on holographic states by applying MMI E 0 ρ faithful; i.e., sq ¼ if and only if AB is a separable state. By taking E ¼ 0, one obtains the upper bound IðAA0∶BB0Þ ≥ IðAA0∶BÞþIðAA0∶B0Þð43Þ 1 I A∶B ≥ E A∶B ; ≥ IðA∶BÞþIðA0∶BÞ 2 ð Þ sqð Þ ð38Þ þ IðA∶B0ÞþIðA0∶B0Þ; ð44Þ which is easily interpreted: There cannot be more quantum correlations than total correlations. after which the positivity of the mutual information implies E While sq is difficult to compute for generic quantum that the optimal extension is the trivial one, so that E E A∶B 1 I A∶B states, it was observed in [32] that the computation of sq Ið Þ¼2 ð Þ. The CEMI is another candidate for holographic states could potentially be made simple entanglement measure that obeys several pleasing axioms

066009-6 OPTIMIZED CORRELATION MEASURES IN HOLOGRAPHY PHYS. REV. D 101, 066009 (2020) such as monotonicity under LOC, faithfulness, asymptotic We can again apply MMI to find continuity, and convexity. The implication for holographic states is similar to that of E ¼ 1 I: Quantum correlations 1 sq 2 E A ∶…∶A I A ∶…∶A dominate the total correlations of the state. sqð 1 nÞ¼2 ð 1 iÞ We can further use (41) to obtain the holographic dual for Xn E − I A …A ∶A ∶E other correlation measures. The saturation of sq for max 3ð 1 i−1 i Þð50Þ ρUE holographic states implies that any bipartite entanglement i¼2 measure defined as a constrained minimization of 1 IðA∶BjEÞ over extensions E will reduce to 1 IðA∶BÞ, 1 2 2 ≥ I A ∶…∶A so long as the constraint is compatible with the trivial 2 ð 1 iÞ extension. One example is the c-squashed entanglement Xn Ec “ ” E sq, which is a classical version of sq, in the sense that − max I3ðA1…Ai−1∶Ai∶EÞð51Þ ρUE the extension E is constrained to be classical [38,42,52]: i¼2

1 Ec A∶B I A∶B E ; 1 sqð Þ¼P min ð j Þ ð45Þ ≥ I A ∶…∶A ; p ρi ⊗ i i 2 ð 1 nÞ ð52Þ i AB j ih jE 2 i

ρ and since the lower bound is achievable by the trivial whereP the minimization isP over all states of the form ABE ¼ n E i i extension, we find that the -party sq is equal to one-half i piρ ⊗ jiihijE with i piρ ¼ ρAB. Because this is a AB AB the n-party mutual information. This confirms the con- constrained minimization, we naturally have Ec ≥ E , sq sq jecture made in [33], and, just as in the bipartite case, and the lower bound is achievable with the single-element Ec 1 I immediately implies that the multipartite sq equal 2 .We ensemble ρAB ⊗ j0ih0jE. We must therefore have Ec E Ec can similarly find the dual to the multipartite CEMI [51]: sq ¼ sq, and a corresponding holographic dual to sq. If Eα is subadditive and its regularization exists, it also 1 EIðA1∶…∶AnÞ collapses to 2 I. Specifically, we consider the class of Eα Eα E 1 measures reg within the set of that equal sq which 0 0 0 0 ¼ min½IðA1A1∶…∶AnAnÞ − IðA1∶…∶AnÞ ð53Þ also satisfy 2 ρUU0

1 α α ⊗n α α Xn E ðA∶BÞ¼ lim E ðρAB Þ; such that E ≥ E : ð46Þ 1 reg n→∞ n reg 0 0 0 ¼ min fIðA1A1…Ai−1Ai−1∶AiAiÞ 2 ρ 0 UU i¼2 E ρ 1 E ρ⊗n Then, by the additivity of sqð ABÞ¼n sqð AB Þ [30],we 0 0 0 find − IðA1…Ai−1∶AiÞg ð54Þ

E ¼ Eα ≥ Eα ≥ E ; ð47Þ sq reg sq Xn α ≥ EIðA1…Ai−1∶AiÞð55Þ and hence, E ¼ E . sq reg i¼2 We are also able to use geometric minimization to find the holographic dual to the multipartite squashed entangle- 1 ment introduced in [53,54]. We first define a multipartite ≥ IðA1∶…∶AnÞ; ð56Þ mutual information [37,55,56]: 2

Xn and the lower bound is again achievable by the trivial I A ∶…∶A I A …A ∶A : ð 1 nÞ¼ ð 1 i−1 iÞ ð48Þ extension, and hence, the multipartite EI is dual to the i¼2 1 multipartite 2 I. We stress here that (41) and other similar equalities are The multipartite conditional mutual information leading order statements in the 1=N expansion—we do not IðA1∶…∶AnjEÞ simply replaces the terms on the rhs of expect equality to hold in the presence of perturbative (48) with their conditional forms. The multipartite E is sq corrections. Indeed, if this were not the case, then (41) then defined to be would imply that holographic states have no classical 1 correlations. Rather, the more accurate statement is that E ðA1∶…∶AnÞ¼ min IðA1∶…∶AnjEÞ: ð49Þ quantum correlations dominate the mutual information, sq 2 ρ A1…AnE while classical ones are subleading.

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C. Axiomatic entanglement measures and EREðρABÞ¼ inf DðρABkσABÞ; ð61Þ the mutual information σAB∈SepðHABÞ We comment here on the apparent degeneracy of holo- where DðρkσÞ¼tr½ρ log ρ − ρ log σ is the relative entropy 1 I A∶B graphic duals to 2 ð Þ. If we work with the heuristic that and SepðHABÞ is the set of separable states on HAB. quantum correlations are the dominant type of correlation E Importantly, sq satisfies all four axioms and is addi- in holographic states, then it is perhaps not too surprising tive, so that any genuine measure of quantum correlations should E ≤ E ≤ E : saturate at the mutual information at leading order. C sq D ð62Þ Therefore, we might, in fact, expect that axiomatic entan- E E glement measures could all equal the mutual information. If C ¼ D, then we naturally have that all entanglement E E One way to formalize this statement is as follows. measures bounded by C and D are equal. For pure states, E E S E E Consider an entanglement measure E that satisfies the C ¼ D ¼ A.If C ¼ D on holographic states (at following axioms: leading order), then because we have already computed E , we would conclude (a) Normalization: EðρABÞ¼log d if ρAB is a maximally sq entangled state of rank d. 1 Λ E E E∞ E I A∶B ; (b) Monotonicity: For any process consisting of LOCC, C ¼ D ¼ ¼ sq ¼ 2 ð Þ ð63Þ we have EðρABÞ ≥ EðΛðρABÞÞ. (c) Continuity: Given two states ρ, σ on H,as leading to a large degeneracy of holographic duals to E ρ −E σ 1 ð ABÞ ð ABÞ IðA∶BÞ. Note again that we do not require EC ¼ ED kρAB − σABk1 → 0, we also have 1 H → 0. 2 þlog dim E ρ⊗n ∞ ð AB Þ exactly, but only approximately. Because of the operational (d) Regularization: The regularization E ρAB ð Þ¼ n E E exists for n → ∞. nature of C and D, they are extremely difficult to Any measure that satisfies these axioms also satisfies the compute for all the simplest of states and can also exhibit bounds [36] counterintuitive behaviors; e.g., there exist states which have nonzero entanglement but zero distillable entangle- reg ment [60]. Operationally defined quantities have not EDðρABÞ ≤ E ðρABÞ ≤ ECðρABÞ; ð57Þ received much attention in the context of holography, E and a very interesting direction for future study would where D is the distillable entanglement [57,58] be a deeper exploration of these objects, leveraging special n h i o properties of holographic states to simplify their analysis. ⊗n EDðρABÞ¼sup r lim inf kΛðρAB Þ − Φ2rn k1 ¼ 0 ; Some recent progress studying ED in holography is n→∞ Λ∈ r LOCC related to the equality of the smooth max and min entropies ð58Þ for holographic states [61,62]. The max and min entropies of a state ρAB are defined as and EC is the entanglement cost [57,59] S ¼ logðrankðρABÞÞ; ð64Þ n h i o max ⊗n −1 E ρ r ρ − Λ Φ rn 0 : S λ ρ ; Cð ABÞ¼inf lim inf k AB ð 2 Þk1 ¼ min ¼ logð ð ABÞÞ ð65Þ r n→∞ Λ∈LOCC min λ ρ ð59Þ where min is the minimal eigenvalue of AB. The smooth versions of these quantities are defined to be Operationally, ED corresponds to the maximal rate that one Sϵ σ ; can obtain Bell pairs from asymptotically many copies of max ¼ min logðrankð ABÞÞ ð66Þ kρ−σk1<ϵ ρAB, while EC corresponds to the minimal rate that one can ρ produce asymptotically many copies of AB from Bell pairs. Sϵ λ−1 σ ; E E∞ ρ E ρ min ¼ max logð minð ABÞÞ ð67Þ Note that if is additive, then ð ABÞ¼ ð ABÞ, and (57) kρ−σk1<ϵ becomes a bound on EðρABÞ itself. Examples of entanglement measures that satisfy these axioms include the that is, they are the minimal max and maximal min entanglement of formation EF, entropies of states that are in an ϵ ball around ρAB, where X the trace distance is the metric on the space of density EFðρABÞ¼P min piSðjψ iihψ ijÞ; ð60Þ matrices. It can be shown [62,63] that evaluating these pijψ iihψ ij¼ρAB i quantities on holographic states yields i pffiffiffi Sϵ S O S ; where the minimization is taken over all pure state max ¼ A þ ð Þ ð68Þ p ; ψ ρ ffiffiffi decompositions f i j iig of AB, and the relative entropy ϵ p E S ¼ SA − Oð SÞ: ð69Þ of entanglement RE, min

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In other words, the smooth max and min entropies are holographic states. Indeed, one could even define entan- equal, at leading order, to the von Neumann entropy. This glement measures of the form (2) such that the minimiza- result was then used in [63] to show that the one-shot tion over geometric extensions is simple to compute using entanglement distillation of a holographic pure state well holographic entropy inequalities. approximates ED, implying that the entanglement spectrum Another direction for finding holographic duals is the bit of a holographic state is due almost entirely to maximal threads formulation of holography [35], which has proven entanglement across the entangling surface. This result very useful for analyzing EP ¼ EW [16–18]. We expect that matches our intuition that quantum correlations are maxi- our result, which implies a bit thread configuration exists E α mal for holographic states, in the sense that sq saturates its for the optimal purification in E , will be a useful tool for upper bound. Then, at the level of entropies, we can finding or proving other holographic dual proposals. imagine replacing ρAB with an appropriate number of A promising direction is to use the saturation of the Bell pairs across the partition, which is quite evocative bipartite squashed entanglement to gain insight into the of EC ¼ ED. fine-grained structure of holographic states. In certain cases, the saturation of entropy inequalities or the bounds V. DISCUSSION of information quantities enforces some structure on a state. By using special properties of holographic states, we For example, it is known that the conditional mutual information vanishes on a bipartite state if and only if have shown that optimized correlation measures in holog- “ ” raphy can be computed, to leading order, by only consid- the state is a so-called quantum Markov chain [67].We ering geometric extensions of a given holographic state, expect that states with maximal squashed entanglement provided the states satisfy a set of given assumptions. The should have constraints on their form, which in turn imply most pressing future direction will be an analysis of the constraints on the form of holographic states. An example assumptions and the degree to which they are reasonable. of such a state can be found in [68]. Determining such We have applied our result to confirm the holographic constraints for generic quantum states is likely a difficult task, but it may be possible to, again, leverage properties duals for various entanglement measures, including EQ, E E Ec E exclusive to holographic states to make progress in this R, sq, sq, and I. Because all our results are leading order statements, it would be interesting to analyze the form direction. We leave such explorations to future work. of the quantum corrections that spoil the degeneracy of Finally, it would be extremely interesting to find evi- 1 dence for or against the conjecture that ED ¼ EC for duals to I and EW. 2 holographic states. Not only would confirmation likely In our argument, we have not made use of any properties lead to strong constraints on their fine-grained structure, it of the optimal purification jϕii, besides the fact that it is would also imply a whole host of correlation measures are geometric. Clarifying the properties of this class of puri- dual to the mutual information. This also motivates work fications would be an interesting future direction. One toward determining the holographic duals of measures such immediate example is understanding the purifications as EF and ERE, whose values could provide evidence in one involved in the EP ¼ EW and ER ¼ EW proposals. Moreover, we know of several other entanglement mea- direction or the other. We leave such explorations to future work. sures related to EW [19–29], for which our result will likely be useful in analyzing their connections to EP and ER. Although our result confirms several conjectured holo- ACKNOWLEDGMENTS graphic duals, it alone does not provide candidate duals for the optimized correlation measures. We have not needed to We thank Ning Bao, Ven Chandrasekar, Masamichi use any properties of the bulk spacetime, other than its Miyaji, Prathik Rath, Grant Remmen, and Koji existence and satisfying certain reasonable assumptions, so Umemoto for useful conversations and helpful comments. generally, more tools are necessary. In determining the We are particularly grateful to Koji Umemoto for a detailed E reading and discussions on a draft of this paper. We thank holographic dual to sq, we have used the MMI property of holographic states. However, MMI is not the only such the anonymous referees for suggestions on where our inequality, and forms one part of the more general “holo- discussion could be improved. We also thank the graphic entropy cone” [64–66] of inequalities that all Yukawa Institute for Theoretical Physics at Kyoto holographic states must satisfy but are not satisfied by University for hospitality during the workshop YITP-T- “ ” generic quantum states. It is likely that these inequalities 19-03 Quantum Information and String Theory 2019, will similarly prove useful for computing entanglement where several discussions were held that motivated measures and their holographic duals of higher-partite this work.

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