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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 mini- mum 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.
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