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arXiv:1909.09334 · Physical review. D/Physical review. D.

Optimized correlation measures in holography

Newton Cheng

Published Mar 12, 2020• 7 citations
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Abstract

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 state. Examples include the entanglement of purification ${E}_{P}$ and squashed entanglement ${E}_{\mathrm{sq}}$. 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

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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 state. Examples include the entanglement of purification EP{E}_{P}EP​ and squashed entanglement Esq{E}_{\mathrm{sq}}Esq​. We show that when evaluating such measures on ``nice" holographic states in the large-NNN limit, the optimal purification has a semiclassical geometric dual. We then apply this result to confirm several holographic dual proposals, including the nnn-party squashed entanglement. Moreover, our result suggests two new techniques for determining holographic duals: holographic entropy inequalities and direct optimization of the dual geometry.
An important class of correlation measures for quantum states are the optimized correlation measures, defined as a minimization of a linear combination of von Neumann entropies over all purifications of a given state. We show that when considering such measures for holographic states, the minimum over all purifications can be achieved by only considering purifications with semi-classical geometric duals. We then apply this result to confirm the conjectured holographic duals of a large class of correlation measures, including the nnn-party squashed entanglement.
We explore ways to quantify multipartite correlations, in quantum information and in holography. We focus on optimized correlation measures, linear combinations of entropies minimized over all possible purifications of a state that satisfy monotonicity conditions. These contain far more information about correlations than entanglement entropy alone. We present a procedure to derive such quantities and construct a menagerie of symmetric optimized correlation measures on three parties. These include tripartite generalizations of the entanglement of purification, the squashed entanglement, and the recently introduced QQQ correlation and RRR correlation. Some correlation measures vanish only on product states and, thus, quantify both classical and quantum correlations; others vanish on any separable state, capturing quantum correlations alone. We then use a procedure motivated by the surface-state correspondence to construct holographic duals for the correlation measures as linear combinations of bulk surfaces. The geometry of the surfaces can preserve, partially break, or fully break the symmetry of the correlation measure. The optimal purification is encoded in the locations of certain points, whose locations are fixed by constraints on the areas of combinations of surfaces. This gives a new concrete connection between information theoretic quantities evaluated on a boundary state and detailed geometric properties of its dual.
Many of the tools and ideas of quantum information theory have been employed to shed light on the holographic principle.The entanglement entropy in the AdS/CFT correspondence tells us that information of the bulk geometries is encoded in the boundary correlations.This thesis is aimed to deepen our understanding of this direction by studying various correlation measures in holography.We first derive a generalization of the holographic entanglement of purification conjecture for multipartite states.It turns out that this proposal is consistently supported in arbitrary n-partite states.We next study the entanglement of purification in quantum many body systems by numerical methods.That reveals a remarkable non-monotonicity of the entanglement of purification, attributed to the different sensitivities to quantum and classical correlations.We finally discuss the holographic duals of two classes of correlation measures: optimized correlation measures and axiomatic entanglement measures.The results suggest that classical correlation also plays an important role in encoding the geometrical information.i
In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purification is now optimized in either a constrained way or over multiple parties. We separately derive new constraints on both the conditional entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. We also show that the multipartite objects we define, despite obeying several of the same inequalities, are not holographic duals of each other. Further, we find inequalities that are true only for the bulk objects and thus could provide additional consistency checks for states dual to (semi-)classical bulk geometries.
Information-theoretic inequalities often impose nontrivial constraints on holographic states. In this work, we study measurement-based classical and quantum correlations in holography, focusing on the proposed duals of classical correlation JWJ_WJW​, quantum discord DWD_WDW​, and one-shot distillable entanglement EDE_DED​, defined in terms of the entanglement wedge cross section (EWCS). We develop a homological framework tailored to inequalities involving multiple EWCSs and Ryu-Takayanagi surfaces, and use it to prove a family of inequalities, including monotonicity and monogamy/polygamy-type relations, as well as one-way strong superadditivity. For strong superadditivity, we additionally confirm its two-way version using Haar random states. We also examine holography-inspired boundary duals in terms of the reflected entropy and provide proofs and counterexamples for their information-theoretic inequalities. Taken together, our results provide further evidence for the duality between the EWCS and its proposed boundary counterparts -- measurement-based correlations and one-shot distillable entanglement -- while also furnishing a unified, rigorous method for proving multi-EWCS inequalities.
We study the holographic realization of optimized correlation measures---measures of quantum correlation that generalize elementary entropic formulas---in two-dimensional thermal states dual to spacetimes with a black hole horizon. We consider the symmetric bipartite optimized correlation measures: the entanglement of purification, Q-correlation, R-correlation, and squashed entanglement, as well as the mutual information, a nonoptimized correlation measure, and identify the bulk surface configurations realizing their geometric duals over the parameter space of boundary region sizes and the black hole radius. This parameter space is divided into phases associated with given topologies for these bulk surface configurations, and first-order phase transitions occur as a new topology of bulk surfaces becomes preferred. The distinct phases can be associated with different degrees of correlation between the boundary regions and the thermal environment. The Q-correlation has the richest behavior, with a structure of nested optimizations leading to two topologically distinct bulk surface configurations being equally valid as geometric duals at generic points in the phase diagram.