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arXiv:2309.15145 · Physical review. D/Physical review. D.

Holographic entropy inequalities and the topology of entanglement wedge nesting

Bartłomiej Czech, Sirui Shuai, Yixu Wang, Daiming Zhang

Published May 10, 2024• 17 citations
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Abstract

We prove two new infinite families of holographic entropy inequalities. A key tool is a graphical arrangement of terms of inequalities that is based on entanglement wedge nesting. It associates the inequalities with tessellations of the torus and the projective plane, which reflect a certain topological aspect of entanglement wedge nesting. The inequalities prove a prior conjecture about the holographic entropy cone. We discuss their relation to black hole physics and differential entropy, and sketch applications to quantum error correction, quantifying randomness of quantum states, and others. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.l101903

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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 introduce a simple geometrical construction similar to covariant holographic entanglement entropy but with the addition of a new term proportional to boundary region volume. This new procedure has properties strongly resembling classical Shannon entropy of probability distributions rather than von Neumann entropy, so we call the quantity holographic Shannon entropy. The holographic Shannon entropy of a region AAA is divergent in AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of the entanglement wedge of AAA with the entanglement wedge of the complement of AAA held fixed. The construction is unambiguous when applied to compact surfaces with convex shape in general spacetimes obeying the null curvature condition (of which AdS with a regulator is a special case). In this context we prove that holographic Shannon entropy obeys monotonicity, a key property of Shannon entropy, as well as all known balanced inequalities of dynamical holographic entanglement entropy. In the static case, we explain why there must exist some classical probability distribution on random variables locally distributed on the boundary with the property that the Shannon entropies of all marginals are exactly reproduced by the holographic Shannon entropy formula.
To elucidate the meaning of holographic entropy inequalities (beyond subadditivity) which characterize the entanglement structure of geometric states in holography, arXiv:2309.06296 proposed the "tripartite form" for these inequalities, consisting of tripartite information and conditional tripartite information terms with unit coefficients. While this provides a compact and useful packaging of the inequalities, it is not a priori guaranteed that all inequalities can be recast in this form. Here we conjecture that they can, and present substantial evidence, by proving that the two known infinite families of holographic entropy inequalities found in arXiv:2309.15145 can indeed be written in the tripartite form. This is significant because such recasting is particularly nontrivial for these families. Apart from providing the explicit tripartite form expressions for every member of these two infinite families, we detail how we arrived at them, in the process deriving several useful identities which may serve as stepping stones to formulate further repackaging of the corresponding information quantities. We also illustrate the power of the tripartite form by proving a number of structural properties satisfied by any holographic entropy inequality.
We verify that the recently proven infinite families of holographic entropy inequalities are maximally tight, i.e. they are facets of the holographic entropy cone. The proof is technical but it offers some heuristic insight. On star graphs, both families of inequalities quantify how concentrated/spread information is with respect to a dihedral symmetry acting on subsystems. In addition, toric inequalities viewed in the K-basis show an interesting interplay between four-party and six-party perfect tensors.
A bstract 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 J W , quantum discord D W , and one-shot distillable entanglement E D , 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.