Zoo of Correlation Inequalities in Holography and Beyond

TL;DR

This paper develops a homological framework to prove a variety of inequalities involving correlation measures in holography, providing evidence for dualities and establishing a unified method for multi-EWCS inequalities.

Contribution

It introduces a new homological approach to prove inequalities among multiple entanglement wedge cross sections and related correlation measures in holography.

Findings

Proved monotonicity and monogamy/polygamy relations for correlation measures.

Confirmed strong superadditivity for Haar random states.

Provided proofs and counterexamples for boundary dual inequalities.

Abstract

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.