Combinatorial properties of holographic entropy inequalities

TL;DR

This paper introduces a new combinatorial framework to analyze holographic entropy inequalities, proving key properties and resolving previous conjectures, thereby advancing understanding of entanglement entropy in holographic theories.

Contribution

It develops a novel combinatorial approach to study HEIs, proving a necessary and sufficient condition for inequalities to be valid and resolving all prior conjectures.

Findings

Proved two conjectures about HEIs.

Disproved two conjectures about HEIs.

Showed null reduction of superbalanced HEIs obeys majorization.

Abstract

A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.