On the foundations and extremal structure of the holographic entropy cone

TL;DR

This paper develops a graph-theoretic framework to analyze the holographic entropy cone, providing new proofs, computational methods, and partial results for its extremal structure, advancing understanding of quantum entropy inequalities.

Contribution

It introduces systematic approaches for identifying facets and extreme rays of the holographic entropy cone using graph theory, including new computational techniques and improved descriptions.

Findings

Recomputed the holographic entropy cone for 5 terminals.

Developed methods to find facets and extreme rays systematically.

Reported partial results and open problems for 6 terminals.

Abstract

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. We also report on some partial results for $6$ terminals. Some interesting open problems are stated throughout.