Towards a complete classification of holographic entropy inequalities

TL;DR

This paper introduces a complete, deterministic method for classifying all holographic entropy inequalities using contraction maps and partial cubes, providing an algorithmic approach and insights into quantum entropy inequalities.

Contribution

The paper presents a novel, complete method for classifying holographic entropy inequalities through contraction maps and partial cubes, with an algorithmic solution and complexity analysis.

Findings

Established a correspondence between entropy inequalities and contraction maps

Developed an algorithm to find all contraction maps and partial cubes

Provided a procedure to generate candidate quantum entropy inequalities

Abstract

We propose a deterministic method to find all holographic entropy inequalities that have corresponding contraction maps and argue the completeness of our method. We use a triality between holographic entropy inequalities, contraction maps and partial cubes. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the argued completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method. We also demonstrate interesting by-products, most notably, a procedure to generate candidate quantum entropy inequalities.