On the completeness of contraction map proof method for holographic entropy inequalities

TL;DR

This paper investigates whether the contraction map proof method is not only sufficient but also necessary for linear holographic entropy inequalities with rational coefficients, revealing structural insights and implications for holographic geometries.

Contribution

It proves the necessity of the contraction map method for a class of linear holographic entropy inequalities and analyzes the geometric implications of non-contraction maps.

Findings

Pre-image of a non-contraction map is a proper cubical subgraph.

Alterations in geodesic structures lead to violations of inequalities.

Contraction map method is necessary for all rational coefficient inequalities.

Abstract

The contraction map proof method is the commonly used method to prove holographic entropy inequalities. Existence of a contraction map corresponding to a holographic entropy inequality is a sufficient condition for its validity. But is it also necessary? In this note, we answer that question in affirmative for all linear holographic entropy inequalities with rational coefficients. We show that the pre-image of a non-contraction map is not a hypercube, but a proper cubical subgraph, and show that this manifests as alterations to the geodesic structure in the bulk, which leads to the violation of inequalities by holographic geometries obeying the RT formula.