Properties of the contraction map for holographic entanglement entropy inequalities

TL;DR

The paper tackles proving holographic entanglement entropy inequalities by refining the contraction-map method through a deterministic, rule-based approach. It introduces three methods that fix image bits on the RHS hypercube, analyzes their algorithmic complexity, and argues for the completeness of the contraction-map proof technique as a necessary and sufficient condition for inequality validity. The authors demonstrate substantial practical gains, including exponential speedups over greedy searches and the discovery of new six-party facet inequalities, while also addressing subtleties like unphysical $H_M$ vertices. They close with a discussion of implications for bulk AdS/CFT geometry and future directions for generating candidate inequalities and understanding bulk rigidity via isometries between contracted subgraphs.

Abstract

We present a deterministic way of finding contraction maps for candidate holographic entanglement entropy inequalities modulo choices due to actual degeneracy. We characterize its complexity and give an argument for the completeness of the contraction map proof method as a necessary and sufficient condition for the validity of an entropy inequality for holographic entanglement.

Theorems & Definitions (16)

• Theorem 1
• Definition 1
• Lemma 1
• proof
• Theorem 2
• proof
• Definition 2
• Lemma 2
• proof
• Theorem 3