Testing holographic entropy inequalities in 2+1 dimensions

TL;DR

This paper investigates whether holographic entropy inequalities valid in static states also hold in time-dependent states within 2+1 dimensional spacetimes, providing theoretical extensions and extensive numerical tests that support their universal validity.

Contribution

The authors extend existing proofs of holographic entropy inequalities to certain multiply-connected spacetimes and perform large-scale numerical tests confirming no violations occur.

Findings

No counterexamples found in extensive numerical tests.

Extended proof applies to specific multiply-connected spacetimes.

Supports the conjecture that inequalities hold universally in 2+1 dimensions.

Abstract

We address the question of whether holographic entropy inequalities obeyed in static states (by the RT formula) are always obeyed in time-dependent states (by the HRT formula), focusing on the case where the bulk spacetime is 2+1 dimensional. An affirmative answer to this question was previously claimed by Czech-Dong. We point out an error in their proof when the bulk is multiply connected. We nonetheless find strong support, of two kinds, for an affirmative answer in that case. We extend the Czech-Dong proof for simply-connected spacetimes to spacetimes with $\pi_1=\mathbb{Z}$ (i.e. 2-boundary, genus-0 wormholes). Specializing to vacuum solutions, we also numerically test thousands of distinct inequalities (including all known RT inequalities for up to 6 regions) on millions of randomly chosen configurations of regions and bulk spacetimes, including three different multiply-connected topologies; we find no counterexamples. In an appendix, we prove some (dimension-independent) facts about degenerate HRT surfaces and symmetry breaking.