The Holographic Entropy Cone

TL;DR

This paper systematically classifies entropy inequalities in holographic theories, establishing the holographic entropy cone and introducing new combinatorial proof techniques for minimal surface relations.

Contribution

It provides a complete characterization of holographic entropy inequalities for up to four regions and introduces an infinite family of inequalities for five or more regions.

Findings

Strong subadditivity and monogamy of mutual information are complete for 2-4 regions.

An infinite family of inequalities applies to 5 or more regions.

The holographic entropy cone is finitely characterized by minimal graph models.

Abstract

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.