# The Holographic Entropy Cone for Five Regions

## TL;DR

This paper fully characterizes the holographic entropy cone for five regions, establishing all known inequalities and confirming no additional inequalities are needed, thus advancing understanding of quantum entanglement in holographic theories.

## Contribution

It provides a complete construction of the five-region holographic entropy cone using graph models, confirming the existing inequalities are sufficient and no new inequalities are necessary.

## Key findings

- Complete description of the $ ext{C}_5$ cone
- All holographic entropy inequalities for five regions are proven to be complete
- Graph models for all extreme rays are constructed

## Abstract

Even though little is known about the quantum entropy cone for $N\geq4$ subsystems, holographic techniques allow one to get a handle on the subspace of entropy vectors corresponding to states with gravity duals. For static spacetimes and $N$ boundary subsystems, this space is a convex polyhedral cone known as the holographic entropy cone $\mathcal{C}_N$ for $N$ regions. While an explicit description of $\mathcal{C}_N$ was accomplished for all $N\leq4$ in the initial study, the information given about larger $N$ was only partial already for $\mathcal{C}_5$. This letter provides a complete construction of $\mathcal{C}_5$ by exhibiting graph models for every extreme ray orbit generating the cone defined by all proven holographic entropy inequalities for $N=5$. The question of whether there exist additional inequalities for $5$ parties is thus settled with a negative answer. The conjecture that $\mathcal{C}_5$ coincides with the analogous cone for dynamical spacetimes is supported by demonstrating that the information quantities defining its facets are primitive.