# Holographic Entropy Relations Repackaged

## TL;DR

This paper introduces a new K-basis for holographic entropy relations, simplifying their structure and proving all inequalities have positive coefficients in this basis, which enhances understanding of holographic entropy cones.

## Contribution

The paper develops a novel K-basis for holographic entropy relations using perfect tensor structures, revealing key positivity properties and simplifying the entropy inequalities.

## Key findings

- Entropy vectors from perfect tensors are extreme rays in the holographic entropy cone.
- All holographic entropy inequalities have positive coefficients in the K-basis.
- The K-basis offers a more compact and advantageous representation of entropy relations.

## Abstract

We explore the structure of holographic entropy relations (associated with 'information quantities' given by a linear combination of entanglement entropies of spatial sub-partitions of a CFT state with geometric bulk dual). Such entropy relations can be recast in multiple ways, some of which have significant advantages. Motivated by the already-noted simplification of entropy relations when recast in terms of multipartite information, we explore additional simplifications when recast in a new basis, which we dub the K-basis, constructed from perfect tensor structures. For the fundamental information quantities such a recasting is surprisingly compact, in part due to the interesting fact that entropy vectors associated to perfect tensors are in fact extreme rays in the holographic entropy cone (as well as the full quantum entropy cone). More importantly, we prove that all holographic entropy inequalities have positive coefficients when expressed in the K-basis, underlying the key advantage over the entropy basis or the multipartite information basis.