Improved proof-by-contraction method and relative homologous entropy inequalities

TL;DR

This paper advances the proof-by-contraction method for holographic entropy inequalities by providing a new rule to simplify the contraction map construction and extends the framework to relative homologous entropy for mixed states.

Contribution

It introduces a general rule to simplify the construction of contraction maps and extends the proof framework to relative homologous entropy.

Findings

Simplifies the proof process for holographic entropy inequalities.

Extends the framework to relative homologous entropy for mixed states.

Provides a more effective approach to analyze holographic entanglement structures.

Abstract

The celebrated holographic entanglement entropy triggered investigations on the connections between quantum information theory and quantum gravity. An important achievement is that we have gained more insights into the quantum states. It allows us to diagnose whether a given quantum state is a holographic state, a state whose bulk dual admits semiclassical geometrical description. The effective tool of this kind of diagnosis is holographic entropy cone (HEC), an entropy space bounded by holographic entropy inequalities allowed by the theory. To fix the HEC and to prove a given holographic entropy inequality, a proof-by-contraction technique has been developed. This method heavily depends on a contraction map $f$, which is very difficult to construct especially for more-region ($n\geq 4$) cases. In this work, we develop a general and effective rule to rule out most of the cases such that $f$ can be obtained in a relatively simple way. In addition, we extend the whole framework to relative homologous entropy, a generalization of holographic entanglement entropy that is suitable for characterizing the entanglement of mixed states.