The holographic entropy zoo

TL;DR

This paper explores the holographic duals of a broad family of information-theoretic divergences, providing explicit bulk expressions for various parameters and extending to arbitrary backgrounds and dimensions.

Contribution

It derives explicit bulk expressions for the holographic duals of the $ ext{α}$-$z$ divergences for specific parameter ranges, generalizing previous results.

Findings

Explicit bulk expressions for boundary divergences at second order in perturbation.

Applicability to arbitrary background states and dimensions.

Results depend on the equality of bulk and boundary modular flows.

Abstract

We study the holographic dual of a two parameter family of quantities known as the $\alpha$-$z$ divergences. Many familiar information theoretic quantities occur within this family, including the relative entropy, fidelity, and collision relative entropy. We find explicit bulk expressions for the boundary divergences to second order in a state perturbation whenever $\alpha$ is an integer and $z\geq0$, as well as when $z\in\{0,\infty\}$ and $\alpha\in \mathbb{R}$. Our results apply for perturbations around an arbitrary background state and in any dimension, under the assumption of the equality of bulk and boundary modular flows.