# The holographic dual of R\'enyi relative entropy

## TL;DR

This paper introduces a generalized form of Rnyi relative entropy in conformal field theories and explores its holographic dual, revealing new insights into conserved charges and positive energy theorems in AdS spacetimes.

## Contribution

It proposes a one-parameter generalization of relative entropy, called refined Rnyi relative entropy, and studies its holographic dual and implications for energy theorems.

## Key findings

- Generalization of relative entropy to refined Rnyi relative entropy.
- Holographic dual relates to conserved charges in AdS spacetime.
- New family of positive energy theorems in asymptotically AdS spaces.

## Abstract

The relative entropy is a measure of the distinguishability of two quantum states. A great deal of progress has been made in the study of the relative entropy between an excited state and the vacuum state of a conformal field theory (CFT) reduced to a spherical region. For example, when the excited state is a small perturbation of the vacuum state, the relative entropy is known to have a universal expression for \textit{all} CFT's \cite{Faulk-GR-entanglement}. Specifically, the perturbative relative entropy can be written as the symplectic flux of a certain scalar field in an \textit{auxiliary} AdS-Rindler spacetime \cite{Faulk-GR-entanglement}. Moreover, if the CFT has a semi-classical holographic dual, the relative entropy is known to be related to conserved charges in the bulk dual spacetime \cite{lashkari2016gravitational}. In this paper, we introduce a one-parameter generalization of the relative entropy which we call \textit{refined} R\'enyi relative entropy. We study this quantity in CFT's and find a one-parameter generalization of the aforementioned known results about the relative entropy. We also discuss a new family of positive energy theorems in asymptotically locally AdS spacetimes that arises from the holographic dual of the refined R\'enyi relative entropy.