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 all CFT’s [1]. Specifically, the perturbative relative entropy can be written as the symplectic flux of a certain scalar field in an auxiliary AdS-Rindler spacetime [1]. 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 [2]. In this paper, we introduce a one-parameter generalization of the relative entropy which we call refined Rényi 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 Rényi relative entropy.

Highlights

  • Our goal in this paper is to present a one-parameter generalization of the known results about the relative entropy in conformal field theory (CFT)’s, eq (1.2) and eq (1.3)

  • We study the refined Renyi relative entropy in section 4 when the reference state is the vacuum of a CFT reduced to a spherical region and the other state is a small perturbation thereof

  • We argue that the holographic dual of the refined Renyi relative entropy between an excited state and a vacuum state reduced to a spherical region is related to conserved charges in the bulk dual of the sandwiched state

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Summary

Background

Our goal in this paper is to present a one-parameter generalization of the known results about the relative entropy in CFT’s, eq (1.2) and eq (1.3). We dedicate this section to a brief review of these known results as well as a brief review of the concept of the conserved charges derived via covariant phase space methods

Perturbative relative entropy in a general CFT
Quasi-local conserved charges and canonical energy
Relative entropy in holographic CFT’s
Refined Renyi relative entropy
Perturbation of the refined Renyi relative entropy
Sandwiched state in a CFT
Perturbative refined Renyi relative entropy in a general CFT
Refined Renyi relative entropy as a symplectic flux
Refined Renyi relative entropy in a holographic CFT
Positive energy theorems
Relation to the perturbative result
Discussion
Data processing inequality
Monotonicity in the Renyi parameter n
Refined Renyi relative entropy under RG flow
Generalization of JLMS formula
A Refined Renyi relative entropy as relative entropy
B Conformal transformation and state ρ
C Refined Renyi relative entropy for thermal states
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