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
Summary
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
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