Relative entropy equals bulk relative entropy

TL;DR

This paper demonstrates that the relative entropy between nearby states in a boundary theory corresponds to the bulk relative entropy, with implications for understanding the duality between boundary and bulk modular flows in holography.

Contribution

It establishes a precise relation between boundary and bulk relative entropy and explores the duality of modular flows within the entanglement wedge.

Findings

Boundary and bulk relative entropy are equivalent to leading order.

Boundary modular flow corresponds to bulk modular flow in the entanglement wedge.

Implications for entanglement wedge reconstruction and holographic duality.

Abstract

We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.