Relative Entropy and Holography

TL;DR

This paper explores the role of relative entropy in holography, demonstrating its fundamental properties, evaluating it in AdS/CFT contexts, and discussing implications for entropy bounds and state reconstruction.

Contribution

It provides a detailed analysis of relative entropy in holographic settings, including new evaluations and applications to entropy bounds and state tomography.

Findings

Relative entropy is positive and increases with system size.

The equation ΔS=ΔH holds for infinitesimally different states in holography.

Supports the holographic entropy formula through inequality checks.

Abstract

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation $\Delta S=\Delta H$ for the first order variation of the entanglement entropy $\Delta S$ and the expectation value of the \modu Hamiltonian $\Delta H$. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation $\Delta S=\Delta H$ for vacuum state tomography and obtain modified versions of the Bekenstein bound.