Relative entropy close to the edge

TL;DR

This paper demonstrates that the relative entropy between vacuum and excited states in quantum field theory near a boundary point is largely shape-independent, with corrections diminishing as the excitation point approaches the boundary.

Contribution

It provides a rigorous, model-independent analysis showing shape insensitivity of relative entropy near boundaries in quantum field theory.

Findings

Relative entropy is insensitive to the global shape of region A.

Corrections vanish as the excitation point approaches the boundary.

Rate of correction decay is slower than sqrt(ell/R).

Abstract

We show that the relative entropy between the reduced density matrix of the vacuum state in some region $A$ and that of an excited state created by a unitary operator localized at a small distance $\ell$ of a boundary point $p$ is insensitive to the global shape of $A$, up to a small correction. This correction tends to zero as $\ell/R$ tends to zero, where $R$ is a measure of the curvature of $\partial A$ at $p$, but at a rate necessarily slower than $\sim \sqrt{\ell/R}$ (in any dimension). Our arguments are mathematically rigorous and only use model-independent, basic assumptions about quantum field theory such as locality and Poincare invariance.