A covariant regulator for entanglement entropy: proofs of the Bekenstein bound and QNEC

TL;DR

This paper introduces a covariant regulator for entanglement entropy in quantum field theory, enabling rigorous definitions of entropy differences in curved spacetime and proving fundamental bounds like the Bekenstein bound and QNEC.

Contribution

It proposes a novel covariant regulator based on modular crossed product, allowing well-defined entropy differences and proofs of key physical bounds in curved spacetime.

Findings

Defines a covariant entropy regulator using type II von Neumann algebras

Reproduces heuristic formulas for entropy differences in quantum field theory

Proves the Bekenstein bound and quantum null energy condition in this framework

Abstract

While von Neumann entropies for subregions in quantum field theory universally contain ultraviolet divergences, differences between von Neumann entropies are finite and well-defined in many physically relevant scenarios. We demonstrate that such a notion of entropy differences can be rigorously defined in quantum field theory in a general curved spacetime by introducing a novel, covariant regulator for the entropy based on the modular crossed product. This regulator associates a type II von Neumann algebra to each spacetime subregion, resulting in well-defined renormalized entropies. This prescription reproduces formulas for entropy differences that coincide with heuristic formulas widely used in the literature, and we prove that it satisfies desirable properties such as unitary invariance and concavity. As an application, we provide proofs of the Bekenstein bound and the quantum null energy condition, formulated directly in terms of vacuum-subtracted von Neumann entropies.