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

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