  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.
