Finite entropy sums in quantum field theory

A bstract Entropies associated with spatial subsystems in conventional local quantum field theories are typically divergent when the spatial regions have boundaries. However, in certain linear combinations of the entropies for various subsystems, these divergences may cancel, giving finite quantities that provide information-theoretic data about the underlying state. In this note, we show that all such quantities can be written as linear combinations of three basic types of quantities: i) the entropy of a spatial subsystem minus the entropy of its complementary subsystem, ii) the mutual information between non-adjacent subsystems, and iii) the tripartite information for triples of disjoint subsystems. For a fixed decomposition of a spatial slice into regions, we describe a basis of sums of entropies for collections of for these regions for which all divergences related to both region boundaries and higher-codimension intersections of regions cancel. Key mathematical technology used in this work (Fourier transforms on the Boolean cube and Möbius transformations of functions on partially ordered sets) and several of the main proof ideas were suggested by AI (ChatGPT5).