# Finite entropy sums in quantum field theory

## TL;DR

This paper demonstrates that certain finite, divergence-free combinations of entropies in quantum field theory can be expressed as linear sums of basic information-theoretic quantities, utilizing advanced mathematical tools and AI assistance.

## Contribution

It introduces a framework to construct finite entropy sums in quantum field theory as linear combinations of fundamental information measures, with a basis that cancels divergences.

## Key findings

- Finite entropy sums can be expressed as linear combinations of basic quantities.
- A basis of entropy sums cancels divergences related to boundaries and intersections.
- Mathematical techniques include Fourier transforms on the Boolean cube and M"obius transformations.

## Abstract

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 sub-systems. 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 M\"obius transformations of functions on partially ordered sets) and several of the main proof ideas were suggested by AI (ChatGPT5). We offer a few comments on the use of AI in physics and mathematics, based on our experience.