Operator-algebraic definition of tripartite information for three adjacent regions

Develop an explicit operator-algebraic construction that expresses the tripartite information I(A, B, C) = S(A) + S(B) + S(C) − S(A ∪ B) − S(A ∪ C) − S(B ∪ C) + S(A ∪ B ∪ C) for three pairwise adjacent spatial regions in a local quantum field theory, in cases where higher-codimension divergences are absent, purely in terms of relative entropies. Establish a definition that applies when each region is adjacent to both of the other regions, analogous to the mutual information representation via relative entropy that holds when at least one region is non-adjacent.

Background

The paper studies finite linear combinations of entanglement entropies for spatial regions in local quantum field theories, focusing on sums whose divergences cancel. To avoid regulator dependence, Section 6 discusses operator-algebraic definitions of these finite quantities, including expressing certain combinations in terms of relative entropies.

Under the split property, the mutual information for two non-adjacent regions can be defined as an Araki relative entropy between the joint state and the product of the marginals, and when at least one region in a triple is non-adjacent, the tripartite information can be built from mutual informations. However, when all three regions are adjacent (but no higher-codimension divergences are present), the authors state that an explicit relative-entropy-based construction for the tripartite information is not known, identifying this as an open question.