Relationship between entanglement cohomology and relative entropy

Determine how the entanglement cohomology groups H_E^*(\mathcal{M},\mathcal{N}) relate to the relative entropy S(\psi||\phi), including whether the cocycle derivative induces the suggested connection through the first cohomology group.

Background

The paper constructs entanglement cohomology H_E^*(\mathcal{M},\mathcal{N}) for an inclusion of von Neumann algebras \mathcal{N} \subset \mathcal{M}. It interprets the first cohomology group as being generated by modular data and connects the Connes–Radon–Nikodym cocycle to relative entropy.

The authors explicitly leave unresolved the precise mathematical relationship between the cohomological groups and relative entropy, noting only that the derivative of the cocycle suggests a connection through first cohomology. Establishing this relationship would clarify whether relative entropy can be recovered or characterized intrinsically as a cohomological invariant.