Variational approach to relative entropies with an application to QFT

We define a new divergence of von Neumann algebras using a variational expression similar in nature to Kosaki's formula for Umegaki's relative entropy. Our divergence satisfies several of the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an orbifolded-in the sense of a conditional expectation-system in terms of the Jones index. We take the opportunity to point out an entropic certainty relation associated with an inclusion of von Neumann factors related to the relative entropy. This certainty relation has an equivalent formulation in terms of error correcting codes.

Automated summary

The paper introduces a new divergence of von Neumann algebras defined using a variational expression similar to Kosaki's formula, which satisfies several desirable properties, upper bounds the sandwiched Renyi entropy, and reduces to fidelity in the limit. As an example, the divergence is computed in quantum field theory between the vacuum in a bipartite system and an orbifolded system, highlighting an entropic certainty relation.