Variational approach to relative entropies (with application to QFT)

TL;DR

This paper introduces a new divergence measure for von Neumann algebras, extending relative entropy concepts, with applications to quantum field theory and connections to entropic certainty relations and error correction.

Contribution

It defines a novel divergence based on a variational formula, which generalizes relative entropy and applies to quantum field theory and von Neumann algebra substructures.

Findings

New divergence satisfies key properties and bounds sandwiched Renyi entropy.

Application to QFT computes divergence between vacuum and orbifolded systems.

Establishes entropic certainty relations linked to von Neumann subalgebras.

Abstract

We define a new divergence of von Neumann algebras using a variational expression that is similar in nature to Kosaki's formula for the relative entropy. Our divergence satisfies 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 conditional expectation -- system in terms of the Jones index. We take the opportunity to point out entropic certainty relation for arbitrary von Neumann subalgebras of a factor related to the relative entropy. This certainty relation has an equivalent formulation in terms of error correcting codes.