Variational approach to relative entropies (with application to QFT)
Stefan Hollands
TL;DR
This work introduces a variational divergence $\Phi_s$ for von Neumann algebras, extending Kosaki's relative entropy framework to a broad operator-algebra setting. It proves that $\Phi_s$ upper-bounds the sandwiched Renyi divergences $D_s$, while recovering fidelity in the limit, and establishes a data-processing inequality for channels. In quantum field theory, the authors show that for a finite-index inclusion $\mathcal A \subset \mathcal F$, the limit $\lim_{n\to\infty} \Phi_s(\omega_\Omega|\omega_\Omega\circ E_n)$ equals $\ln[\mathcal F: \mathcal A]$, connecting entropic quantities to the Jones index and fidelity via dual relations and entropic certainty. The framework suggests avenues for generalization to $C^*$-algebras and unbounded operators, and bridges subfactor theory with information-theoretic notions in QFT and holography.
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.