Relative entropy for von Neumann subalgebras

TL;DR

This paper explores the relationship between index and relative entropy in finite von Neumann algebras, introducing a new generalized relative entropy concept with applications in quantum decoherence analysis.

Contribution

It establishes a connection between the Pimsner-Popa index and sandwiched Renyi p-relative entropy, generalizing the subfactor index and applying it to quantum decoherence estimation.

Findings

Connects index with sandwiched Renyi p-relative entropy for all p in [1/2, ∞]

Introduces a generalized relative entropy to a subalgebra

Provides applications in estimating decoherence time of quantum Markov semigroups

Abstract

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.