Strong converse rate for asymptotic hypothesis testing in type III

TL;DR

This paper extends the operational interpretation of sandwiched Re9nyi entropy in the strong converse of quantum hypothesis testing from hyperfinite to general von Neumann algebras, revealing its fundamental nature.

Contribution

It generalizes the strong converse hypothesis testing results to arbitrary von Neumann algebras using approximation techniques, broadening the theoretical framework.

Findings

Operational meaning of sandwiched Re9nyi entropy holds beyond matrices

Approximation methods connect finite and infinite von Neumann algebras

Potential links to random matrix theory and fundamental physics

Abstract

We extend from the hyperfinite setting to general von Neumann algebras Mosonyi and Ogawa's (2015) and Mosonyi and Hiai's (2023) results showing the operational interpretation of sandwiched relative R\'enyi entropy in the strong converse of hypothesis testing. The specific task is to distinguish between two quantum states given many copies. We use a reduction method of Haagerup, Junge, and Xu (2010) to approximate relative entropy inequalities in an arbitrary von Neumann algebra by those in finite von Neumann algebras. Within these finite von Neumann algebras, it is possible to approximate densities via finite spectrum operators, after which the quantum method of types reduces them to effectively commuting subalgebras. Generalizing beyond the hyperfinite setting shows that the operational meaning of sandwiched R\'enyi entropy is not restricted to the matrices but is a more fundamental property of quantum information. Furthermore, applicability in general von Neumann algebras opens potential new connections to random matrix theory and the quantum information theory of fundamental physics.