Ke Li's lemma for quantum hypothesis testing in general von Neumann algebras

TL;DR

This paper extends Ke Li's lemma for quantum hypothesis testing from finite and type I von Neumann algebras to general von Neumann algebras using modular theory, providing new proofs and generalizations.

Contribution

It generalizes Ke Li's lemma to all von Neumann algebras, offering a more transparent understanding and broader applicability in quantum hypothesis testing.

Findings

Proves Ke Li's lemma for general von Neumann algebras.

Provides a new proof of quantum Stein's lemma with weaker assumptions.

Extends second order asymptotics results to general von Neumann algebras.

Abstract

A lemma stated by Ke Li in [arXiv:1208.1400] has been used in e.g. [arXiv:1510.04682,arXiv:1706.04590,arXiv:1612.01464,arXiv:1308.6503,arXiv:1602.08898] for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein's lemma with slightly weaker assumption, as well as immediate generalizations of its second order asymptotics, for example the main results in [arXiv:1510.04682] and [arXiv:1208.1400].