Ke Li's lemma for quantum hypothesis testing in 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].

v2: minor changes following the referees' suggestions; in particular, a few more details are added in the proof, as well as in the discussions around the quantum Stein lemma and its second asymptotics. A new proof of quantum Stein lemma with slightly weaker assumption is also illustrated in Section 4. To appear in Annales Henri Poincaré