  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].
