  The direct part of Stein's lemma in quantum hypothesis testing is revisited
based on a key operator inequality between a density operator and its pinching.
The operator inequality is used to show a simple proof of the direct part of
Stein's lemma without using Hiai-Petz's theorem, along with an operator
monotone function, and in addition it is also used to show a new proof of
Hiai-Petz's theorem.

Quantum Physics