2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/89785The 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.6 pages, submitted to ISIT2002Quantum Physicstext