2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/90301In the simple quantum hypothesis testing problem, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and its pinching. Concerning the error exponents, the upper bounds lead to a noncommutative analogue of the Hoeffding bound, which is identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.13 pages, 2 figures, submitted to IEEE trans. Inform. TheoryQuantum Physicstext