On error exponents in quantum hypothesis testing

On error exponents in the simple quantum hypothesis testing problem, a noncommutative analogue of the Hoeffding bound (W. Hoeffding, 1991) is presented. If the hypotheses, namely two density operators, are mutually commutative, the bound is identical with the classical counter part and thus is optimal. As a by-product, a simple proof of the direct part of the quantum Stein´s lemma [F. Hiai (1991)] [T. Ogawa (2000)] is also given.