Error Exponents in Quantum Hypothesis Testing

In the simple quantum hypothesis testing problem, upper bounds on the error probabilities are shown based on a key matrix inequality. Concerning the error exponents, the upper bounds lead to noncommutative analogues of the Chernoff bound and the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our bound improves the bound by Ogawa-Hayashi. Our upper bounds also provide a simpler proof of the direct part of the quantum Stein’s lemma. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding.