On Error Exponents in Quantum Hypothesis Testing

In the simple quantum hypothesis testing problem for two density operators, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and a conditional expectation of it. Concerning the error exponents, the upper bounds lead to a noncommutative analog of the Hoeffding bound, which is identical with the classical counterpart if two density operators commute. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.