Symmetric hypothesis testing in quantum statistics is considered, where the hy potheses are density operators on a finite dimensional complex Hilbert space, rep resenting states of a finite quantum system.A lower bound on the asymptotic rate exponents of Bayesian error probabilities is proved.The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable er ror exponent in classical discrimination between two probability measures on a finite set.In the present framework the classical result is reproduced if the two hypothctic density operators commute.Recently it has been shown that the lower bound is achievable also in the generic quantum (noncommutative) case.This implies that the result is one part of the definitive quantum Chernoff bound.We will also dis cuss the quantum version of the Hoeffding bound, where errors of first and second kind are treated nonsymmetrically such that Steins lemma for classical alpha testing appears as a limiting case.