  We prove the ergodic version of the quantum Stein's lemma which was
conjectured by Hiai and Petz. The result provides an operational and
statistical interpretation of the quantum relative entropy as a statistical
measure of distinguishability, and contains as a special case the quantum
version of the Shannon-McMillan theorem for ergodic states. A version of the
quantum relative Asymptotic Equipartition Property (AEP) is given.
