A Generalization of Quantum Stein's Lemma

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state rho or sigma (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying rho instead of sigma, while requiring that the probability that sigma is identified in the place of rho is bounded by a small fixed number. Quantum Stein's Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of rho and sigma. We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.