Entanglement manipulation under non-entangling operations

We demonstrate that entanglement shared by two or more parties can be asymptotically reversibly interconverted when one considers the set of operations which asymptotically cannot generate entanglement. In this scenario we find that the entanglement of every quantum state is uniquely characterized by a single quantity: the regularized relative entropy of entanglement. The main technical tool is a generalization of quantum Stein's Lemma, which gives optimal discrimination rates in quantum hypothesis testing, to the case in which the alternative hypothesis might vary over sets of correlated states. We analyze the connection of our approach to recent rigorous formulations of the second law of thermodynamics.