Classification of nonasymptotic bipartite pure-state entanglement transformations

We show that deterministic and conclusive transformation properties of bipartite entanglement in the nonasymptotic scenario (when many but a finite number of copies of a source state are collectively manipulated) are fundamentally different from those in both the single-copy and asymptotic limits. For instance, by generalizing the notion of local comparability of entanglement in the single-copy case, we provide a complete classification of bipartite entanglement transformations in the nonasymptotic scenario. We also show that, unlike the asymptotic case, collective operations need not always be advantageous for the many-copy case. In particular, we show that (1) there exists a class of states for which the optimal conclusive transformation probability decreases exponentially with increasing number of copies, even if the source state has more entropy of entanglement, and (2) optimal conclusive transformation probability need not be a monotonic function of the number of copies