Discrete And Differentiable Entanglement Transformations.

The study of transformations among pure states via Local Operations assisted by Classical Communication (LOCC) plays a central role in entanglement theory. The main emphasis of these investigations is on the deterministic, or probabilistic transformations between two states and mainly tools from linear algebra are employed. Here, we go one step beyond that and analyze all optimal protocols. We show that for all bipartite and almost all multipartite (of arbitrarily many d-level systems) pairs of states, there exist infinitely many optimal intermediate states to which the initial state can first be transformed (locally) before it is transformed to the final state. The success probability of this transformation is nevertheless optimal. We provide a simple characterization of all intermediate states. We generalize this concept to differentiable paths in Hilbert space that connect two states of interest. With the help of survival analysis we determine the success probability for the continuous transformation along such a path and derive necessary and sufficient conditions on the optimality. We show, in strong contrast to previous results on state transformations, that optimal paths are characterized as solutions of a differential equation. Whereas, for almost all pairs of states, there exist infinitely many optimal paths, we present examples of pairs of states for which not even a single optimal intermediate state exists. Furthermore, we introduce a physically motivated distance measure, the interconversion metric, and show that (generically) any minimal geodesic with respect to the interconversion metric is an optimal path. Moreover, we identify infinitely many easily computable entanglement monotones for generic multipartite pure states. We show that a given finite set of these entanglement monotones can be used to completely characterize the entanglement contained in a generic state.