In this talk I will focus on the local manipulation of multipartite entanglement contained in systems which are composed of n d-level subsystems. I will explain that non-trivial LOCC (Local operations assisted by classical communication) transformations among generic fully entangled pure states are almost never possible. Hence, almost all multipartite states are isolated. They can neither be deterministically obtained from local unitary (LU)-inequivalent states via local operations, nor can they be deterministically transformed to pure fully entangled LU-inequivalent states. I will then present a simple and elegant expression for the maximal probability to convert one multi-qudit fully entangled state to another for this generic set of states. The consequences of these findings in the context of entanglement theory will be discussed. Moreover, I will present some recent results about probabilistic transformations from a higher dimensional Hilbert space to a lower dimensional one.
Part of the Mathematics of Quantum Information workshop