Local distinguishability with preservation of entanglement
  I consider deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (1) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum; (2) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed; and (3) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are effected by the resources used by the parties are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of "non-locality without entanglement", a phenomenon originally discussed by Bennett and co-workers several years ago. 
Scott M. Cohen