Local distinguishability with preservation of entanglement

I consider the 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 (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) 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 (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. 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 affected by the resources used by the parties (amount of classical communication and types of local operations), 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 nonlocality without entanglement, a phenomenon originally discussed by Bennett and co-workers several years ago.