  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.
