  We consider the question of perfect local distinguishability of mutually
orthogonal bipartite quantum states, with the property that every state can be
specified by a unitary operator acting on the local Hilbert space of Bob. We
show that if the states can be exactly discriminated by one-way LOCC where
Alice goes first, then the unitary operators can also be perfectly
distinguished by an orthogonal measurement on Bob's Hilbert space. We give
examples of sets of N<=d maximally entangled states in $d \otimes d $ for
d=4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly
for d=5,6 our examples consist of four and five states respectively. We
conjecture that these states cannot be perfectly discriminated by two-way LOCC.
