Two types of results are presented for distinguishing pure bipartite quantum states using Local Operations and Classical Communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C^3 tensor C^3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in C^n tensor C^m. In the

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... process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.