Characterizing Locally Indistinguishable Orthogonal Product States

Bennett [Physical Review A, vol. 59, no. 2, p. 1070, 1999] identified a set of orthogonal product states in the Hilbert space BBC ³ otimesBBC ³ such that reliably distinguishing those states requires nonlocal quantum operations. While more examples have been found for this counterintuitive ldquononlocality without entanglementrdquo phenomenon, a complete and computationally verifiable characterization for all such sets of states remains unknown. In this paper, we give such a characterization for both BBC ³ otimesBBC ³ and BBC ² otimesBBC ² otimesBBC ². As a consequence, we show that in both spaces, there is no additional set of a fundamentally different structure than those of the known instances.