Characterizing Locally Indistinguishable Orthogonal Product States

Bennett [<i>Physical</i> <i>Review</i> <i>A</i>, vol. 59, no. 2, p. 1070, 1999] identified a set of orthogonal product states in the Hilbert space <i>\BBC</i> ³ <i>otimes\BBC</i> 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 <i>\BBC</i> ³ <i>otimes\BBC</i> ³ and <i>\BBC</i> ² <i>otimes\BBC</i> ² <i>otimes\BBC</i> 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.