Characterizing Locally Indistinguishable Orthogonal Product States

Bennett [PhysicalReviewA, vol. 59, no. 2, p. 1070, 1999] identified a set of orthogonal product states in the Hilbert space3otimes3such 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 both3otimes3and2otimes2otimes2. 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.