Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are (mathcal {C}^motimes mathcal {C}^{n_1}) with (5le mle n_1) and (mathcal {C}^motimes mathcal {C}^{n_2}) with (5le m le n_2), where (n_1) is odd and (n_2) is even. The second one is in (mathcal {C}^motimes mathcal {C}^n) ((m, nge 4)). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.