Indistinguishability of pure orthogonal product states by LOCC

Abstract

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 $$\mathcal {C}^m\otimes \mathcal {C}^{n_1}$$Cm?Cn1 with $$5\le m\le n_1$$5≤m≤n1 and $$\mathcal {C}^m\otimes \mathcal {C}^{n_2}$$Cm?Cn2 with $$5\le m \le n_2$$5≤m≤n2, where $$n_1$$n1 is odd and $$n_2$$n2 is even. The second one is in $$\mathcal {C}^m\otimes \mathcal {C}^n$$Cm?Cn$$(m, n\ge 4)$$(m,n?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.