LOCC indistinguishable orthogonal product quantum states

We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of [inline-graphic not available: see fulltext]2k+i ⊗ [inline-graphic not available: see fulltext]2l+j (i, j ∈ {0, 1} and i ≥ j ) and [inline-graphic not available: see fulltext]3k+i ⊗ [inline-graphic not available: see fulltext]3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of [inline-graphic not available: see fulltext]3k+i ⊗ [inline-graphic not available: see fulltext]3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only [inline-graphic not available: see fulltext]2k ⊗ [inline-graphic not available: see fulltext]2l and [inline-graphic not available: see fulltext]2k+1 ⊗ [inline-graphic not available: see fulltext]2l but also [inline-graphic not available: see fulltext]2k ⊗ [inline-graphic not available: see fulltext]2l+1 and [inline-graphic not available: see fulltext]2k+1 ⊗ [inline-graphic not available: see fulltext]2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of [inline-graphic not available: see fulltext]2k ⊗ [inline-graphic not available: see fulltext]2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.