2015/12/24 by Guang-Bao Xu, Ying-Hui Yang, Qiao-Yan Wen +2
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1038/srep31048
published as Scientific Reports, 6:31048 (2016)
arxiv created 2015/12/24 · arxiv updated 2020/04/06
Recently, Zhang et al [Phys. Rev. A 92, 012332 (2015)] presented 4d-4 orthogonal product states that are locally indistinguishable and completable in a d⊗ d quantum system. Later, Zhang et al. [arXiv: 1509.01814v2 (2015)] constructed 2n-1 orthogonal product states that are locally indistinguishable in m⊗ n (3≤ m ≤ n). In this paper, we construct a locally indistinguishable and completable orthogonal product basis with 4p-4 members in a general m⊗ n (3≤ m ≤ n) quantum system, where p is an arbitrary integer from 3 to m, and give a very simple but quite effective proof for its local indistinguishability. Specially, we get a completable orthogonal product basis with 8 members that cannot be locally distinguished in m⊗ n (3≤ m ≤ n) when p=3. It is so far the smallest completable orthogonal product basis that cannot be locally distinguished in a m⊗ n quantum system. On the other hand, we construct a small locally indistinguishable orthogonal product basis with 2p-1 members, which is maybe uncompletable, in m⊗ n (3≤ m ≤ n and p is an arbitrary integer from 3 to m). We also prove its local indistinguishability. As a corollary, we give an uncompletable orthogonal product basis with 5 members that are locally indistinguishable in m⊗ n (3≤ m ≤ n). All the results can lead us to a better understanding of the structure of a locally indistinguishable product basis in m ⊗ n.