Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system.

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

众所周知，不可扩展乘积基（UPB）是一种不完备基，其元素无法通过局域操作和经典通信完美区分。然而，对于那些不是UPB的不完备且局域不可区分的乘积基，人们知之甚少。在本文中，我们首先在一般的\(m\otimes n\)（\(m\geq3\)且\(n\geq3\)）量子系统中构造了一系列可完备但局域不可区分的正交乘积基。特别地，我们给出了迄今为止任意量子系统中可完备正交乘积基的局域不可区分态的最小数量。此外，我们在\(m\otimes n\)（\(m\geq3\)且\(n\geq3\)）中构造了一系列小规模且局域不可区分的正交乘积基。所有这些结果都有助于更好地理解任意二分量子系统中局域不可区分乘积基的结构。