Local distinguishability of orthogonal product states is an area of active research in quantum information theory. However, most of the relevant results about local distinguishability are found in bipartite quantum systems and very few are known in multipartite cases. In this paper, we construct a locally indistinguishable subset in \({\mathbb {C}}^{2d}\bigotimes {\mathbb {C}}^{2d}\bigotimes {\mathbb {C}}^{2d}\), \(d\ge 2\) that contains \(18(d-1)\) orthogonal product states. Further, we generalize our method to arbitrary tripartite quantum systems \({\mathbb {C}}^{k}\bigotimes {\mathbb {C}}^{l}\bigotimes {\mathbb {C}}^{m}\). This result helps us to understand the role of nonlocality without entanglement in multipartite quantum systems. Finally, we prove that a three-qubit GHZ state is sufficient as a resource to distinguish each of the above classes of states.

正交乘积态的局部可区分性是量子信息理论中积极研究的领域。但是，有关局部可区分性的大多数相关结果是在二分量子系统中发现的，而在多分情况下则鲜为人知。在本文中，我们在\（{\ mathbb {C}} ^ {2d} \ bigotimes {\ mathbb {C}} ^ {2d} \ bigotimes {\ mathbb {C}} ^ {2d}中构造局部不可区分的子集\），\（d \ ge 2 \）包含\（18（d-1）\）正交乘积状态。此外，我们将方法推广到任意的三方量子系统\（{\ mathbb {C}} ^ {k} \ bigotimes {\ mathbb {C}} ^ {l} \ bigotimes {\ mathbb {C}} ^ {m} \）。这一结果有助于我们理解非局部性的作用而不会在多粒子量子系统中发生纠缠。最后，我们证明了一个三量子位的GHZ状态足以作为区分上述每种状态的资源。