Nonlocality of tripartite orthogonal product states

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