We study the constructions of nonlocal orthogonal product states in multipartite systems that cannot be distinguished by local operations and classical communication. We first present two constructions of nonlocal orthogonal product states in tripartite systems
Cd⊗Cd⊗Cd (d≥3) and
Cd⊗Cd+1⊗Cd+2 (d≥3). Then for general tripartite quantum system
Cn1⊗Cn2⊗Cn3 (3≤n1≤n2≤n3), we obtain
2(n2+n3−1)−n1 nonlocal orthogonal product states. Finally, we put forward a new construction approach in
Cd1⊗Cd2⊗⋯⊗Cdn (d1,d2,⋯dn≥3,n>6) multipartite systems. Remarkably, our indistinguishable sets contain less nonlocal product states than the existing ones, which improves the recent results and highlights their related applications in quantum information processing.
The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.