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Nonlocal sets of orthogonal product states (OPSs) are widely used in quantumprotocols owing to their good property. Thus a lot of attention are paid to howto construct a nonlocal set of orthogonal product states though it is adifficult problem. In this paper, we propose a novel general method toconstruct a nonlocal set of orthogonal product states in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ for $d\geq3$. We give an ingenious proof for the localindistinguishability of those product states. The set of product states, whichare constructed by our method, has a very good structure. Subsequently, we givea construction of nonlocal set of OPSs with smaller members in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ for $d\geq3$. On the other hand, we present twoconstruction methods of nonlocal sets of OPSs in $\mathbb{C}^{m} \otimes\mathbb{C}^{n}$, where $m\geq3$ and $n\geq3.$ Furthermore, we propose theconcept of isomorphism for two nonlocal sets of OPSs. Our work is of great helpto understand the structure and classification of locally indistinguishableOPSs.

Novel methods to construct nonlocal sets of orthogonal product states in arbitrary bipartite high-dimensional system