Strong quantum nonlocality without entanglement in every -partition.

The orthogonal product set with quantum nonlocality can enhance the confidentiality of information without consuming entanglement resources. The confidentiality increases with the reinforcement of its nonlocality. However, the orthogonal product sets with the strongest nonlocality need an enormous number of quantum states. In this paper, we propose a sufficient condition for orthogonal product set being strongly nonlocal and construct some strongly nonlocal sets in every -partition on -qudit systems, where is greater than 3. These newly constructed sets possess stronger properties than nonlocality and reduce the exponential number of required quantum states compared with the strongest nonlocal sets. Our results not only partially answer the open question "how to construct different strength nonlocality of orthogonal product states for general multipartite and high-dimensional quantum systems" but also provide significant theoretical fundamental for quantum secure communication.