Strong quantum nonlocality without entanglement in every $(n-1)$-partition

Abstract: Orthogonal product sets that are locally irreducible in every bipartition have the strongest nonlocality while also need a large number of quantum states. In this paper, we construct the orthogonal product sets with strong quantum nonlocality in any possible $n$-partite systems, where $n$ is greater than three. Rigorous proofs show that these sets are locally irreducible in every $(n-1)$-partition. They not only possess stronger properties than nonlocality and fewer quantum states than the strongest nonlocal sets, but also are positive answers to the open question "how to construct different strength nonlocality of orthogonal product states for general multipartite and high-dimensional quantum systems" of Zhang et al. [{Phys. Rev. A \textbf{99}, 062108 (2019)}]. Our results can also enhance one understanding for the nonlocality without entanglement.