Constructing the three-qudit unextendible product bases with strong nonlocality

Unextendible product bases (UPBs) are interesting members of a family of orthogonal product bases. Here, we investigate the construction of 3-qudit UPBs with strong nonlocality. First, a UPB set in C^3\otimes C^3\otimes C^3 of size 19 is presented based on the shift UPBs. By mapping the system to a Rubik's cube, we provide a general method of constructing UPBs in C^d\otimes C^d\otimes C^d of size \left(d-1 \right)^3+2d+5, whose corresponding Rubik's cube is composed of four parts. Second, for the more general case where the dimensions of parties are different, we extend the classical tile structure to the 3-qudit system and propose the tri-tile structure. By means of this structure, a C^4\otimes C^4\otimes C^5 system of size 38 is obtained based on a C^3\otimes C^3\otimes C^4 system of size 19. Then, we generalize this approach to the C^d_1\otimes C^d_2\otimes C^d_3 system which also consists of four parts. Our research provides a positive answer to the open question raised in by Halder et al. Phys. Rev. Lett. 122 040403 (2019), indicating that there do exist UPBs that can exhibit strong quantum nonlocality without entanglement.