Constructing the three-qudit unextendible product bases with strong nonlocality

Abstract

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 ⊗ C 3 ⊗ 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 Cd ⊗ Cd ⊗ Cd of size ( d – 1 )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 ⊗ C 4 ⊗ C 5 system of size 38 is obtained based on a C 3 ⊗ C 3 ⊗ C 4 system of size 19. Then, we generalize this approach to the C d 1 ⊗ C d 2 ⊗ 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.