Chin. Phys. B, 2022, Vol. 31(6): 060302    DOI: 10.1088/1674-1056/ac4a62
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# Constructing the three-qudit unextendible product bases with strong nonlocality

Bichen Che(车碧琛)1, Zhao Dou(窦钊)1,†, Xiubo Chen(陈秀波)1, Yu Yang(杨榆)2, Jian Li(李剑)2, and Yixian Yang(杨义先)1
1 Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China;
2 Information Security Center, Beijing University of Posts and Telecommunications, Beijing 100876, China
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 ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}$ 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.
Keywords:  strong nonlocality      unextendible product bases      tri-tile structure      construction method
Received:  10 September 2021      Revised:  20 December 2021      Accepted manuscript online:  12 January 2022
 PACS: 03.67.-a (Quantum information) 03.65.Aa (Quantum systems with finite Hilbert space) 03.65.Ud (Entanglement and quantum nonlocality)
Fund: This work was supported by the National Key R&D Program of China (Grant No. 2020YFB1805405), the 111 Project (Grant No. B21049), the Foundation of Guizhou Provincial Key Laboratory of Public Big Data (Grant No. 2019BDKFJJ014), and the Fundamental Research Funds for the Central Universities (Grant Nos. 2019XD-A02 and 2020RC38).
Corresponding Authors:  Zhao Dou     E-mail:  dou@bupt.edu.cn