Constructing the three-qudit unextendible product bases with strong nonlocality

doi:10.1088/1674-1056/ac4a62

中国物理B ›› 2022, Vol. 31 ›› Issue (6): 60302-060302.doi: 10.1088/1674-1056/ac4a62

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

收稿日期:2021-09-10 修回日期:2021-12-20 接受日期:2022-01-12 出版日期:2022-05-17 发布日期:2022-05-17
通讯作者: Zhao Dou E-mail:dou@bupt.edu.cn
基金资助:
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).

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

Received:2021-09-10 Revised:2021-12-20 Accepted:2022-01-12 Online:2022-05-17 Published:2022-05-17
Contact: Zhao Dou E-mail:dou@bupt.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: strong nonlocality, unextendible product bases, tri-tile structure, construction method

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.

Key words: strong nonlocality, unextendible product bases, tri-tile structure, construction method

中图分类号: (Quantum information)

03.67.-a

03.65.Aa (Quantum systems with finite Hilbert space) 03.65.Ud (Entanglement and quantum nonlocality)

Bichen Che(车碧琛), Zhao Dou(窦钊), Xiubo Chen(陈秀波), Yu Yang(杨榆), Jian Li(李剑), and Yixian Yang(杨义先). Constructing the three-qudit unextendible product bases with strong nonlocality[J]. 中国物理B, 2022, 31(6): 60302-060302.

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Constructing the three-qudit unextendible product bases with strong nonlocality

Constructing the three-qudit unextendible product bases with strong nonlocality

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