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Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems

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Abstract

Many achievements have been made since quantum nonlocality without entanglement was found. A natural question is how many orthogonal product states (OPSs) are needed at least to form a nonlocal set in a given quantum system. In this paper, we achieve some interesting results about nonlocal sets with the less amount of OPSs. Firstly, we construct new nonlocal sets of OPSs with only 6 and 9 members in the \(\mathbb {C}^{3} \otimes \mathbb {C}^{3} \otimes \mathbb {C}^{3}\) and \(\mathbb {C}^{4} \otimes \mathbb {C}^{4} \otimes \mathbb {C}^{4}\) quantum systems, respectively. Secondly, we give a general construction of nonlocal set of OPSs with only \(3(d-1)\) members in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d}\otimes \mathbb {C}^{d}\) quantum system for \(d\ge 3\). Finally, we compare our work with existing results and make a series of generalizations. The comparison result shows that the nonlocal sets of OPSs constructed by us have the least number of elements in their corresponding quantum systems.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 62171264, 62071015) and Shandong Provincial Natural Science Foundation (Grant No. ZR2019MF023).

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Yan-Ying Zhu, Dong-Huan Jiang & Xiang-Qian Liang

  2. College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Guang-Bao Xu

  3. Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China

    Yu-Guang Yang

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  1. Yan-Ying Zhu
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  2. Dong-Huan Jiang
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  3. Xiang-Qian Liang
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Correspondence to Guang-Bao Xu.

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Zhu, YY., Jiang, DH., Liang, XQ. et al. Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems. Quantum Inf Process 21, 252 (2022). https://doi.org/10.1007/s11128-022-03601-x

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