Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$

Yiyun He1, Fei Shi2, and Xiande Zhang3,4

1Department of Mathematics, University of California, Irvine, 92697, CA, United States
2QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
3School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People's Republic of China
4Hefei National Laboratory, University of Science and Technology of China, Hefei, 230088, China

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Abstract

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the $N$-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in $N$-partite systems for all odd $N\geq 3$. Based on this decomposition, we give explicit constructions of unextendible product bases in $N$-partite systems for odd $N\geq 3$. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the $N$-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in $N$-partite systems for all odd $N\geq 3$. Based on this decomposition, we give explicit constructions of unextendible product bases in $N$-partite systems for odd $N\geq 3$. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.

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[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, Jun 2009. 10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[3] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters. Quantum nonlocality without entanglement. Phys. Rev. A, 59: 1070–1091, Feb 1999a. 10.1103/​PhysRevA.59.1070.
https:/​/​doi.org/​10.1103/​PhysRevA.59.1070

[4] Jonathan Walgate, Anthony J. Short, Lucien Hardy, and Vlatko Vedral. Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett., 85: 4972–4975, Dec 2000. 10.1103/​PhysRevLett.85.4972.
https:/​/​doi.org/​10.1103/​PhysRevLett.85.4972

[5] Sibasish Ghosh, Guruprasad Kar, Anirban Roy, Aditi Sen(De), and Ujjwal Sen. Distinguishability of bell states. Phys. Rev. Lett., 87: 277902, Dec 2001. 10.1103/​PhysRevLett.87.277902.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.277902

[6] Michał Horodecki, Aditi Sen(De), Ujjwal Sen, and Karol Horodecki. Local indistinguishability: More nonlocality with less entanglement. Phys. Rev. Lett., 90: 047902, Jan 2003. 10.1103/​PhysRevLett.90.047902.
https:/​/​doi.org/​10.1103/​PhysRevLett.90.047902

[7] David P Divincenzo, Tal Mor, Peter W Shor, John A Smolin, and Barbara M Terhal. Unextendible product bases, uncompletable product bases and bound entanglement. Commun. Math. Phys., 238 (3): 379–410, 2003. 10.1007/​s00220-003-0877-6.
https:/​/​doi.org/​10.1007/​s00220-003-0877-6

[8] S. De Rinaldis. Distinguishability of complete and unextendible product bases. Phys. Rev. A, 70: 022309, Aug 2004. 10.1103/​PhysRevA.70.022309.
https:/​/​doi.org/​10.1103/​PhysRevA.70.022309

[9] Sibasish Ghosh, Guruprasad Kar, Anirban Roy, and Debasis Sarkar. Distinguishability of maximally entangled states. Phys. Rev. A, 70: 022304, Aug 2004. 10.1103/​PhysRevA.70.022304.
https:/​/​doi.org/​10.1103/​PhysRevA.70.022304

[10] Heng Fan. Distinguishability and indistinguishability by local operations and classical communication. Phys. Rev. Lett., 92: 177905, Apr 2004. 10.1103/​PhysRevLett.92.177905.
https:/​/​doi.org/​10.1103/​PhysRevLett.92.177905

[11] J. Niset and N. J. Cerf. Multipartite nonlocality without entanglement in many dimensions. Phys. Rev. A, 74: 052103, Nov 2006. 10.1103/​PhysRevA.74.052103.
https:/​/​doi.org/​10.1103/​PhysRevA.74.052103

[12] Heng Fan. Distinguishing bipartite states by local operations and classical communication. Phys. Rev. A, 75: 014305, Jan 2007. 10.1103/​PhysRevA.75.014305.
https:/​/​doi.org/​10.1103/​PhysRevA.75.014305

[13] Yuan Feng and Yaoyun Shi. Characterizing locally indistinguishable orthogonal product states. IEEE Trans. Inf. Theory, 55 (6): 2799–2806, 2009. 10.1109/​TIT.2009.2018330.
https:/​/​doi.org/​10.1109/​TIT.2009.2018330

[14] Nengkun Yu, Runyao Duan, and Mingsheng Ying. Four locally indistinguishable ququad-ququad orthogonal maximally entangled states. Phys. Rev. Lett., 109: 020506, Jul 2012. 10.1103/​PhysRevLett.109.020506.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.020506

[15] Somshubhro Bandyopadhyay. Entanglement, mixedness, and perfect local discrimination of orthogonal quantum states. Phys. Rev. A, 85: 042319, Apr 2012. 10.1103/​PhysRevA.85.042319.
https:/​/​doi.org/​10.1103/​PhysRevA.85.042319

[16] Alessandro Cosentino. Positive-partial-transpose-indistinguishable states via semidefinite programming. Phys. Rev. A, 87: 012321, Jan 2013. 10.1103/​PhysRevA.87.012321.
https:/​/​doi.org/​10.1103/​PhysRevA.87.012321

[17] Yan-Ling Wang, Mao-Sheng Li, Zhu-Jun Zheng, and Shao-Ming Fei. Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A, 92: 032313, Sep 2015. 10.1103/​PhysRevA.92.032313.
https:/​/​doi.org/​10.1103/​PhysRevA.92.032313

[18] Mao-Sheng Li, Yan-Ling Wang, Shao-Ming Fei, and Zhu-Jun Zheng. $d$ locally indistinguishable maximally entangled states in ${C^{d}{\bigotimes}C^{d}}$. Phys. Rev. A, 91: 042318, Apr 2015. 10.1103/​PhysRevA.91.042318.
https:/​/​doi.org/​10.1103/​PhysRevA.91.042318

[19] Zhi-Chao Zhang, Fei Gao, Ya Cao, Su-Juan Qin, and Qiao-Yan Wen. Local indistinguishability of orthogonal product states. Phys. Rev. A, 93: 012314, Jan 2016. 10.1103/​PhysRevA.93.012314.
https:/​/​doi.org/​10.1103/​PhysRevA.93.012314

[20] Guangbao Xu, Qiaoyan Wen, Fei Gao, Sujuan Qin, and Huijuan Zuo. Local indistinguishability of multipartite orthogonal product bases. Quantum Inf. Process., 16 (11): 276, 2017. 10.1007/​s11128-017-1725-5.
https:/​/​doi.org/​10.1007/​s11128-017-1725-5

[21] Yan-Ling Wang, Mao-Sheng Li, Zhu-Jun Zheng, and Shao-Ming Fei. The local indistinguishability of multipartite product states. Quantum Inf. Process., 16: 1–13, 2017. 10.1007/​s11128-016-1477-7.
https:/​/​doi.org/​10.1007/​s11128-016-1477-7

[22] Zhi-Chao Zhang, Ke-Jia Zhang, Fei Gao, Qiao-Yan Wen, and C. H. Oh. Construction of nonlocal multipartite quantum states. Phys. Rev. A, 95: 052344, May 2017. 10.1103/​PhysRevA.95.052344.
https:/​/​doi.org/​10.1103/​PhysRevA.95.052344

[23] Saronath Halder. Several nonlocal sets of multipartite pure orthogonal product states. Phys. Rev. A, 98: 022303, Aug 2018. 10.1103/​PhysRevA.98.022303.
https:/​/​doi.org/​10.1103/​PhysRevA.98.022303

[24] Guang-Bao Xu and Dong-Huan Jiang. Novel methods to construct nonlocal sets of orthogonal product states in an arbitrary bipartite high-dimensional system. Quantum Inf. Process., 20: 1–38, 2021. 10.1007/​s11128-021-03062-8.
https:/​/​doi.org/​10.1007/​s11128-021-03062-8

[25] Zong-Xing Xiong, Mao-Sheng Li, Zhu-Jun Zheng, Chuan-Jie Zhu, and Shao-Ming Fei. Positive-partial-transpose distinguishability for lattice-type maximally entangled states. Phys. Rev. A, 99: 032346, Mar 2019. 10.1103/​PhysRevA.99.032346.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032346

[26] Hui-Juan Zuo, Jia-Huan Liu, Xiao-Fan Zhen, and Shao-Ming Fei. Nonlocal sets of orthogonal multipartite product states with less members. Quantum Inf. Process., 20: 1–15, 2021. 10.1007/​s11128-021-03320-9.
https:/​/​doi.org/​10.1007/​s11128-021-03320-9

[27] Mao-Sheng Li, Yan-Ling Wang, Fei Shi, and Man-Hong Yung. Local distinguishability based genuinely quantum nonlocality without entanglement. J. Phys. A, 54 (44): 445301, 2021. 10.1088/​1751-8121/​ac28cd.
https:/​/​doi.org/​10.1088/​1751-8121/​ac28cd

[28] Yan-Ying Zhu, Dong-Huan Jiang, Xiang-Qian Liang, Guang-Bao Xu, and Yu-Guang Yang. Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems. Quantum Inf. Process., 21 (7): 252, 2022. 10.1007/​s11128-022-03601-x.
https:/​/​doi.org/​10.1007/​s11128-022-03601-x

[29] Xiao-Fan Zhen, Shao-Ming Fei, and Hui-Juan Zuo. Nonlocality without entanglement in general multipartite quantum systems. Phys. Rev. A, 106 (6), December 2022. 10.1103/​physreva.106.062432.
https:/​/​doi.org/​10.1103/​physreva.106.062432

[30] Yan-Ling Wang, Wei Chen, and Mao-Sheng Li. Small set of orthogonal product states with nonlocality. Quantum Inf. Process., 22 (1): 15, 2022. 10.1007/​s11128-022-03764-7.
https:/​/​doi.org/​10.1007/​s11128-022-03764-7

[31] Mao-Sheng Li and Yan-Ling Wang. Bounds on the smallest sets of quantum states with special quantum nonlocality. Quantum, 7: 1101, September 2023. ISSN 2521-327X. 10.22331/​q-2023-09-07-1101.
https:/​/​doi.org/​10.22331/​q-2023-09-07-1101

[32] Hai-Qing Cao, Mao-Sheng Li, and Hui-Juan Zuo. Locally stable sets with minimum cardinality. Phys. Rev. A, 108: 012418, Jul 2023. 10.1103/​PhysRevA.108.012418.
https:/​/​doi.org/​10.1103/​PhysRevA.108.012418

[33] Saronath Halder, Manik Banik, Sristy Agrawal, and Somshubhro Bandyopadhyay. Strong quantum nonlocality without entanglement. Phys. Rev. Lett., 122: 040403, Feb 2019. 10.1103/​PhysRevLett.122.040403.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.040403

[34] Fei Shi, Mengyao Hu, Lin Chen, and Xiande Zhang. Strong quantum nonlocality with entanglement. Phys. Rev. A, 102: 042202, Oct 2020a. 10.1103/​PhysRevA.102.042202.
https:/​/​doi.org/​10.1103/​PhysRevA.102.042202

[35] Pei Yuan, Guojing Tian, and Xiaoming Sun. Strong quantum nonlocality without entanglement in multipartite quantum systems. Phys. Rev. A, 102: 042228, Oct 2020. 10.1103/​PhysRevA.102.042228.
https:/​/​doi.org/​10.1103/​PhysRevA.102.042228

[36] Yan-Ling Wang, Mao-Sheng Li, and Man-Hong Yung. Graph-connectivity-based strong quantum nonlocality with genuine entanglement. Phys. Rev. A, 104 (012424), July 2021. 10.1103/​physreva.104.012424.
https:/​/​doi.org/​10.1103/​physreva.104.012424

[37] Fei Shi, Mao-Sheng Li, Lin Chen, and Xiande Zhang. Strong quantum nonlocality for unextendible product bases in heterogeneous systems. J. Phys. A, 55 (1): 015305, December 2021a. 10.1088/​1751-8121/​ac3bea.
https:/​/​doi.org/​10.1088/​1751-8121/​ac3bea

[38] Fei Shi, Mao-Sheng Li, Mengyao Hu, Lin Chen, Man-Hong Yung, Yan-Ling Wang, and Xiande Zhang. Strongly nonlocal unextendible product bases do exist. Quantum, 6: 619, January 2022a. 10.22331/​q-2022-01-05-619.
https:/​/​doi.org/​10.22331/​q-2022-01-05-619

[39] Huaqi Zhou, Ting Gao, and Fengli Yan. Orthogonal product sets with strong quantum nonlocality on a plane structure. Phys. Rev. A, 106: 052209, Nov 2022. 10.1103/​PhysRevA.106.052209.
https:/​/​doi.org/​10.1103/​PhysRevA.106.052209

[40] Fei Shi, Zuo Ye, Lin Chen, and Xiande Zhang. Strong quantum nonlocality in $N$-partite systems. Phys. Rev. A, 105 (022209), February 2022b. 10.1103/​physreva.105.022209.
https:/​/​doi.org/​10.1103/​physreva.105.022209

[41] Jicun Li, Fei Shi, and Xiande Zhang. Strongest nonlocal sets with small sizes. Phys. Rev. A, 108: 062407, Dec 2023. 10.1103/​PhysRevA.108.062407.
https:/​/​doi.org/​10.1103/​PhysRevA.108.062407

[42] Zong-Xing Xiong and Mao-Sheng Li. Existence of strongly nonlocal sets of three states in any $n$-partite system. arXiv preprint arXiv:2403.10969, 2024. 10.48550/​arXiv.2403.10969.
https:/​/​doi.org/​10.48550/​arXiv.2403.10969
arXiv:2403.10969

[43] Mengying Hu, Ting Gao, and Fengli Yan. Strong quantum nonlocality with genuine entanglement in an $n$-qutrit system. Phys. Rev. A, 109: 022220, Feb 2024. 10.1103/​PhysRevA.109.022220.
https:/​/​doi.org/​10.1103/​PhysRevA.109.022220

[44] Atanu Bhunia, Subrata Bera, Indranil Biswas, Indrani Chattopadhyay, and Debasis Sarkar. Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis. arXiv preprint arXiv:2404.05882, 2024. 10.48550/​arXiv.2404.05882.
https:/​/​doi.org/​10.48550/​arXiv.2404.05882
arXiv:2404.05882

[45] Fei Shi, Mao-Sheng Li, Mengyao Hu, Lin Chen, Man-Hong Yung, Yan-Ling Wang, and Xiande Zhang. Strong quantum nonlocality from hypercubes. arXiv:2110.08461, 2021b. 10.48550/​arXiv.2110.08461.
https:/​/​doi.org/​10.48550/​arXiv.2110.08461
arXiv:2110.08461

[46] Charles H. Bennett, David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, and Barbara M. Terhal. Unextendible product bases and bound entanglement. Phys. Rev. Lett., 82: 5385–5388, Jun 1999b. 10.1103/​PhysRevLett.82.5385.
https:/​/​doi.org/​10.1103/​PhysRevLett.82.5385

[47] R. Augusiak, T. Fritz, Ma. Kotowski, Mi. Kotowski, M. Pawłowski, M. Lewenstein, and A. Acín. Tight bell inequalities with no quantum violation from qubit unextendible product bases. Phys. Rev. A, 85: 042113, Apr 2012. 10.1103/​PhysRevA.85.042113.
https:/​/​doi.org/​10.1103/​PhysRevA.85.042113

[48] R. Augusiak, J. Stasińska, C. Hadley, J. K. Korbicz, M. Lewenstein, and A. Acín. Bell inequalities with no quantum violation and unextendable product bases. Phys. Rev. Lett., 107: 070401, Aug 2011. 10.1103/​PhysRevLett.107.070401.
https:/​/​doi.org/​10.1103/​PhysRevLett.107.070401

[49] Jianxin Chen, Lin Chen, and Bei Zeng. Unextendible product basis for fermionic systems. J. Math. Phys., 55 (8), 2014. 10.1063/​1.4893358.
https:/​/​doi.org/​10.1063/​1.4893358

[50] Fei Shi, Xiande Zhang, and Lin Chen. Unextendible product bases from tile structures and their local entanglement-assisted distinguishability. Phys. Rev. A, 101: 062329, Jun 2020b. 10.1103/​PhysRevA.101.062329.
https:/​/​doi.org/​10.1103/​PhysRevA.101.062329

[51] Sristy Agrawal, Saronath Halder, and Manik Banik. Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition. Phys. Rev. A, 99: 032335, Mar 2019. 10.1103/​PhysRevA.99.032335.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032335

[52] Fei Shi, Mao-Sheng Li, Xiande Zhang, and Qi Zhao. Unextendible and uncompletable product bases in every bipartition. New J. Phys., 24 (11): 113025, 2022c. 10.1088/​1367-2630/​ac9e14.
https:/​/​doi.org/​10.1088/​1367-2630/​ac9e14

[53] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Mixed-state entanglement and distillation: Is there a ``bound'' entanglement in nature? Phys. Rev. Lett., 80: 5239–5242, Jun 1998. 10.1103/​PhysRevLett.80.5239.
https:/​/​doi.org/​10.1103/​PhysRevLett.80.5239

[54] Hari krishnan S V, Ashish Ranjan, and Manik Banik. State space structure of tripartite quantum systems. Phys. Rev. A, 104: 022437, Aug 2021. 10.1103/​PhysRevA.104.022437.
https:/​/​doi.org/​10.1103/​PhysRevA.104.022437

[55] Kalyanapuram Rangachari Parthasarathy. On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Math. Sci., 114 (4): 365–374, 2004. 10.1007/​BF02829441.
https:/​/​doi.org/​10.1007/​BF02829441

[56] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 2010. 10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[57] Huaqi Zhou, Ting Gao, and Fengli Yan. Strong quantum nonlocality without entanglement in an $n$-partite system with even $n$. Phys. Rev. A, 107: 042214, Apr 2023. 10.1103/​PhysRevA.107.042214.
https:/​/​doi.org/​10.1103/​PhysRevA.107.042214

Cited by

[1] Huaqi Zhou, Ting Gao, and Fengli Yan, "Orthogonal product sets with strong quantum nonlocality on a plane structure", Physical Review A 106 5, 052209 (2022).

[2] Jicun Li, Fei Shi, and Xiande Zhang, "Strongest nonlocal sets with small sizes", Physical Review A 108 6, 062407 (2023).

[3] Huaqi Zhou, Ting Gao, and Fengli Yan, "Strong quantum nonlocality without entanglement in an n -partite system with even n", Physical Review A 107 4, 042214 (2023).

[4] Mengying Hu, Ting Gao, and Fengli Yan, "Strong quantum nonlocality with genuine entanglement in an N -qutrit system", Physical Review A 109 2, 022220 (2024).

[5] Siwen You, Chen Wang, Fei Shi, Sihuang Hu, and Yiwei Zhang, "Unextendible product bases from tile structures in bipartite systems", Journal of Physics A Mathematical General 56 1, 015303 (2023).

[6] Xiao-Fan Zhen, Mao-Sheng Li, and Hui-Juan Zuo, "Strongest nonlocal sets with minimum cardinality in tripartite systems", Physical Review A 109 5, 052422 (2024).

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