Bounds on the smallest sets of quantum states with special quantum nonlocality

Mao-Sheng Li1 and Yan-Ling Wang2

1School of Mathematics, South China University of Technology, Guangzhou 510641, China
2School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, China

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Abstract

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems [46]. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_N} $ for any $d_i \geq 2$ and $1\leq i\leq N$?) raised in a recent paper [54]. Compared with all previous relevant proofs, our proof here is quite concise.

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[1] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, U.K., 2004).

[2] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters. Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.59.1070

[3] B. M. Terhal, D. P. DiVincenzo, and D. W. Leung. Hiding Bits in Bell States. Phys. Rev. Lett. 86, 5807 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5807

[4] D. P. DiVincenzo, D.W. Leung and B.M. Terhal. Quantum data hiding. IEEE Trans. Inf. Theory 48, 580 (2002).
https:/​/​doi.org/​10.1109/​18.985948

[5] D. Markham and B. C. Sanders. Graph States for Quantum Secret Sharing. Phys. Rev. A 78, 042309 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.042309

[6] R. Rahaman and M. G. Parker. Quantum scheme for secret sharing based on local distinguishability. Phys. Rev. A 91, 022330 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.022330

[7] J. Wang, L. Li, H. Peng, and Y. Yang. Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states. Phys. Rev. A 95, 022320 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.022320

[8] J. Walgate and L. Hardy. Nonlocality Asymmetry and Distinguishing Bipartite States. Phys. Rev. Lett. 89, 147901 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.147901

[9] J. Walgate, A. J. Short, L. Hardy, and V. Vedral. Local Distinguishability of Multipartite Orthogonal Quantum States. Phys. Rev. Lett. 85, 4972 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.85.4972

[10] S. Ghosh, G. Kar, A. Roy, A. Sen(De), and U. Sen. Distinguishability of Bell States. Phys. Rev. Lett. 87, 277902 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.87.277902

[11] H. Fan. Distinguishability and Indistinguishability by Local Operations and Classical Communication. Phys. Rev. Lett. 92, 177905 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.92.177905

[12] M. Nathanson. Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases. J. Math. Phys. (N.Y.) 46, 062103 (2005).
https:/​/​doi.org/​10.1063/​1.1914731

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

[14] S. M. Cohen. Local distinguishability with preservation of entanglement. Phys. Rev. A 75, 052313 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.052313

[15] S. Bandyopadhyay, S. Ghosh, and G. Kar. LOCC distinguishability of unilaterally transformable quantum states. New J. Phys. 13 123013 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​12/​123013

[16] N. Yu, R. Duan, and M. Ying. Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States. Phys. Rev. Lett. 109, 020506 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.020506

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

[18] M.-S. Li, Y.-L. Wang, S.-M. Fei and Z.-J. Zheng. $d$ locally indistinguishable maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d$. Phys. Rev. A 91, 042318 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.042318

[19] S. X. Yu and C. H. Oh, Detecting the local indistinguishability of maximally entangled states. arXiv:1502.01274v1.
https:/​/​doi.org/​10.48550/​arXiv.1502.01274
arXiv:1502.01274v1

[20] Y.-L. Wang, M.-S. Li, and Z.-X. Xiong. One-way local distinguishability of generalized Bell states in arbitrary dimension. Phys. Rev. A 99, 022307 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.022307

[21] Z.-X. Xiong, M.-S. Li, Z.-J. Zheng, C.-J. Zhu, and S.-M. Fei. Positive-partial-transpose distinguishability for lattice-type maximally entangled states. Phys. Rev. A 99, 032346 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.032346

[22] M.-S. Li and Y.-L. Wang. Alternative method for deriving nonlocal multipartite product states. Phys. Rev. A 98, 052352 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.052352

[23] M.-S. Li, S.-M. Fei, Z.-X. Xiong, and Y.-L. Wang. Twist-teleportation-based local discrimination of maximally entangled states. SCIENCE CHINA Physics, Mechnics $\&$ Astronomy 63 8, 280312 (2020).
https:/​/​doi.org/​10.1007/​s11433-020-1562-4

[24] M. Banik, T. Guha, M. Alimuddin, G. Kar, S. Halder, and S. S. Bhattacharya. Multicopy Adaptive Local Discrimination: Strongest Possible Two-Qubit Nonlocal Bases. Phys. Rev. Lett. 126, 210505 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.210505

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

[26] M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki. Local Indistinguishability: More Nonlocality with Less Entanglement. Phys. Rev. Lett. 90, 047902 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.90.047902

[27] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal. Unextendible Product Bases and Bound Entanglement. Phys. Rev. Lett. 82, 5385 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.82.5385

[28] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal. Unextendible product bases, uncompletable product bases and bound entanglement. Comm. Math. Phys. 238, 379 (2003).
https:/​/​doi.org/​10.1007/​s00220-003-0877-6

[29] Y. Feng and Y.-Y. Shi. Characterizing Locally Indistinguishable Orthogonal Product States. IEEE Trans. Inf. Theory 55, 2799 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2018330

[30] Y.-H. Yang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen. Local distinguishability of orthogonal quantum states in a $2\otimes 2\otimes 2$ system. Phys. Rev. A 88, 024301 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.024301

[31] Z.-C. Zhang, F. Gao, G.-J. Tian, T.-Q. Cao and Q.-Y. Wen. Nonlocality of orthogonal product basis quantum states. Phys. Rev. A 90, 022313 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.022313

[32] Z.-C. Zhang, F. Gao, S.-J. Qin, Y.-H. Yang, and Q.-Y. Wen. Nonlocality of orthogonal product states. Phys. Rev. A 92, 012332 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.012332

[33] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei. Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A 92, 032313 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.032313

[34] Z.-C. Zhang, F. Gao, Y. Cao, S.-J. Qin, and Q.-Y. Wen. Local indistinguishability of orthogonal product states. Phys. Rev. A 93, 012314 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.012314

[35] G.-B. Xu, Y.-H. Yang, Q.-Y. Wen, S.-J. Qin, and F. Gao. Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system. Sci. Rep. 6, 31048 (2016).
https:/​/​doi.org/​10.1038/​srep31048

[36] G.-B. Xu, Q.-Y. Wen, S.-J. Qin, Y.-H. Yang, and F. Gao. Quantum nonlocality of multipartite orthogonal product states. Phys. Rev. A 93, 032341 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.032341

[37] X.-Q. Zhang, X.-Q. Tan, J. Weng, and Y.-J. Li. LOCC indistinguishable orthogonal product quantum states. Sci. Rep. 6, 28864 (2016).
https:/​/​doi.org/​10.1038/​srep28864

[38] Z.-C. Zhang, K.-J. Zhang, F. Gao, Q.-Y. Wen, and C. H. Oh. Construction of nonlocal multipartite quantum states. Phys. Rev. A 95, 052344 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052344

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

[40] S. Rout, A. G. Maity, A. Mukherjee, S. Halder, and M. Banik. Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination. Phys. Rev. A 100, 032321 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.032321

[41] S. Halder, and C. Srivastava. Locally distinguishing quantum states with limited classical communication. Phys. Rev. A 101, 052313 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.052313

[42] D.-H. Jiang, and G.-B. Xu. Nonlocal sets of orthogonal product states in arbitrary multipartite quantum system. Phys. Rev. A 102, 032211 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.032211

[43] G.-B. Xu, and D.-H. Jiang. Novel methods to construct nonlocal sets of orthogonal product states in any bipartite high-dimensional system. Quantum Inf. Process. 20, 128 (2021).
https:/​/​doi.org/​10.1007/​s11128-021-03062-8

[44] S. Halder, R. Sengupta. Distinguishability classes, resource sharing, and bound entanglement distribution. Phys. Rev. A 101, 012311 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.012311

[45] M.-S. Li, Y.-L. Wang, F. Shi, and M.-H. Yung. Local distinguishability based genuinely quantum nonlocality without entanglement. J. Phys. A: Math. Theor. 54 445301 (2021).
https:/​/​doi.org/​10.1088/​1751-8121/​ac28cd

[46] S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay. Strong Quantum Nonlocality without Entanglement. Phys. Rev. Lett. 122, 040403 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.040403

[47] P. Yuan, G. J. Tian, and X. M. Sun. Strong quantum nonlocality without entanglement in multipartite quantum systems. Phys. Rev. A 102, 042228 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.042228

[48] Z.-C. Zhang and X. Zhang. Strong quantum nonlocality in multipartite quantum systems. Phys. Rev. A 99, 062108 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.062108

[49] F. Shi, M. Hu, L. Chen, and X. Zhang. Strong quantum nonlocality with entanglement. Phys. Rev. A 102, 042202 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.042202

[50] Y.-L. Wang, M.-S. Li, and M.-H. Yung. Graph-connectivity-based strong quantum nonlocality with genuine entanglement, Phys. Rev. A 104, 012424 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.104.012424

[51] F. Shi, M.-S. Li, M. Hu, L. Chen, M.-H. Yung, Y.-L. Wang and X. Zhang. Strongly nonlocal unextendible product bases do exist. Quantum 6, 619 (2022).
https:/​/​doi.org/​10.22331/​q-2022-01-05-619

[52] F. Shi, M.-S. Li, M. Hu, L. Chen, M.-H. Yung, Y.-L. Wang and X. Zhang. Strong quantum nonlocality from hypercubes. arXiv:2110.08461.
https:/​/​doi.org/​10.48550/​arXiv.2110.08461
arXiv:2110.08461

[53] F. Shi, M.-S. Li, L. Chen and X. Zhang. Strong quantum nonlocality for unextendible product bases in heterogeneous systems. J. Phys. A: Math. Theor. 55, 015305 (2022).
https:/​/​doi.org/​10.1088/​1751-8121/​ac3bea

[54] F. Shi, Z. Ye, L. Chen, and X. Zhang. Strong quantum nonlocality in $N$-partite systems. Phys. Rev. A 105, 022209 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.022209

[55] A. Miyake and H. J. Briegel. Distillation of Multipartite Entanglement by Complementary Stabilizer Measurements. Phys. Rev. Lett. 95, 220501 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.220501

[56] S. M. Cohen. Local approximation for perfect discrimination of quantum states. Phys. Rev. A 107, 012401 (2023).
https:/​/​doi.org/​10.1103/​PhysRevA.107.012401

[57] H.-Q. Cao, M.-S. Li , and H.-J. Zuo. Locally stable sets with minimum cardinality. Phys. Rev. A 108, 012418 (2023).
https:/​/​doi.org/​10.1103/​PhysRevA.108.012418

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

[3] Zong-Xing Xiong, Yongli Zhang, Mao-Sheng Li, and Lvzhou Li, "Small sets of genuinely nonlocal Greenberger-Horne-Zeilinger states in multipartite systems", Physical Review A 109 2, 022428 (2024).

[4] Zong-Xing Xiong, Mao-Sheng Li, Zhu-Jun Zheng, and Lvzhou Li, "Distinguishability-based genuine nonlocality with genuine multipartite entanglement", Physical Review A 108 2, 022405 (2023).

[5] Hai-Qing Cao and Hui-Juan Zuo, "Locally distinguishing nonlocal sets with entanglement resource", Physica A Statistical Mechanics and its Applications 623, 128852 (2023).

[6] 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).

[7] Yan-Ling Wang, Wei Chen, and Mao-Sheng Li, "Small set of orthogonal product states with nonlocality", Quantum Information Processing 22 1, 15 (2023).

[8] Hai-Qing Cao, Mao-Sheng Li, and Hui-Juan Zuo, "Locally stable sets with minimum cardinality", Physical Review A 108 1, 012418 (2023).

[9] Ying-Hui Yang, Guang-Wei Mi, Shi-Jiao Geng, Qian-Qian Liu, and Hui-Juan Zuo, "Strong nonlocality with genuine entanglement based on GHZ-like states in multipartite quantum systems", Physica Scripta 98 1, 015104 (2023).

[10] Yan-Ying Zhu, Dong-Huan Jiang, Guang-Bao Xu, and Yu-Guang Yang, "Completable sets of orthogonal product states with minimal nonlocality", Physica A Statistical Mechanics and its Applications 624, 128956 (2023).

[11] Wang Yan-Ling, Chen Wei, and Li Mao-Sheng, "Small set of orthogonal product states with nonlocality", arXiv:2207.04603, (2022).

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