Inequality relations for the hierarchy of quantum correlations in two-qubit systems

Abstract

Entanglement, quantum steering and Bell nonlocality can be used to describe the distinct quantum correlations of quantum systems. Because of their different characteristics and application fields, how to divide them quantitatively and accurately becomes particularly important. Based on the sufficient and necessary criterion for quantum steering of an arbitrary two-qubit T-state, we derive the inequality relations between quantum steering and entanglement as well as between quantum steering and Bell nonlocality for the T-state. Additionally, we have verified those relations experimentally.

Graphical abstract

Keywords

entanglement / quantum steering / Bell nonlocality / inequality relation

References

[1] A. Einstein, B. Podolsky, N. Rosen. Can quantum-mechanical description of physical reality be considered complete. Phys. Rev., 1935, 47(10): 777

[2] E. Schrödinger. Discussion of relations between separated systems. Math. Proc. Camb. Philos. Soc., 1935, 31(4): 555

[3] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki. Quantum entanglement. Rev. Mod. Phys., 2009, 81(2): 865

[4] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 1998, 80(10): 2245

[5] M. Piani. Relative entropy of entanglement and restricted measurements. Phys. Rev. Lett., 2009, 103(16): 160504

[6] A. Miranowicz, A. Grudka. Ordering two-qubit states with concurrence and negativity. Phys. Rev. A, 2004, 70(3): 032326

[7] A. Ekert, R. Jozsa. Quantum computation and Shor’s factoring algorithm. Rev. Mod. Phys., 1996, 68(3): 733

[8] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 1993, 70(13): 1895

[9] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden. Quantum cryptography. Rev. Mod. Phys., 2002, 74(1): 145

[10] C. H. Bennett, D. P. DiVincenzo, J. Smolin, W. K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 1996, 54(5): 3824

[11] H. M. Wiseman, S. J. Jones, A. C. Doherty. Steering, entanglement, nonlocality, and the Einstein−Podolsky−Rosen paradox. Phys. Rev. Lett., 2007, 98(14): 140402

[12] R. Uola, A. C. S. Costa, H. C. Nguyen, O. Gühne. Quantum steering. Rev. Mod. Phys., 2020, 92(1): 015001

[13] J. S. Bell. On the Einstein−Podolsky−Rosen paradox. Physics, 1964, 1(3): 195

[14] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner. Bell nonlocality. Rev. Mod. Phys., 2014, 86(2): 419

[15] W. X. Zhong, G. L. Cheng, X. M. Hu. One-way Einstein−Podolsky−Rosen steering via atomic coherence. Opt. Express, 2017, 25(10): 11584

[16] C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, H. M. Wiseman. One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering. Phys. Rev. A, 2012, 85(1): 010301(R)

[17] B. Opanchuk, L. Arnaud, M. D. Reid. Detecting faked continuous-variable entanglement using one-sided device independent entanglement witnesses. Phys. Rev. A, 2014, 89(6): 062101

[18] N. Walk, S. Hosseini, J. Geng, O. Thearle, J. Y. Haw, S. Armstrong, S. M. Assad, J. Janousek, T. C. Ralph, T. Symul, H. M. Wiseman, P. K. Lam. Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution. Optica, 2016, 3(6): 634

[19] C. M. Zhang, M. Li, H. W. Li, Z. Q. Yin, D. Wang, J. Z. Huang, Y. G. Han, M. L. Xu, W. Chen, S. Wang, P. Treeviriyanupab, G. C. Guo, Z. F. Han. Decoy-state measurement-device independent quantum key distribution based on the Clauser−Horne−Shimony−Holt inequality. Phys. Rev. A, 2014, 90(3): 034302

[20] Č. Brukner, M. Żukowski, J. W. Pan, A. Zeilinger. Bell’s inequalities and quantum communication complexity. Phys. Rev. Lett., 2004, 92(12): 127901

[21] S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe. Random numbers certified by Bell’s theorem. Nature, 2010, 464(7291): 1021

[22] M. T. Quintino, T. Vertesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acín, N. Brunner. Inequivalence of entanglement, steering, and Bell nonlocality for general measurements. Phys. Rev. A, 2015, 92(3): 032107

[23] X. G. Fan, H. Yang, F. Ming, D. Wang, L. Ye. Constraint relation between steerability and concurrence for two-qubit states. Ann. Phys., 2021, 533(8): 2100098

[24] C. Chen, C. L. Ren, X. J. Ye, J. L. Chen. Mapping criteria between nonlocality and steerability in qudit−qubit systems and between steerability and entanglement in qubit-qudit systems. Phys. Rev. A, 2018, 98(5): 052114

[25] D. Das, S. Sasmal, S. Roy. Detecting Einstein−Podolsky−Rosen steering through entanglement detection. Phys. Rev. A, 2019, 99(5): 052109

[26] F. Verstraete, M. M. Wolf. Entanglement versus bell violations and their behavior under local filtering operations. Phys. Rev. Lett., 2002, 89(17): 170401

[27] K. Bartkiewicz, B. Horst, K. Lemr, A. Miranowicz. Entanglement estimation from Bell inequality violation. Phys. Rev. A, 2013, 88(5): 052105

[28] B. Horst, K. Bartkiewicz, A. Miranowicz. Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality. Phys. Rev. A, 2013, 87(4): 042108

[29] K. Bartkiewicz, K. Lemr, A. Černoch, A. Miranowicz. Bell nonlocality and fully entangled fraction measured in an entanglement-swapping device without quantum state tomography. Phys. Rev. A, 2017, 95(3): 030102(R)

[30] Z. F. Su, H. S. Tan, X. Y. Li. Entanglement as upper bound for the nonlocality of a general two-qubit system. Phys. Rev. A, 2020, 101(4): 042112

[31] M. Li, M. J. Zhao, S. M. Fei, Z. X. Wang. Experimental detection of quantum entanglement. Front. Phys., 2013, 8(4): 357

[32] Z. R. Zhong, X. Wang, W. Qin. Towards quantum entanglement of micromirrors via a two-level atom and radiation pressure. Front. Phys., 2018, 13(5): 130319

[33] Q. Dong, A. J. Torres-Arenas, G. H. Sun, W. C. Qiang, S. H. Dong. Entanglement measures of a new type pseudo-pure state in accelerated frames. Front. Phys., 2019, 14(2): 21603

[34] P. Zhang. Quantum entanglement in the Sachdev−Ye−Kitaev model and its generalizations. Front. Phys., 2022, 17(4): 43201

[35] Y. Y. Yang, W. Y. Sun, W. N. Shi, F. Ming, D. Wang, L. Ye. Dynamical characteristic of measurement uncertainty under Heisenberg spin models with Dzyaloshinskii−Moriya interactions. Front. Phys., 2019, 14(3): 31601

[36] L. Y. Cheng, F. Ming, F. Zhao, L. Ye, D. Wang. The uncertainty and quantum correlation of measurement in double quantum-dot systems. Front. Phys., 2022, 17(6): 61504

[37] Y. Cao, D. Wang, X. G. Fan, F. Ming, Z. Y. Wang, L. Ye. Complementary relation between quantum entanglement and entropic uncertainty. Commum. Theor. Phys., 2021, 73(1): 015101

[38] H. C. Nguyen, T. Vu. Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes. Europhys. Lett., 2016, 115(1): 10003

[39] H. C. Nguyen, H. V. Nguyen, O. Gühne. Geometry of Einstein−Podolsky−Rosen correlations. Phys. Rev. Lett., 2019, 122(24): 240401

[40] X. G. Fan, W. Y. Sun, Z. Y. Ding, H. Yang, F. Ming, D. Wang, L. Ye. Universal complementarity between coherence and intrinsic concurrence for two-qubit states. New J. Phys., 2019, 21(9): 093053

[41] S. Jevtic, M. J. W. Hall, M. R. Anderson, M. Zwierz, H. M. Wiseman. Einstein−Podolsky−Rosen steering and the steering ellipsoid. J. Opt. Soc. Am. B, 2015, 32(4): A40

[42] X.G. FanH. YangF.MingX.K. SongD.Wang L.Ye, Necessary and sufficient criterion of steering for two-qubit T states, arXiv: 2103.04280v1 (2021)

[43] R. Horodecki, P. Horodecki, M. Horodecki. Violating bell inequality by mixed spin-1/2 states: Necessary and sufficient condition. Phys. Lett. A, 1995, 200(5): 340

[44] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, P. H. Eberhard. Ultrabright source of polarization-entangled photons. Phys. Rev. A, 1999, 60(2): R773

[45] A. Aiello, G. Puentes, D. Voigt, J. P. Woerdman. Maximally entangled mixed-state generation via local operations. Phys. Rev. A, 2007, 75(6): 062118

Acknowledgements

We acknowledge Prof. Shu-Ming Cheng for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12175001 and 12075001).

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