Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Violation of a noncontextuality inequality or the phenomenon referred to `quantum contextuality’ is a fundamental feature of quantum theory. In this article, we derive a novel family of noncontextuality inequalities along with their sum-of-squares decompositions in the simplest (odd-cycle) sequential-measurement scenario capable to demonstrate Kochen-Specker contextuality. The sum-of-squares decompositions allow us to obtain the maximal quantum violation of these inequalities and a set of algebraic relations necessarily satisfied by any state and measurements achieving it. With their help, we prove that our inequalities can be used for self-testing of three-dimensional quantum state and measurements. Remarkably, the presented self-testing results rely on weaker assumptions than the ones considered in Kochen-Specker contextuality.
► BibTeX data
► References
[1] B. Amaral and M. T. Cunha. Contextuality: The Compatibility-Hypergraph Approach, pages 13–48. Springer Briefs in Mathematics. Springer, Cham, 2018. DOI: 10.1007/978-3-319-93827-1_2.
https://doi.org/10.1007/978-3-319-93827-1_2
[2] M. Araújo, M. T. Quintino, C. Budroni, M. T. Cunha, and A. Cabello. All noncontextuality inequalities for the $n$-cycle scenario. Phys. Rev. A, 88: 022118, 2013. DOI: 10.1103/PhysRevA.88.022118.
https://doi.org/10.1103/PhysRevA.88.022118
[3] R. Augusiak, A. Salavrakos, J. Tura, and A. Acín. Bell inequalities tailored to the Greenberger-Horne-Zeilinger states of arbitrary local dimension. New J. Phys., 21(11): 113001, 2019. DOI: 10.1088/1367-2630/ab4d9f.
https://doi.org/10.1088/1367-2630/ab4d9f
[4] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1: 195–200, 1964. DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[5] C. Bamps and S. Pironio. Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing. Phys. Rev. A, 91: 052111, 2015. DOI: 10.1103/PhysRevA.91.052111.
https://doi.org/10.1103/PhysRevA.91.052111
[6] K. Bharti, M. Ray, A. Varvitsiotis, A. Cabello, and L. Kwek. Local certification of programmable quantum devices of arbitrary high dimensionality. 2019. https://arxiv.org/abs/1911.09448.
arXiv:1911.09448
[7] K. Bharti, M. Ray, A. Varvitsiotis, N. Warsi, A. Cabello, and L. Kwek. Robust Self-Testing of Quantum Systems via Noncontextuality Inequalities. Phys. Rev. Lett., 122: 250403, 2019. DOI: 10.1103/PhysRevLett.122.250403.
https://doi.org/10.1103/PhysRevLett.122.250403
[8] P. Busch and J. Singh. Lüders theorem for unsharp quantum measurements. Physics Letters A, 249(1): 10–12, 1998. DOI: 10.1016/S0375-9601(98)00704-X.
https://doi.org/10.1016/S0375-9601(98)00704-X
[9] P. Busch. Unsharp reality and joint measurements for spin observables. Phys. Rev. D, 33: 2253–2261, 1986. DOI: 10.1103/PhysRevD.33.2253.
https://doi.org/10.1103/PhysRevD.33.2253
[10] A. Cabello. Experimentally Testable State-Independent Quantum Contextuality. Phys. Rev. Lett., 101: 210401, 2008. DOI: 10.1103/PhysRevLett.101.210401.
https://doi.org/10.1103/PhysRevLett.101.210401
[11] A. Cabello. Simple Explanation of the Quantum Violation of a Fundamental Inequality. Phys. Rev. Lett., 110: 060402, 2013. DOI: 10.1103/PhysRevLett.110.060402.
https://doi.org/10.1103/PhysRevLett.110.060402
[12] A. Coladangelo, K. Goh, and V. Scarani. All pure bipartite entangled states can be self-tested. Nature Communications, 8(1): 15485, 2017. DOI: 10.1038/ncomms15485.
https://doi.org/10.1038/ncomms15485
[13] D. Cui, A. Mehta, H. Mousavi, and S. Nezhadi. A generalization of CHSH and the algebraic structure of optimal strategies. 2019.
[14] A. Cabello, S. Severini, and A. Winter. Graph-Theoretic Approach to Quantum Correlations. Phys. Rev. Lett., 112: 040401, 2014. DOI: 10.1103/PhysRevLett.112.040401.
https://doi.org/10.1103/PhysRevLett.112.040401
[15] M. Farkas and J. Kaniewski. Self-testing mutually unbiased bases in the prepare-and-measure scenario. Phys. Rev. A, 99: 032316, 2019. DOI: 10.1103/PhysRevA.99.032316.
https://doi.org/10.1103/PhysRevA.99.032316
[16] O. Gühne, C. Budroni, A. Cabello, M. Kleinmann, and J. Larsson. Bounding the quantum dimension with contextuality. Phys. Rev. A, 89: 062107, 2014. DOI: 10.1103/PhysRevA.89.062107.
https://doi.org/10.1103/PhysRevA.89.062107
[17] A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, P. Joshi, W. Kłobus, and A. Wójcik. Quantifying Contextuality. Phys. Rev. Lett., 112: 120401, 2014. DOI: 10.1103/PhysRevLett.112.120401.
https://doi.org/10.1103/PhysRevLett.112.120401
[18] M. Howard, J. Wallman, V. Veitch, and J. Emerson. Contextuality supplies the “magic” for quantum computation. Nature, 510(7505): 351–355, 2014. DOI: 10.1038/nature13460.
https://doi.org/10.1038/nature13460
[19] A. Irfan, K. Mayer, G. Ortiz, and E. Knill. Certified quantum measurement of Majorana fermions. Phys. Rev. A, 101: 032106, 2020. DOI: 10.1103/PhysRevA.101.032106.
https://doi.org/10.1103/PhysRevA.101.032106
[20] J. Kaniewski. A weak form of self-testing. 2019. https://arxiv.org/abs/1910.00706.
arXiv:1910.00706
[21] P. Kurzyński, A. Cabello, and D. Kaszlikowski. Fundamental Monogamy Relation between Contextuality and Nonlocality. Phys. Rev. Lett., 112: 100401, 2014. DOI: 10.1103/PhysRevLett.112.100401.
https://doi.org/10.1103/PhysRevLett.112.100401
[22] A. Klyachko, M. Can, S. Binicioğlu, and A. Shumovsky. Simple Test for Hidden Variables in Spin-1 Systems. Phys. Rev. Lett., 101: 020403, 2008. DOI: 10.1103/PhysRevLett.101.020403.
https://doi.org/10.1103/PhysRevLett.101.020403
[23] S. Kochen and E. Specker. The Problem of Hidden Variables in Quantum Mechanics. In The Logico-Algebraic Approach to Quantum Mechanics, The Western Ontario Series in Philosophy of Science, pages 293–328. Springer Netherlands, 1975. DOI: 10.1007/978-94-010-1795-4.
https://doi.org/10.1007/978-94-010-1795-4
[24] J. Kaniewski, I. Šupić, J. Tura, F. Baccari, A. Salavrakos, and R. Augusiak. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems. Quantum, 3: 198, 2019. DOI: 10.22331/q-2019-10-24-198.
https://doi.org/10.22331/q-2019-10-24-198
[25] Y. Liang, R. Spekkens, and H. Wiseman. Specker$’$s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Phys. Rep., 506(1): 1–39, 2011. DOI: 10.1016/j.physrep.2011.05.001.
https://doi.org/10.1016/j.physrep.2011.05.001
[26] D. Mayers and A. Yao. Self testing quantum apparatus. Quantum Inf. Comput., 4(4): 273–286, 2004. DOI: doi.org/10.26421/QIC4.4.
https://doi.org/10.26421/QIC4.4
[27] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys., 70: 101–144, 1998. DOI: 10.1103/RevModPhys.70.101.
https://doi.org/10.1103/RevModPhys.70.101
[28] R. Raussendorf. Contextuality in measurement-based quantum computation. Phys. Rev. A, 88: 022322, 2013. DOI: 10.1103/PhysRevA.88.022322.
https://doi.org/10.1103/PhysRevA.88.022322
[29] I. Šupić, R. Augusiak, A. Salavrakos, and A. Acín. Self-testing protocols based on the chained bell inequalities. New J. Phys., 18(3): 035013, 2016. DOI: 10.1088/1367-2630/18/3/035013.
https://doi.org/10.1088/1367-2630/18/3/035013
[30] A. Salavrakos, R. Augusiak, J. Tura, P. Wittek, A. Acín, and S. Pironio. Bell Inequalities Tailored to Maximally Entangled States. Phys. Rev. Lett., 119: 040402, 2017. DOI: 10.1103/PhysRevLett.119.040402.
https://doi.org/10.1103/PhysRevLett.119.040402
[31] J. Singh, K. Bharti, and Arvind. Quantum key distribution protocol based on contextuality monogamy. Phys. Rev. A, 95: 062333, 2017. DOI: 10.1103/PhysRevA.95.062333.
https://doi.org/10.1103/PhysRevA.95.062333
[32] D. Saha, P. Horodecki, and M. Pawłowski. State independent contextuality advances one-way communication. New J. Phys., 21(9): 093057, 2019. DOI: 10.1088/1367-2630/ab4149.
https://doi.org/10.1088/1367-2630/ab4149
[33] D. Saha and R. Ramanathan. Activation of monogamy in nonlocality using local contextuality. Phys. Rev. A, 95: 030104, 2017. DOI: 10.1103/PhysRevA.95.030104.
https://doi.org/10.1103/PhysRevA.95.030104
[34] S. Sarkar, D. Saha, J. Kaniewski, and R. Augusiak. Self-testing quantum systems of arbitrary local dimension with minimal number of measurements. 2019. https://arxiv.org/abs/1909.12722v2.
arXiv:1909.12722v2
[35] A. Tavakoli, J. Kaniewski, T. Vértesi, D. Rosset, and N. Brunner. Self-testing quantum states and measurements in the prepare-and-measure scenario. Phys. Rev. A, 98: 062307, 2018. DOI: 10.1103/PhysRevA.98.062307.
https://doi.org/10.1103/PhysRevA.98.062307
[36] Z. Xu, D. Saha, H. Su, M. Pawłowski, and J. Chen. Reformulating noncontextuality inequalities in an operational approach. Phys. Rev. A, 94: 062103, 2016. DOI: 10.1103/PhysRevA.94.062103.
https://doi.org/10.1103/PhysRevA.94.062103
[37] T. Yang, T. Vértesi, J. Bancal, V. Scarani, and M. Navascués. Robust and Versatile Black-Box Certification of Quantum Devices. Phys. Rev. Lett., 113: 040401, 2014. DOI: 10.1103/PhysRevLett.113.040401.
https://doi.org/10.1103/PhysRevLett.113.040401
Cited by
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Source: https://quantum-journal.org/papers/q-2020-08-03-302/