Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices

Debashis Saha, Rafael Santos, and Remigiusz Augusiak

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.


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.

[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.

[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.

[4] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1: 195–200, 1964. DOI: 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.

[6] K. Bharti, M. Ray, A. Varvitsiotis, A. Cabello, and L. Kwek. Local certification of programmable quantum devices of arbitrary high dimensionality. 2019. https:/​/​​abs/​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.

[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.

[9] P. Busch. Unsharp reality and joint measurements for spin observables. Phys. Rev. D, 33: 2253–2261, 1986. DOI: 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20] J. Kaniewski. A weak form of self-testing. 2019. https:/​/​​abs/​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.

[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.

[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.

[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.

[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.

[26] D. Mayers and A. Yao. Self testing quantum apparatus. Quantum Inf. Comput., 4(4): 273–286, 2004. DOI:​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.

[28] R. Raussendorf. Contextuality in measurement-based quantum computation. Phys. Rev. A, 88: 022322, 2013. DOI: 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.

[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.

[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.

[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.

[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.

[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:/​/​​abs/​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.

[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.

[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.

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2020-09-22 11:12:28). On SAO/NASA ADS no data on citing works was found (last attempt 2020-09-22 11:12:28).