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 a single assumption about the measurement device that is much weaker than the assumptions considered in Kochen-Specker contextuality.
 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.
 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.
 R. Augusiak, A. Salavrakos, J. Tura, and A. Acín. Bell inequalities tailored to the \textGreenberger–\textHorne–\textZeilinger states of arbitrary local dimension. New J. Phys., 21(11): 113001, 2019. DOI: 10.1088/1367-2630/ab4d9f.
 J. S. Bell. On the \textEinstein \textPodolsky \textRosen paradox. Physics Physique Fizika, 1: 195–200, 1964. DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
 C. Bamps and S. Pironio. Sum-of-squares decompositions for a family of \textClauser-\textHorne-\textShimony-\textHolt-like inequalities and their application to self-testing. Phys. Rev. A, 91: 052111, 2015. DOI: 10.1103/PhysRevA.91.052111.
 K. Bharti, M. Ray, A. Varvitsiotis, A. Cabello, and L. Kwek. Local certification of programmable quantum devices of arbitrary high dimensionality. 2019.
 K. Bharti, M. Ray, A. Varvitsiotis, N. Warsi, A. Cabello, and L. Kwek. Robust \textSelf-\textTesting of \textQuantum \textSystems via \textNoncontextuality \textInequalities. Phys. Rev. Lett., 122: 250403, 2019. DOI: 10.1103/PhysRevLett.122.250403.
 A. Cabello. Experimentally \textTestable \textState-\textIndependent \textQuantum \textContextuality. Phys. Rev. Lett., 101: 210401, 2008. DOI: 10.1103/PhysRevLett.101.210401.
 A. Cabello. Simple \textExplanation of the \textQuantum \textViolation of a \textFundamental \textInequality. Phys. Rev. Lett., 110: 060402, 2013. DOI: 10.1103/PhysRevLett.110.060402.
 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.
 D. Cui, A. Mehta, H. Mousavi, and S. Nezhadi. A generalization of \textCHSH and the algebraic structure of optimal strategies. 2019.
 A. Cabello, S. Severini, and A. Winter. Graph-\textTheoretic \textApproach to \textQuantum \textCorrelations. Phys. Rev. Lett., 112: 040401, 2014. DOI: 10.1103/PhysRevLett.112.040401.
 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.
 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.
 A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, P. Joshi, W. Kłobus, and A. Wójcik. Quantifying \textContextuality. Phys. Rev. Lett., 112: 120401, 2014. DOI: 10.1103/PhysRevLett.112.120401.
 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.
 A. Irfan, K. Mayer, G. Ortiz, and E. Knill. Certified quantum measurement of \textMajorana fermions. Phys. Rev. A, 101: 032106, 2020. DOI: 10.1103/PhysRevA.101.032106.
 J. Kaniewski. A weak form of self-testing. 2019.
 P. Kurzyński, A. Cabello, and D. Kaszlikowski. Fundamental \textMonogamy \textRelation between \textContextuality and \textNonlocality. Phys. Rev. Lett., 112: 100401, 2014. DOI: 10.1103/PhysRevLett.112.100401.
 A. Klyachko, M. Can, S. Binicioğlu, and A. Shumovsky. Simple \textTest for \textHidden \textVariables in \textSpin-1 \textSystems. Phys. Rev. Lett., 101: 020403, 2008. DOI: 10.1103/PhysRevLett.101.020403.
 S. Kochen and E. Specker. The \textProblem of \textHidden \textVariables in \textQuantum \textMechanics. 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.
 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.
 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.
 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.
 A. Salavrakos, R. Augusiak, J. Tura, P. Wittek, A. Acín, and S. Pironio. Bell \textInequalities \textTailored to \textMaximally \textEntangled \textStates. Phys. Rev. Lett., 119: 040402, 2017. DOI: 10.1103/PhysRevLett.119.040402.
 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.
 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.
 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.
 S. Sarkar, D. Saha, J. Kaniewski, and R. Augusiak. Self-testing quantum systems of arbitrary local dimension with minimal number of measurements. 2019.
 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.
 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.
 T. Yang, T. Vértesi, J. Bancal, V. Scarani, and M. Navascués. Robust and \textVersatile \textBlack-\textBox \textCertification of \textQuantum \textDevices. Phys. Rev. Lett., 113: 040401, 2014. DOI: 10.1103/PhysRevLett.113.040401.
 Owidiusz Makuta and Remigiusz Augusiak, "Self-testing maximally-dimensional genuinely entangled subspaces within the stabilizer formalism", New Journal of Physics 23 4, 043042 (2021).
 Li-Yi Hsu and Ching-Hsu Chen, "Exploring Bell nonlocality of quantum networks with stabilizing and logical operators", Physical Review Research 3 2, 023139 (2021).
 Debarshi Das, Ananda G. Maity, Debashis Saha, and A. S. Majumdar, "Robust certification of arbitrary outcome quantum measurements from temporal correlations", Quantum 6, 716 (2022).
 Rafael Santos, Chellasamy Jebarathinam, and Remigiusz Augusiak, "Scalable noncontextuality inequalities and certification of multiqubit quantum systems", Physical Review A 106 1, 012431 (2022).
 Shashank Gupta, Debashis Saha, Zhen-Peng Xu, Adán Cabello, and A. S. Majumdar, "Quantum Contextuality Provides Communication Complexity Advantage", Physical Review Letters 130 8, 080802 (2023).
 Ananda G. Maity, Shiladitya Mal, Chellasamy Jebarathinam, and A. S. Majumdar, "Self-testing of binary Pauli measurements requiring neither entanglement nor any dimensional restriction", Physical Review A 103 6, 062604 (2021).
 Adel Sohbi, Damian Markham, Jaewan Kim, and Marco Túlio Quintino, "Certifying dimension of quantum systems by sequential projective measurements", Quantum 5, 472 (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-27 19:43:03) and SAO/NASA ADS (last updated successfully 2023-09-27 19:43:04). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.