Efficient Computation of Generalized Noncontextual Polytopes and Quantum violation of their Facet Inequalities
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AbstractFinding a set of empirical criteria fulfilled by any theory satisfying the generalized notion of noncontextuality is a challenging task of both operational and foundational importance. This work presents a methodology for constructing the noncontextual polytope while ensuring that the dimension of the polytope associated with the preparations remains constant regardless of the number of measurements and their outcome size. The facet inequalities of the noncontextual polytope can thus be obtained in a computationally efficient manner. We illustrate the efficacy of our methodology through several distinct contextuality scenarios. Our investigation uncovers several hitherto unexplored noncontextuality inequalities and demonstrates applications of quantum contextual correlations in certification of non-projective measurements, witnessing the dimension of quantum systems, and randomness certification.► BibTeX data@article{Hazra2026efficient, doi = {10.22331/q-2026-03-09-2015}, url = {https://doi.org/10.22331/q-2026-03-09-2015}, title = {Efficient {C}omputation of {G}eneralized {N}oncontextual {P}olytopes and {Q}uantum violation of their {F}acet {I}nequalities}, author = {Hazra, Soumyabrata and Saha, Debashis and Chaturvedi, Anubhav and Bera, Subhankar and Majumdar, A. S.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2015}, month = mar, year = {2026} }► References [1] A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai. Parity oblivious d-level random access codes and class of noncontextuality inequalities.
Quantum Information Processing, 18(4): 111, 2019. DOI: 10.1007/s11128-019-2228-3. https://doi.org/10.1007/s11128-019-2228-3 [2] D. Avis and K. Fukuda. Reverse search for enumeration.
Discrete Applied Mathematics, 65(1-3): 21–46, 1996. DOI: 10.1016/0166-218X(95)00026-N. https://doi.org/10.1016/0166-218X(95)00026-N [3] E. D. Andersen. Certificates of primal or dual infeasibility in linear programming. Computational Optimization and Applications, 20(2): 171–183, 2001. [4] C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-A. Larsson. Kochen-specker contextuality. Rev. Mod. Phys., 94: 045007, 2022. DOI: 10.1103/RevModPhys.94.045007. https://doi.org/10.1103/RevModPhys.94.045007 [5] A. Chaturvedi, M. Farkas, and V. J. Wright. Characterising and bounding the set of quantum behaviours in contextuality scenarios. Quantum, 5: 484, 2021. DOI: 10.22331/q-2021-06-29-484. https://doi.org/10.22331/q-2021-06-29-484 [6] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora. Optimal bounds for parity-oblivious random access codes. New Journal of Physics, 18(4): 045003, 2016. DOI: 10.1088/1367-2630/18/4/045003. https://doi.org/10.1088/1367-2630/18/4/045003 [7] L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens. What is nonclassical about uncertainty relations? Phys. Rev. Lett., 129: 240401, 2022. DOI: 10.1103/PhysRevLett.129.240401. https://doi.org/10.1103/PhysRevLett.129.240401 [8] L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens. Aspects of the phenomenology of interference that are genuinely nonclassical. Phys. Rev. A, 108: 022207, 2023. DOI: 10.1103/PhysRevA.108.022207. https://doi.org/10.1103/PhysRevA.108.022207 [9] L. Catani, M. Leifer, D. Schmid, and R. W. Spekkens. Why interference phenomena do not capture the essence of quantum theory. Quantum, 7: 1119, 2023. DOI: 10.22331/q-2023-09-25-1119. https://doi.org/10.22331/q-2023-09-25-1119 [10] https://github.com/soumya-s3/noncontextual polytope and quantum advantage, 2024. Online: https://github.com/soumya-s3/NonContextual_polytope_and_quantum_advantage. https://github.com/soumya-s3/NonContextual_polytope_and_quantum_advantage [11] A. Chaturvedi, M. Pawłowski, and D. Saha. Quantum description of reality is empirically incomplete, 2021. [12] A. Chaturvedi and D. Saha. Quantum prescriptions are more ontologically distinct than they are operationally distinguishable. Quantum, 4: 345, 2020. DOI: 10.22331/q-2020-10-21-345. https://doi.org/10.22331/q-2020-10-21-345 [13] K. Flatt, H. Lee, C. R. I. Carceller, J. B. Brask, and J. Bae. Contextual advantages and certification for maximum-confidence discrimination. PRX Quantum, 3: 030337, 2022. DOI: 10.1103/PRXQuantum.3.030337. https://doi.org/10.1103/PRXQuantum.3.030337 [14] O. Gühne, C. Budroni, A. Cabello, M. Kleinmann, and J.-Å. 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 [15] S. Ghorai and A. K. Pan. Optimal quantum preparation contextuality in an $n$-bit parity-oblivious multiplexing task. Phys. Rev. A, 98: 032110, 2018. DOI: 10.1103/PhysRevA.98.032110. https://doi.org/10.1103/PhysRevA.98.032110 [16] S. Gupta, D. Saha, Z.-P. Xu, A. Cabello, and A. S. Majumdar. Quantum contextuality provides communication complexity advantage. Phys. Rev. Lett., 130(8): 080802, 2023. Online: https://doi.org/10.1103/PhysRevLett.130.080802. https://doi.org/10.1103/PhysRevLett.130.080802 [17] T. Giordani, R. Wagner, C. Esposito, A. Camillini, F. Hoch, G. Carvacho, C. Pentangelo, F. Ceccarelli, S. Piacentini, A. Crespi, N. Spagnolo, R. Osellame, E. F. Galvão, and F. Sciarrino. Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor. Science Advances, 9(44): eadj4249, 2023. DOI: 10.1126/sciadv.adj4249. https://doi.org/10.1126/sciadv.adj4249 [18] M. Henk, J. Richter-Gebert, and G. M. Ziegler. Basic properties of convex polytopes. In Handbook of discrete and computational geometry, pages 383–413. Chapman and Hall/CRC, 2017. [19] N. Harrigan and R. W. Spekkens. Einstein, incompleteness, and the epistemic view of quantum states. Foundations of Physics, 40(2): 125–157, 2010. DOI: 10.1007/s10701-009-9347-0. https://doi.org/10.1007/s10701-009-9347-0 [20] A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane. Communication games reveal preparation contextuality. Phys. Rev. Lett., 119: 220402, 2017. DOI: 10.1103/PhysRevLett.119.220402. https://doi.org/10.1103/PhysRevLett.119.220402 [21] M. Howard, J. Wallman, V. Veitch, and J. Emerson. Contextuality supplies the ‘magic’for quantum computation. Nature, 510(7505): 351–355, 2014. Online: https://doi.org/10.1038/nature13460. https://doi.org/10.1038/nature13460 [22] M. S. Leifer. Is the quantum state real? an extended review of $\psi$-ontology theorems. Quanta, 3: 67–155, 2014. DOI: 10.12743/quanta.v3i1.22. https://doi.org/10.12743/quanta.v3i1.22 [23] M. Lostaglio. Quantum fluctuation theorems, contextuality, and work quasiprobabilities. Phys. Rev. Lett., 120: 040602, 2018. DOI: 10.1103/PhysRevLett.120.040602. https://doi.org/10.1103/PhysRevLett.120.040602 [24] M. Lostaglio and G. Senno. Contextual advantage for state-dependent cloning. Quantum, 4: 258, 2020. DOI: 10.22331/q-2020-04-27-258. https://doi.org/10.22331/q-2020-04-27-258 [25] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Physics Reports, 506(1): 1–39, 2011. Online: https://doi.org/10.1016/j.physrep.2011.05.001. https://doi.org/10.1016/j.physrep.2011.05.001 [26] D. G. Luenberger, Y. Ye, et al. Linear and nonlinear programming, volume 2. Springer, 1984. [27] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens. An experimental test of noncontextuality without unphysical idealizations. Nature communications, 7: 11780, 2016. DOI: 10.1038/ncomms11780. https://doi.org/10.1038/ncomms11780 [28] M. Navascués, S. Pironio, and A. Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98: 010401, 2007. DOI: 10.1103/PhysRevLett.98.010401. https://doi.org/10.1103/PhysRevLett.98.010401 [29] M. F. Pusey. Anomalous weak values are proofs of contextuality. Phys. Rev. Lett., 113: 200401, 2014. DOI: 10.1103/PhysRevLett.113.200401. https://doi.org/10.1103/PhysRevLett.113.200401 [30] M. F. Pusey. Robust preparation noncontextuality inequalities in the simplest scenario. Phys. Rev. A, 98: 022112, 2018. DOI: 10.1103/PhysRevA.98.022112. https://doi.org/10.1103/PhysRevA.98.022112 [31] K. F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome bell inequality using infinite-dimensional quantum systems. Phys. Rev. A, 82: 022116, 2010. DOI: 10.1103/PhysRevA.82.022116. https://doi.org/10.1103/PhysRevA.82.022116 [32] C. Roch i Carceller, K. Flatt, H. Lee, J. Bae, and J. B. Brask. Quantum vs noncontextual semi-device-independent randomness certification. Phys. Rev. Lett., 129: 050501, 2022. DOI: 10.1103/PhysRevLett.129.050501. https://doi.org/10.1103/PhysRevLett.129.050501 [33] 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 [34] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde. Preparation contextuality powers parity-oblivious multiplexing. Phys. Rev. Lett., 102: 010401, 2009. DOI: 10.1103/PhysRevLett.102.010401. https://doi.org/10.1103/PhysRevLett.102.010401 [35] D. Schmid, R. D. Baldijão, J. H. Selby, A. B. Sainz, and R. W. Spekkens. Noncontextuality inequalities for prepare-transform-measure scenarios, 2024. Online: https://arxiv.org/abs/2407.09624. arXiv:2407.09624 [36] D. Saha and A. Chaturvedi. Preparation contextuality as an essential feature underlying quantum communication advantage. Phys. Rev. A, 100: 022108, 2019. DOI: 10.1103/PhysRevA.100.022108. https://doi.org/10.1103/PhysRevA.100.022108 [37] D. Schmid, H. Du, J. H. Selby, and M. F. Pusey. Uniqueness of noncontextual models for stabilizer subtheories. Phys. Rev. Lett., 129: 120403, 2022. DOI: 10.1103/PhysRevLett.129.120403. https://doi.org/10.1103/PhysRevLett.129.120403 [38] 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 [39] R. W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A, 71: 052108, 2005. DOI: 10.1103/PhysRevA.71.052108. https://doi.org/10.1103/PhysRevA.71.052108 [40] R. W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A, 75: 032110, 2007. DOI: 10.1103/PhysRevA.75.032110. https://doi.org/10.1103/PhysRevA.75.032110 [41] R. W. Spekkens. Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett., 101: 020401, 2008. DOI: 10.1103/PhysRevLett.101.020401. https://doi.org/10.1103/PhysRevLett.101.020401 [42] R. W. Spekkens. The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of einstein, 2019. Online: https://arxiv.org/abs/1909.04628. arXiv:1909.04628 [43] D. Schmid and R. W. Spekkens. Contextual advantage for state discrimination. Phys. Rev. X, 8: 011015, 2018. DOI: 10.1103/PhysRevX.8.011015. https://doi.org/10.1103/PhysRevX.8.011015 [44] D. Schmid, J. H. Selby, M. F. Pusey, and R. W. Spekkens. A structure theorem for generalized-noncontextual ontological models, 2020. DOI: 10.48550/ARXIV.2005.07161. https://doi.org/10.48550/ARXIV.2005.07161 [45] D. Schmid, J. H. Selby, and R. W. Spekkens. Unscrambling the omelette of causation and inference: The framework of causal-inferential theories, 2020. DOI: 10.48550/ARXIV.2009.03297. https://doi.org/10.48550/ARXIV.2009.03297 [46] D. Schmid, R. W. Spekkens, and E. Wolfe. All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences. Phys. Rev. A, 97: 062103, 2018. DOI: 10.1103/PhysRevA.97.062103. https://doi.org/10.1103/PhysRevA.97.062103 [47] J. H. Selby, E. Wolfe, D. Schmid, and A. B. Sainz. An open-source linear program for testing nonclassicality. arXiv preprint arXiv:2204.11905, 2022. Online: https://doi.org/10.1103/PhysRevLett.132.050202. https://doi.org/10.1103/PhysRevLett.132.050202 arXiv:2204.11905 [48] A. Tavakoli, E. Z. Cruzeiro, R. Uola, and A. A. Abbott. Bounding and simulating contextual correlations in quantum theory. PRX Quantum, 2: 020334, 2021. DOI: 10.1103/PRXQuantum.2.020334. https://doi.org/10.1103/PRXQuantum.2.020334 [49] R. Wagner, R. S. Barbosa, and E. F. Galvão. Inequalities witnessing coherence, nonlocality, and contextuality. Physical Review A, 109(3), 2024. DOI: 10.1103/physreva.109.032220. https://doi.org/10.1103/physreva.109.032220 [50] R. Wagner, A. Camillini, and E. F. Galvão. Coherence and contextuality in a Mach-Zehnder interferometer. Quantum, 8: 1240, 2024. DOI: 10.22331/q-2024-02-05-1240. https://doi.org/10.22331/q-2024-02-05-1240 [51] V. J. Wright and M. Farkas. Invertible map between bell nonlocal and contextuality scenarios. Phys. Rev. Lett., 131: 220202, 2023. DOI: 10.1103/PhysRevLett.131.220202. https://doi.org/10.1103/PhysRevLett.131.220202 [52] Z.-P. Xu, D. Saha, H.-Y. Su, M. Pawłowski, and J.-L. 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.94.062103Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-09 11:31:13: Could not fetch cited-by data for 10.22331/q-2026-03-09-2015 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-09 11:31:13: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractFinding a set of empirical criteria fulfilled by any theory satisfying the generalized notion of noncontextuality is a challenging task of both operational and foundational importance. This work presents a methodology for constructing the noncontextual polytope while ensuring that the dimension of the polytope associated with the preparations remains constant regardless of the number of measurements and their outcome size. The facet inequalities of the noncontextual polytope can thus be obtained in a computationally efficient manner. We illustrate the efficacy of our methodology through several distinct contextuality scenarios. Our investigation uncovers several hitherto unexplored noncontextuality inequalities and demonstrates applications of quantum contextual correlations in certification of non-projective measurements, witnessing the dimension of quantum systems, and randomness certification.► BibTeX data@article{Hazra2026efficient, doi = {10.22331/q-2026-03-09-2015}, url = {https://doi.org/10.22331/q-2026-03-09-2015}, title = {Efficient {C}omputation of {G}eneralized {N}oncontextual {P}olytopes and {Q}uantum violation of their {F}acet {I}nequalities}, author = {Hazra, Soumyabrata and Saha, Debashis and Chaturvedi, Anubhav and Bera, Subhankar and Majumdar, A. S.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2015}, month = mar, year = {2026} }► References [1] A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai. Parity oblivious d-level random access codes and class of noncontextuality inequalities.
Quantum Information Processing, 18(4): 111, 2019. DOI: 10.1007/s11128-019-2228-3. https://doi.org/10.1007/s11128-019-2228-3 [2] D. Avis and K. Fukuda. Reverse search for enumeration.
Discrete Applied Mathematics, 65(1-3): 21–46, 1996. DOI: 10.1016/0166-218X(95)00026-N. https://doi.org/10.1016/0166-218X(95)00026-N [3] E. D. Andersen. Certificates of primal or dual infeasibility in linear programming. Computational Optimization and Applications, 20(2): 171–183, 2001. [4] C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-A. Larsson. Kochen-specker contextuality. Rev. Mod. Phys., 94: 045007, 2022. DOI: 10.1103/RevModPhys.94.045007. https://doi.org/10.1103/RevModPhys.94.045007 [5] A. Chaturvedi, M. Farkas, and V. J. Wright. Characterising and bounding the set of quantum behaviours in contextuality scenarios. Quantum, 5: 484, 2021. DOI: 10.22331/q-2021-06-29-484. https://doi.org/10.22331/q-2021-06-29-484 [6] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora. Optimal bounds for parity-oblivious random access codes. New Journal of Physics, 18(4): 045003, 2016. DOI: 10.1088/1367-2630/18/4/045003. https://doi.org/10.1088/1367-2630/18/4/045003 [7] L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens. What is nonclassical about uncertainty relations? Phys. Rev. Lett., 129: 240401, 2022. DOI: 10.1103/PhysRevLett.129.240401. https://doi.org/10.1103/PhysRevLett.129.240401 [8] L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens. Aspects of the phenomenology of interference that are genuinely nonclassical. Phys. Rev. A, 108: 022207, 2023. DOI: 10.1103/PhysRevA.108.022207. https://doi.org/10.1103/PhysRevA.108.022207 [9] L. Catani, M. Leifer, D. Schmid, and R. W. Spekkens. Why interference phenomena do not capture the essence of quantum theory. Quantum, 7: 1119, 2023. DOI: 10.22331/q-2023-09-25-1119. https://doi.org/10.22331/q-2023-09-25-1119 [10] https://github.com/soumya-s3/noncontextual polytope and quantum advantage, 2024. Online: https://github.com/soumya-s3/NonContextual_polytope_and_quantum_advantage. https://github.com/soumya-s3/NonContextual_polytope_and_quantum_advantage [11] A. Chaturvedi, M. Pawłowski, and D. Saha. Quantum description of reality is empirically incomplete, 2021. [12] A. Chaturvedi and D. Saha. Quantum prescriptions are more ontologically distinct than they are operationally distinguishable. Quantum, 4: 345, 2020. DOI: 10.22331/q-2020-10-21-345. https://doi.org/10.22331/q-2020-10-21-345 [13] K. Flatt, H. Lee, C. R. I. Carceller, J. B. Brask, and J. Bae. Contextual advantages and certification for maximum-confidence discrimination. PRX Quantum, 3: 030337, 2022. DOI: 10.1103/PRXQuantum.3.030337. https://doi.org/10.1103/PRXQuantum.3.030337 [14] O. Gühne, C. Budroni, A. Cabello, M. Kleinmann, and J.-Å. 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 [15] S. Ghorai and A. K. Pan. Optimal quantum preparation contextuality in an $n$-bit parity-oblivious multiplexing task. Phys. Rev. A, 98: 032110, 2018. DOI: 10.1103/PhysRevA.98.032110. https://doi.org/10.1103/PhysRevA.98.032110 [16] S. Gupta, D. Saha, Z.-P. Xu, A. Cabello, and A. S. Majumdar. Quantum contextuality provides communication complexity advantage. Phys. Rev. Lett., 130(8): 080802, 2023. Online: https://doi.org/10.1103/PhysRevLett.130.080802. https://doi.org/10.1103/PhysRevLett.130.080802 [17] T. Giordani, R. Wagner, C. Esposito, A. Camillini, F. Hoch, G. Carvacho, C. Pentangelo, F. Ceccarelli, S. Piacentini, A. Crespi, N. Spagnolo, R. Osellame, E. F. Galvão, and F. Sciarrino. Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor. Science Advances, 9(44): eadj4249, 2023. DOI: 10.1126/sciadv.adj4249. https://doi.org/10.1126/sciadv.adj4249 [18] M. Henk, J. Richter-Gebert, and G. M. Ziegler. Basic properties of convex polytopes. In Handbook of discrete and computational geometry, pages 383–413. Chapman and Hall/CRC, 2017. [19] N. Harrigan and R. W. Spekkens. Einstein, incompleteness, and the epistemic view of quantum states. Foundations of Physics, 40(2): 125–157, 2010. DOI: 10.1007/s10701-009-9347-0. https://doi.org/10.1007/s10701-009-9347-0 [20] A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane. Communication games reveal preparation contextuality. Phys. Rev. Lett., 119: 220402, 2017. DOI: 10.1103/PhysRevLett.119.220402. https://doi.org/10.1103/PhysRevLett.119.220402 [21] M. Howard, J. Wallman, V. Veitch, and J. Emerson. Contextuality supplies the ‘magic’for quantum computation. Nature, 510(7505): 351–355, 2014. Online: https://doi.org/10.1038/nature13460. https://doi.org/10.1038/nature13460 [22] M. S. Leifer. Is the quantum state real? an extended review of $\psi$-ontology theorems. Quanta, 3: 67–155, 2014. DOI: 10.12743/quanta.v3i1.22. https://doi.org/10.12743/quanta.v3i1.22 [23] M. Lostaglio. Quantum fluctuation theorems, contextuality, and work quasiprobabilities. Phys. Rev. Lett., 120: 040602, 2018. DOI: 10.1103/PhysRevLett.120.040602. https://doi.org/10.1103/PhysRevLett.120.040602 [24] M. Lostaglio and G. Senno. Contextual advantage for state-dependent cloning. Quantum, 4: 258, 2020. DOI: 10.22331/q-2020-04-27-258. https://doi.org/10.22331/q-2020-04-27-258 [25] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Physics Reports, 506(1): 1–39, 2011. Online: https://doi.org/10.1016/j.physrep.2011.05.001. https://doi.org/10.1016/j.physrep.2011.05.001 [26] D. G. Luenberger, Y. Ye, et al. Linear and nonlinear programming, volume 2. Springer, 1984. [27] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens. An experimental test of noncontextuality without unphysical idealizations. Nature communications, 7: 11780, 2016. DOI: 10.1038/ncomms11780. https://doi.org/10.1038/ncomms11780 [28] M. Navascués, S. Pironio, and A. Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98: 010401, 2007. DOI: 10.1103/PhysRevLett.98.010401. https://doi.org/10.1103/PhysRevLett.98.010401 [29] M. F. Pusey. Anomalous weak values are proofs of contextuality. Phys. Rev. Lett., 113: 200401, 2014. DOI: 10.1103/PhysRevLett.113.200401. https://doi.org/10.1103/PhysRevLett.113.200401 [30] M. F. Pusey. Robust preparation noncontextuality inequalities in the simplest scenario. Phys. Rev. A, 98: 022112, 2018. DOI: 10.1103/PhysRevA.98.022112. https://doi.org/10.1103/PhysRevA.98.022112 [31] K. F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome bell inequality using infinite-dimensional quantum systems. Phys. Rev. A, 82: 022116, 2010. DOI: 10.1103/PhysRevA.82.022116. https://doi.org/10.1103/PhysRevA.82.022116 [32] C. Roch i Carceller, K. Flatt, H. Lee, J. Bae, and J. B. Brask. Quantum vs noncontextual semi-device-independent randomness certification. Phys. Rev. Lett., 129: 050501, 2022. DOI: 10.1103/PhysRevLett.129.050501. https://doi.org/10.1103/PhysRevLett.129.050501 [33] 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 [34] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde. Preparation contextuality powers parity-oblivious multiplexing. Phys. Rev. Lett., 102: 010401, 2009. DOI: 10.1103/PhysRevLett.102.010401. https://doi.org/10.1103/PhysRevLett.102.010401 [35] D. Schmid, R. D. Baldijão, J. H. Selby, A. B. Sainz, and R. W. Spekkens. Noncontextuality inequalities for prepare-transform-measure scenarios, 2024. Online: https://arxiv.org/abs/2407.09624. arXiv:2407.09624 [36] D. Saha and A. Chaturvedi. Preparation contextuality as an essential feature underlying quantum communication advantage. Phys. Rev. A, 100: 022108, 2019. DOI: 10.1103/PhysRevA.100.022108. https://doi.org/10.1103/PhysRevA.100.022108 [37] D. Schmid, H. Du, J. H. Selby, and M. F. Pusey. Uniqueness of noncontextual models for stabilizer subtheories. Phys. Rev. Lett., 129: 120403, 2022. DOI: 10.1103/PhysRevLett.129.120403. https://doi.org/10.1103/PhysRevLett.129.120403 [38] 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 [39] R. W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A, 71: 052108, 2005. DOI: 10.1103/PhysRevA.71.052108. https://doi.org/10.1103/PhysRevA.71.052108 [40] R. W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A, 75: 032110, 2007. DOI: 10.1103/PhysRevA.75.032110. https://doi.org/10.1103/PhysRevA.75.032110 [41] R. W. Spekkens. Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett., 101: 020401, 2008. DOI: 10.1103/PhysRevLett.101.020401. https://doi.org/10.1103/PhysRevLett.101.020401 [42] R. W. Spekkens. The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of einstein, 2019. Online: https://arxiv.org/abs/1909.04628. arXiv:1909.04628 [43] D. Schmid and R. W. Spekkens. Contextual advantage for state discrimination. Phys. Rev. X, 8: 011015, 2018. DOI: 10.1103/PhysRevX.8.011015. https://doi.org/10.1103/PhysRevX.8.011015 [44] D. Schmid, J. H. Selby, M. F. Pusey, and R. W. Spekkens. A structure theorem for generalized-noncontextual ontological models, 2020. DOI: 10.48550/ARXIV.2005.07161. https://doi.org/10.48550/ARXIV.2005.07161 [45] D. Schmid, J. H. Selby, and R. W. Spekkens. Unscrambling the omelette of causation and inference: The framework of causal-inferential theories, 2020. DOI: 10.48550/ARXIV.2009.03297. https://doi.org/10.48550/ARXIV.2009.03297 [46] D. Schmid, R. W. Spekkens, and E. Wolfe. All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences. Phys. Rev. A, 97: 062103, 2018. DOI: 10.1103/PhysRevA.97.062103. https://doi.org/10.1103/PhysRevA.97.062103 [47] J. H. Selby, E. Wolfe, D. Schmid, and A. B. Sainz. An open-source linear program for testing nonclassicality. arXiv preprint arXiv:2204.11905, 2022. Online: https://doi.org/10.1103/PhysRevLett.132.050202. https://doi.org/10.1103/PhysRevLett.132.050202 arXiv:2204.11905 [48] A. Tavakoli, E. Z. Cruzeiro, R. Uola, and A. A. Abbott. Bounding and simulating contextual correlations in quantum theory. PRX Quantum, 2: 020334, 2021. DOI: 10.1103/PRXQuantum.2.020334. https://doi.org/10.1103/PRXQuantum.2.020334 [49] R. Wagner, R. S. Barbosa, and E. F. Galvão. Inequalities witnessing coherence, nonlocality, and contextuality. Physical Review A, 109(3), 2024. DOI: 10.1103/physreva.109.032220. https://doi.org/10.1103/physreva.109.032220 [50] R. Wagner, A. Camillini, and E. F. Galvão. Coherence and contextuality in a Mach-Zehnder interferometer. Quantum, 8: 1240, 2024. DOI: 10.22331/q-2024-02-05-1240. https://doi.org/10.22331/q-2024-02-05-1240 [51] V. J. Wright and M. Farkas. Invertible map between bell nonlocal and contextuality scenarios. Phys. Rev. Lett., 131: 220202, 2023. DOI: 10.1103/PhysRevLett.131.220202. https://doi.org/10.1103/PhysRevLett.131.220202 [52] Z.-P. Xu, D. Saha, H.-Y. Su, M. Pawłowski, and J.-L. 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.94.062103Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-09 11:31:13: Could not fetch cited-by data for 10.22331/q-2026-03-09-2015 from Crossref. This is normal if the DOI was registered recently. 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