Simplicial quantum contextuality

Cihan Okay, Aziz Kharoof, and Selman Ipek

Department of Mathematics, Bilkent University, Ankara, Turkey

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen--Specker theorem can be expressed naturally within this new language.

Contextuality is a fundamental feature of quantum mechanics in which measurement statistics cannot be reproduced by hidden variable models with pre-existing outcome assignments, a phenomenon which is captured by violations of Bell inequalities. We introduce a new topological framework for contextuality in which simplicial sets, central objects in modern homotopy theory, are used to model measurement statistics as distributions on spaces. Quantum statistics satisfy a principle known as nonsignaling, or nondisturbance. In this simplicial framework this compatibility property is encoded topologically as the gluing of simplices, giving rise to simplicial distributions. Topological notions, such as gluing and extension, can be used to systematize the analysis of contextuality, leading to novel characterizations of Bell inequalities. The theory of simplicial distributions unifies the sheaf-theoretic and the group cohomological approaches to contextuality and goes beyond by allowing for new scenarios and novel proof techniques that cannot be realized in earlier frameworks. Such a topological perspective brings new insight into the study of Bell inequalities.

► BibTeX data

► References

[1] E. Schrödinger, ``Discussion of probability relations between separated systems,'' Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 555–563, 1935. doi: 10.1017/​S0305004100013554.

[2] A. Einstein, B. Podolsky, and N. Rosen, ``Can quantum-mechanical description of physical reality be considered complete?,'' Physical Review, vol. 47, p. 777, 1935. doi: 10.1103/​PhysRev.47.777.

[3] J. S. Bell, ``On the Einstein Podolsky Rosen paradox,'' Physics Physique Fizika, vol. 1, pp. 195–200, Nov 1964. doi: 10.1103/​PhysicsPhysiqueFizika.1.195.

[4] S. Kochen and E. P. Specker, ``The problem of hidden variables in quantum mechanics,'' Journal of Mathematics and Mechanics, vol. 17, pp. 59–87, 1967. doi: 10.1007/​978-94-010-1795-4_17.

[5] J. S. Bell, ``On the problem of hidden variables in quantum mechanics,'' Reviews of Modern Physics, vol. 38, p. 447, 1966. doi: 10.1103/​RevModPhys.38.447.

[6] N. D. Mermin, ``Hidden variables and the two theorems of John Bell,'' Reviews of Modern Physics, vol. 65, no. 3, p. 803, 1993. doi: 10.1103/​RevModPhys.65.803.

[7] C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-Å. Larsson, ``Quantum contextuality,'' arXiv preprint arXiv:2102.13036, 2021. doi: 10.48550/​arXiv.2102.13036. arXiv: 2102.13036.

[8] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality,'' Review of Modern Physics, vol. 86, p. 419, 2014. doi: 10.1103/​RevModPhys.86.419. arXiv: 1303.2849.

[9] B. Amaral, ``Resource theory of contextuality,'' Philosophical Transactions of the Royal Society A, vol. 377, no. 2157, p. 20190010, 2019. doi: 10.1098/​rsta.2019.0010. arXiv: 1904.04182.

[10] R. W. Spekkens, ``Contextuality for preparations, transformations, and unsharp measurements,'' Physical Review A, vol. 71, p. 052108, May 2005. doi: 10.1103/​PhysRevA.71.052108. arXiv: quant-ph/​0406166.

[11] S. Abramsky and A. Brandenburger, ``The sheaf-theoretic structure of non-locality and contextuality,'' New Journal of Physics, vol. 13, no. 11, p. 113036, 2011. doi: 10.1088/​1367-2630/​13/​11/​113036. arXiv: 1102.0264.

[12] A. Cabello, S. Severini, and A. Winter, ``(non-)contextuality of physical theories as an axiom,'' arXiv preprint arXiv:1010.2163, 2010. doi: 10.48550/​arXiv.1010.2163.

[13] A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, ``A combinatorial approach to nonlocality and contextuality,'' Communications in Mathematical Physics, vol. 334, no. 2, pp. 533–628, 2015. doi: 10.1007/​s00220-014-2260-1. arXiv: 1212.4084.

[14] C. Okay, S. Roberts, S. D. Bartlett, and R. Raussendorf, ``Topological proofs of contextuality in quantum mechanics,'' Quantum Information & Computation, vol. 17, no. 13-14, pp. 1135–1166, 2017. doi: 10.26421/​QIC17.13-14-5. arXiv: 1701.01888.

[15] R. Raussendorf, ``Cohomological framework for contextual quantum computations,'' Quantum Information and Computation, vol. 19, no. 13&14, pp. 1141–1170, 2019. doi: 10.26421/​QIC19.13-14-4. arXiv: 1602.04155.

[16] K. Beer and T. J. Osborne, ``Contextuality and bundle diagrams,'' Physical Review A, vol. 98, p. 052124, Nov 2018. doi: 10.1103/​PhysRevA.98.052124. arXiv: 1802.08424.

[17] T. C. Marcelo, ``On measures and measurements: a fibre bundle approach to contextuality,'' Philosophical Transactions of the Royal Society A, vol. 377(15), p. 20190146. doi: 10.1098/​rsta.2019.0146. arXiv: 1903.08819.

[18] G. Caru, Logical and topological contextuality in quantum mechanics and beyond. PhD thesis, University of Oxford, 2019.

[19] S. Aasnæss, ``Cohomology and the algebraic structure of contextuality in measurement based quantum computation,'' in Proceedings 16th International Conference on Quantum Physics and Logic (B. Coecke and M. Leifer, eds.), vol. 318, pp. 242–253, Open Publishing Association, 2020. doi: 10.4204/​EPTCS.318.15. arXiv: 2005.00213.

[20] S. B. Montanhano, ``Contextuality in the fibration approach and the role of holonomy,'' arXiv preprint arXiv:2105.14132, 2021. doi: 10.48550/​arXiv.2105.14132. arXiv: 2105.14132.

[21] S. Popescu and D. Rohrlich, ``Quantum nonlocality as an axiom,'' Foundations of Physics, vol. 24, no. 3, pp. 379–385, 1994. doi: 10.1007/​BF02058098.

[22] R. F. Werner and M. M. Wolf, ``All-multipartite Bell-correlation inequalities for two dichotomic observables per site,'' Physical Review A, vol. 64, no. 3, p. 032112, 2001. doi: 10.1103/​PhysRevA.64.032112. arXiv: quant-ph/​0102024.

[23] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, ``Nonlocal correlations as an information-theoretic resource,'' Physical Review A, vol. 71, p. 022101, Feb 2005. doi: 10.1103/​PhysRevA.71.022101. arXiv: quant-ph/​0404097.

[24] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, ``Proposed experiment to test local hidden-variable theories,'' Physical review letters, vol. 23, no. 15, p. 880, 1969. doi: 10.1103/​PhysRevLett.23.880.

[25] A. Fine, ``Hidden variables, joint probability, and the Bell inequalities,'' Physical Review Letters, vol. 48, no. 5, p. 291, 1982. doi: 10.1103/​PhysRevLett.48.291.

[26] A. Fine, ``Joint distributions, quantum correlations, and commuting observables,'' Journal of Mathematical Physics, vol. 23, no. 7, pp. 1306–1310, 1982. doi: 10.1063/​1.525514.

[27] A. M. Gleason, ``Measures on the closed subspaces of a Hilbert space,'' in The Logico-Algebraic Approach to Quantum Mechanics, pp. 123–133, Springer, 1975. doi: 10.1007/​978-94-010-1795-4_7.

[28] J. J. Halliwell and C. Mawby, ``Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times,'' Physical Review A, vol. 100, no. 4, p. 042103, 2019. doi: 10.1103/​PhysRevA.100.042103. arXiv: 1906.04865.

[29] C. Okay, E. Tyhurst, and R. Raussendorf, ``The cohomological and the resource-theoretic perspective on quantum contextuality: common ground through the contextual fraction,'' Quantum Information and Computation, vol. 18, no. 15&16, pp. 1272–1294, 2018. doi: 10.26421/​QIC18.15-16-2. arXiv: 1806.04657.

[30] L. Masanes, A. Acín, and N. Gisin, ``General properties of nonsignaling theories,'' Physical Review A, vol. 73, no. 1, p. 012112, 2006. doi: 10.1103/​PhysRevA.73.012112. arXiv: quant-ph/​0508016.

[31] A. Krishna, R. W. Spekkens, and E. Wolfe, ``Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the peres–mermin square,'' New Journal of Physics, vol. 19, no. 12, p. 123031, 2017. doi: 10.1088/​1367-2630/​aa9168. arXiv: 1704.01153.

[32] 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, vol. 506, no. 1-2, pp. 1–39, 2011. doi: 10.1016/​j.physrep.2011.05.001. arXiv: 1010.1273.

[33] R. Cleve, P. Hoyer, B. Toner, and J. Watrous, ``Consequences and limits of nonlocal strategies,'' in Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pp. 236–249, IEEE, 2004. doi: 10.1109/​CCC.2004.1313847. arXiv: quant-ph/​0404076.

[34] S. Abramsky, R. S. Barbosa, G. Carù, and S. Perdrix, ``A complete characterization of all-versus-nothing arguments for stabilizer states,'' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 375, no. 2106, p. 20160385, 2017. doi: 10.1098/​rsta.2016.0385. arXiv: 1705.08459.

[35] I. Pitowsky, Quantum Probability Quantum Logic. Springer, 1989. doi: 10.1007/​BFb0021186.

[36] P. G. Goerss and J. F. Jardine, Simplicial homotopy theory. Springer Science & Business Media, 2009. doi: 10.1007/​978-3-0346-0189-4.

[37] G. Friedman, ``An elementary illustrated introduction to simplicial sets,'' arXiv preprint arXiv:0809.4221, 2008. doi: 10.48550/​arXiv.0809.4221.

[38] B. Jacobs, ``Convexity, duality and effects,'' in IFIP International Conference on Theoretical Computer Science, pp. 1–19, Springer, 2010. doi: 10.1007/​978-3-642-15240-5_1.

[39] A. Adem and R. J. Milgram, Cohomology of finite groups, vol. 309. Springer Science & Business Media, 2013. doi: 10.1007/​978-3-662-06280-7.

[40] S. Abramsky and L. Hardy, ``Logical Bell inequalities,'' Physical Review A, vol. 85, p. 062114, Jun 2012. doi: 10.1103/​PhysRevA.85.062114. arXiv: 1203.1352.

[41] L. Hardy, ``Nonlocality for two particles without inequalities for almost all entangled states,'' Physical Review Letters, vol. 71, pp. 1665–1668, Sep 1993. doi: 10.1103/​PhysRevLett.71.1665.

[42] C. Flori and T. Fritz, ``Compositories and gleaves,'' Theory Appl. Categ., vol. 31, pp. Paper No. 33, 928–988, 2016. doi: 10.48550/​arXiv.1308.6548.

[43] C. Okay, H. Y. Chung, and S. Ipek, ``Mermin polytopes in quantum computation and foundations,'' arXiv preprint arXiv:2210.10186, 2022. doi: 10.48550/​arXiv.2210.10186.

[44] C. A. Weibel, An introduction to homological algebra. No. 38, Cambridge university press, 1995. doi: 10.1017/​CBO9781139644136.

[45] S. Abramsky, R. S. Barbosa, K. Kishida, R. Lal, and S. Mansfield, ``Contextuality, cohomology and paradox,'' in 24th EACSL Annual Conference on Computer Science Logic (CSL 2015) (S. Kreutzer, ed.), vol. 41, (Dagstuhl, Germany), pp. 211–228, Schloss Dagstuhl, 2015. doi: 10.4230/​LIPIcs.CSL.2015.211. arXiv: 1502.03097.

[46] J. Watrous, The theory of quantum information. Cambridge university press, 2018. doi: 10.1017/​9781316848142.

[47] A. Adem, F. R. Cohen, and E. T. Giese, ``Commuting elements, simplicial spaces and filtrations of classifying spaces,'' in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 152, pp. 91–114, Cambridge University Press, 2012. doi: 10.1017/​S0305004111000570. arXiv: 0901.0137.

[48] S. Mansfield, The Mathematical Structure of Non-locality and Contextuality. PhD thesis, University of Oxford, 2013.

[49] V. Moretti, Fundamental Mathematical Structures of Quantum Theory. Springer, 2019. doi: 10.1007/​978-3-030-18346-2.

[50] H. Kleisli, ``Every standard construction is induced by a pair of adjoint functors,'' Proceedings of the American Mathematical Society, vol. 16, no. 3, pp. 544–546, 1965. doi: 10.2307/​2034693.

[51] A. Kharoof and C. Okay, ``Simplicial distributions, convex categories and contextuality,'' arXiv preprint arXiv:2211.00571, 2022. doi: 10.48550/​arXiv.2211.00571.

[52] M. Karvonen, ``Categories of empirical models,'' arXiv preprint arXiv:1804.01514, 2018. doi: 10.4204/​EPTCS.287.14.

[53] R. S. Barbosa, M. Karvonen, and S. Mansfield, ``Closing bell: Boxing black box simulations in the resource theory of contextuality,'' arXiv preprint arXiv:2104.11241, 2021. doi: 10.48550/​arXiv.2104.11241.

[54] M. Araújo, M. T. Quintino, C. Budroni, M. T. Cunha, and A. Cabello, ``All noncontextuality inequalities for the n-cycle scenario,'' Physical Review A, vol. 88, no. 2, p. 022118, 2013. doi: 10.1103/​PhysRevA.88.022118. arXiv: 1206.3212.

[55] S. Abramsky, R. S. Barbosa, M. Karvonen, and S. Mansfield, ``A comonadic view of simulation and quantum resources,'' in 2019 34th Annual ACM/​IEEE Symposium on Logic in Computer Science (LICS), pp. 1–12, IEEE, 2019. doi: 10.1109/​LICS.2019.8785677. arXiv: 1904.10035.

Cited by

[1] Aziz Kharoof, Selman Ipek, and Cihan Okay, "Topological Methods for Studying Contextuality: N-Cycle Scenarios and Beyond", Entropy 25 8, 1127 (2023).

[2] Rui Soares Barbosa, Aziz Kharoof, and Cihan Okay, "A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios", arXiv:2308.06336, (2023).

[3] Kin Ian Lo, Mehrnoosh Sadrzadeh, and Shane Mansfield, "Generalised Winograd Schema and its Contextuality", arXiv:2308.16498, (2023).

[4] Aziz Kharoof and Cihan Okay, "Simplicial distributions, convex categories and contextuality", arXiv:2211.00571, (2022).

[5] Cihan Okay, Ho Yiu Chung, and Selman Ipek, "Mermin polytopes in quantum computation and foundations", arXiv:2210.10186, (2022).

[6] Sivert Aasnæss, "Comparing two cohomological obstructions for contextuality, and a generalised construction of quantum advantage with shallow circuits", arXiv:2212.09382, (2022).

[7] Cihan Okay and Igor Sikora, "Equivariant simplicial distributions and quantum contextuality", arXiv:2310.18135, (2023).

[8] Aziz Kharoof and Cihan Okay, "Homotopical characterization of strongly contextual simplicial distributions on cone spaces", arXiv:2311.14111, (2023).

[9] Ho Yiu Chung, Cihan Okay, and Igor Sikora, "Simplicial techniques for operator solutions of linear constraint systems", arXiv:2305.07974, (2023).

[10] Selman Ipek and Cihan Okay, "The degenerate vertices of the $2$-qubit $\Lambda$-polytope and their update rules", arXiv:2312.10734, (2023).

[11] Cihan Okay, "On the rank of two-dimensional simplicial distributions", arXiv:2312.15794, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-02-26 08:55:06) and SAO/NASA ADS (last updated successfully 2024-02-26 08:55:07). The list may be incomplete as not all publishers provide suitable and complete citation data.