Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada
Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC, Canada
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We consider the phenomenon of quantum mechanical contextuality, and specifically parity-based proofs thereof. Mermin’s square and star are representative examples. Part of the information invoked in such contextuality proofs is the commutativity structure among the pertaining observables. We investigate to which extent this commutativity structure alone determines the viability of a parity-based contextuality proof. We establish a topological criterion for this, generalizing an earlier result by Arkhipov.
► BibTeX data
► References
[1] John S Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1 (3): 195, 1964.
[2] Simon Kochen and E. P. Specker. The problem of hidden variables in quantum mechanics. In The Logico-Algebraic Approach to Quantum Mechanics, pages 293-328. Springer Netherlands, 1975. 10.1007/978-94-010-1795-4_17.
https://doi.org/10.1007/978-94-010-1795-4_17
[3] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A, 71 (2), feb 2005. 10.1103/physreva.71.022316.
https://doi.org/10.1103/physreva.71.022316
[4] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Physical Review Letters, 86 (22): 5188-5191, may . 10.1103/physrevlett.86.5188.
https://doi.org/10.1103/physrevlett.86.5188
[5] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the `magic’ for quantum computation. Nature, 510 (7505): 351-355, jun 2014. 10.1038/nature13460.
https://doi.org/10.1038/nature13460
[6] Nicolas Delfosse, Philippe Allard Guerin, Jacob Bian, and Robert Raussendorf. Wigner function negativity and contextuality in quantum computation on rebits. Physical Review X, 5 (2), apr 2015. 10.1103/physrevx.5.021003.
https://doi.org/10.1103/physrevx.5.021003
[7] Robert Raussendorf, Dan E. Browne, Nicolas Delfosse, Cihan Okay, and Juan Bermejo-Vega. Contextuality and wigner-function negativity in qubit quantum computation. Physical Review A, 95 (5), may 2017. 10.1103/physreva.95.052334.
https://doi.org/10.1103/physreva.95.052334
[8] Janet Anders and Dan E. Browne. Computational power of correlations. Physical Review Letters, 102 (5), feb 2009. 10.1103/physrevlett.102.050502.
https://doi.org/10.1103/physrevlett.102.050502
[9] Matty J. Hoban and Dan E. Browne. Stronger quantum correlations with loophole-free postselection. Physical Review Letters, 107 (12), sep 2011. 10.1103/physrevlett.107.120402.
https://doi.org/10.1103/physrevlett.107.120402
[10] Robert Raussendorf. Contextuality in measurement-based quantum computation. Physical Review A, 88 (2), aug 2013. 10.1103/physreva.88.022322.
https://doi.org/10.1103/physreva.88.022322
[11] Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield. Contextual fraction as a measure of contextuality. Physical Review Letters, 119 (5), aug 2017. 10.1103/physrevlett.119.050504.
https://doi.org/10.1103/physrevlett.119.050504
[12] Markus Frembs, Sam Roberts, and Stephen D Bartlett. Contextuality as a resource for measurement-based quantum computation beyond qubits. New Journal of Physics, 20 (10): 103011, oct 2018. 10.1088/1367-2630/aae3ad.
https://doi.org/10.1088/1367-2630/aae3ad
[13] Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics, 13 (11): 113036, nov 2011. 10.1088/1367-2630/13/11/113036.
https://doi.org/10.1088/1367-2630/13/11/113036
[14] A. Döring and C. Isham. “what is a thing?”: Topos theory in the foundations of physics. In New Structures for Physics, pages 753-937. Springer Berlin Heidelberg, 2010. 10.1007/978-3-642-12821-9_13.
https://doi.org/10.1007/978-3-642-12821-9_13
[15] A. Döring and C. J. Isham. A topos foundation for theories of physics: I. formal languages for physics. Journal of Mathematical Physics, 49 (5): 053515, may 2008. 10.1063/1.2883740.
https://doi.org/10.1063/1.2883740
[16] Samson Abramsky, Shane Mansfield, and Rui Soares Barbosa. The cohomology of non-locality and contextuality. Electronic Proceedings in Theoretical Computer Science, 95: 1-14, oct 2012. 10.4204/eptcs.95.1.
https://doi.org/10.4204/eptcs.95.1
[17] Alex Arkhipov. Extending and characterizing quantum magic games. arXiv preprint arXiv:1209.3819, 2012.
arXiv:1209.3819
[18] Cihan Okay, Sam Roberts, Stephen D Bartlett, and Robert Raussendorf. Topological proofs of contextuality in qunatum mechanics. Quantum Information & Computation, 17 (13-14): 1135-1166, 2017.
[19] Antonio Acín, Tobias Fritz, Anthony Leverrier, and Ana Belén Sainz. A combinatorial approach to nonlocality and contextuality. Communications in Mathematical Physics, 334 (2): 533-628, jan 2015. 10.1007/s00220-014-2260-1.
https://doi.org/10.1007/s00220-014-2260-1
[20] R. W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Physical Review A, 71 (5), may 2005. 10.1103/physreva.71.052108.
https://doi.org/10.1103/physreva.71.052108
[21] Adan Cabello, Simone Severini, and Andreas Winter. (non-) contextuality of physical theories as an axiom. arXiv preprint arXiv:1010.2163, 2010.
arXiv:1010.2163
[22] Nadish de Silva and Rui Soares Barbosa. Contextuality and noncommutative geometry in quantum mechanics. Communications in Mathematical Physics, 365 (2): 375-429, jan 2019. 10.1007/s00220-018-3222-9.
https://doi.org/10.1007/s00220-018-3222-9
[23] Frank Roumen. Cohomology of effect algebras. Electronic Proceedings in Theoretical Computer Science, 236: 174-201, jan 2017. 10.4204/eptcs.236.12.
https://doi.org/10.4204/eptcs.236.12
[24] Giovanni Carù. On the cohomology of contextuality. Electronic Proceedings in Theoretical Computer Science, 236: 21-39, jan 2017. 10.4204/eptcs.236.2.
https://doi.org/10.4204/eptcs.236.2
[25] Samson Abramsky, Rui Soares Barbosa, Kohei Kishi
da, Raymond Lal, and Shane Mansfield. Contextuality, cohomology and paradox. arXiv preprint arXiv:1502.03097, 2015.
arXiv:1502.03097
[26] Giovanni Caru. Towards a complete cohomology invariant for non-locality and contextuality. arXiv preprint arXiv:1807.04203, 2018.
arXiv:1807.04203
[27] Sivert Aasaess. Cohomology and the algebraic structure of contextuality in measurement based quantum computation. presented at QPL, 2019.
[28] N. David Mermin. Hidden variables and the two theorems of john bell. Reviews of Modern Physics, 65 (3): 803-815, jul 1993. 10.1103/revmodphys.65.803.
https://doi.org/10.1103/revmodphys.65.803
[29] Petr Lisonek, Robert Raussendorf, and Vijaykumar Singh. Generalized parity proofs of the kochen-specker theorem. arXiv preprint arXiv:1401.3035, 2014.
arXiv:1401.3035
[30] Richard Cleve, Li Liu, and William Slofstra. Perfect commuting-operator strategies for linear system games. Journal of Mathematical Physics, 58 (1): 012202, jan 2017. 10.1063/1.4973422.
https://doi.org/10.1063/1.4973422
[31] 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. IEEE. 10.1109/ccc.2004.1313847.
https://doi.org/10.1109/ccc.2004.1313847
[32] Cynthia Hog-Angeloni and Wolfgang Metzler, editors. Two-dimensional homotopy and combinatorial group theory, volume 197 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1993. ISBN 0-521-44700-3. 10.1017/cbo9780511629358.
https://doi.org/10.1017/cbo9780511629358
[33] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. ISBN 0-521-79160-X; 0-521-79540-0.
[34] Joseph J Rotman. An introduction to the theory of groups, volume 148. Springer Science & Business Media, 2012.
[35] Derek F Holt, Bettina Eick, and Eamonn A O’Brien. Handbook of computational group theory. Chapman and Hall/CRC, 2005. 10.1201/9781420035216.
https://doi.org/10.1201/9781420035216
Cited by
[1] Hammam Qassim and Joel. J. Wallman, “Classical vs. quantum satisfiability in linear constraint systems modulo an integer”, arXiv:1911.11171.
The above citations are from SAO/NASA ADS (last updated successfully 2020-01-22 18:18:08). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2020-01-22 18:18:06).
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-2020-01-05-217/
