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Contextuality, Cohomology and Paradox

Authors Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, Shane Mansfield

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Samson Abramsky
Rui Soares Barbosa
Kohei Kishida
Raymond Lal
Shane Mansfield

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Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contextuality, Cohomology and Paradox. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 211-228, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CSL.2015.211

Abstract

Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.
Keywords
  • Quantum mechanics
  • contextuality
  • sheaf theory
  • cohomology
  • logical paradoxes

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References

  1. Samson Abramsky. Relational databases and Bell’s theorem. In Val Tannen, Limsoon Wong, Leonid Libkin, Wenfei Fan, Wang-Chiew Tan, and Michael Fourman, editors, In search of elegance in the theory and practice of computation, volume 8000 of LNCS, pages 13-35. Springer, 2013. Google Scholar
  2. Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contextuality, cohomology and paradox. arXiv preprint arXiv:1502.03097, 2015. Google Scholar
  3. Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New J. Phys., 13(11):113036, 2011. Google Scholar
  4. Samson Abramsky, Georg Gottlob, and Phokion G. Kolaitis. Robust constraint satisfaction and local hidden variables in quantum mechanics. In Francesca Rossi, editor, Proceedings of the Twenty-Third IJCAI, pages 440-446. AAAI Press, 2013. Google Scholar
  5. Samson Abramsky, Shane Mansfield, and Rui Soares Barbosa. The cohomology of non-locality and contextuality. In Bart Jacobs, Peter Selinger, and Bas Spitters, editors, Proc. 8th International Workshop on Quantum Physics and Logic 2011, volume 95 of EPTCS, pages 1-14, 2012. Google Scholar
  6. John S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1(3):195-200, 1964. Google Scholar
  7. Adán Cabello, José M. Estebaranz, and Guillermo García-Alcaine. Bell-Kochen-Specker theorem. Phys. Lett. A, 212(4):183-187, 1996. Google Scholar
  8. Adan Cabello, Simone Severini, and Andreas Winter. (Non-)Contextuality of Physical Theories as an Axiom. arXiv:1010.2163, 2010. Google Scholar
  9. Carlton Caves. Stabilizer formalism for qubits. Available at http://info.phys.unm.edu/~caves/reports/stabilizer.ps, 2006.
  10. Roy T Cook. Patterns of paradox. The Journal of Symbolic Logic, 69(03):767-774, 2004. Google Scholar
  11. Ronald Fagin, Alberto O. Mendelzon, and Jeffrey D. Ullman. A simplified universal relation assumption and its properties. ACM Transactions on Database Systems (TODS), 7(3):343-360, 1982. Google Scholar
  12. Daniel M. Greenberger, Michael A. Horne, Abner Shimony, and Anton Zeilinger. Bell’s theorem without inequalities. Am. J. Phys., 58(12):1131-1143, 1990. Google Scholar
  13. Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett., 71(11):1665-1668, 1993. Google Scholar
  14. Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the `magic' for quantum computation. Nature, 510(7505):351-355, 06 2014. Google Scholar
  15. Raymond Lal. A sheaf-theoretic approach to cluster states, 2011. Private communication. Google Scholar
  16. Yeong-Cherng Liang, Robert W. Spekkens, and Howard M. Wiseman. Specker’s parable of the overprotective seer. Phys. Rep., 506(1):1-39, 2011. Google Scholar
  17. Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1992. Google Scholar
  18. David Maier, Jeffrey D. Ullman, and Moshe Y. Vardi. On the foundations of the universal relation model. ACM Transactions on Database Systems (TODS), 9(2):283-308, 1984. Google Scholar
  19. Shane Mansfield. Completeness of Hardy non-locality: Consequences & applications. In Informal Proceedings of 11th International Workshop on Quantum Physics & Logic, 2014. Google Scholar
  20. N. David Mermin. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett., 65(15):1838-1840, 1990. Google Scholar
  21. N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65(27):3373-3376, 1990. Google Scholar
  22. N. David Mermin. Hidden variables and the two theorems of John Bell. Rev. Mod. Phys., 65(3):803-815, 1993. Google Scholar
  23. M.Q.C. Nielsen and I. Chuang. Quantum computation and quantum information. Cambridge University Press, 2000. Google Scholar
  24. Roger Penrose. On the cohomology of impossible figures. Leonardo, 25(3/4):245-247, 1992. Google Scholar
  25. A. Peres. Incompatible results of quantum measurements. Phys. Lett. A, 151(3-4):107-108, 1990. Google Scholar
  26. Stefano Pironio, Jean-Daniel Bancal, and Valerio Scarani. Extremal correlations of the tripartite no-signaling polytope. J. Phys. A-Math. Theor., 44(6):065303, 2011. Google Scholar
  27. Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Found. Phys., 24(3):379-385, 1994. Google Scholar
  28. Robert Raussendorf and Hans J Briegel. A one-way quantum computer. Phys. Rev. Lett., 86(22):5188, 2001. Google Scholar
  29. Ernst Specker. Die Logik nicht gleichzeitig entscheidbarer Aussagen. Dialectica, 14:239-246, 1960. Google Scholar
  30. Michał Walicki. Reference, paradoxes and truth. Synthese, 171(1):195-226, 2009. Google Scholar
  31. Lan Wen. Semantic paradoxes as equations. Math. Intell., 23(1):43-48, 2001. Google Scholar
  32. Xiang Zhang, Mark Um, Junhua Zhang, Shuoming An, Ye Wang, Dong-ling Deng, Chao Shen, Lu-Ming Duan, and Kihwan Kim. State-independent experimental test of quantum contextuality with a single trapped ion. Phys. Rev. Lett., 110(7):070401, 2013. Google Scholar
  33. Chong Zu, Y-X Wang, D-L Deng, X-Y Chang, Ke Liu, P-Y Hou, H-X Yang, and L-M Duan. State-independent experimental test of quantum contextuality in an indivisible system. Phys. Rev. Lett., 109(15):150401, 2012. Google Scholar
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