Logic of Local Inference for Contextuality in Quantum Physics and Beyond

Author Kohei Kishida

Thumbnail PDF


  • Filesize: 0.5 MB
  • 14 pages

Document Identifiers

Author Details

Kohei Kishida

Cite AsGet BibTex

Kohei Kishida. Logic of Local Inference for Contextuality in Quantum Physics and Beyond. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 113:1-113:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [Abramsky and Brandenburger, New J. Phys., 2011, Abramsky et al., CSL 2015, 2015] characterizes contextuality as "global inconsistency" coupled with "local consistency", revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the "global inconsistency" part of contextuality. Our goal is to capture the other, "local consistency" part, which requires a novel approach to logic that is sensitive to the topology of contexts. To achieve this, we formulate a logic of local inference by using context-sensitive theories and models in regular categories. This provides a uniform framework for local consistency, and lays a foundation for high-level methods of detecting, proving, and moreover using contextuality as computational resource.
  • Contextuality
  • quantum mechanics
  • regular category
  • regular logic
  • separated presheaf


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Samson Abramsky. Relational databases and Bell’s theorem. In Val Tannen, Limsoon Wong, Leonid Libkin, Wenfei Fan, Wang-Chiew Tan, and Michael P. Fourman, editors, In Search of Elegance in the Theory and Practice of Computation: Essays Dedicated to Peter Buneman, volume 8000 of Lecture Notes in Computer Science, pages 13-35. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-41660-6_2.
  2. Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contextuality, cohomology and paradox. In Stephan Kreutzer, editor, 24th EACSL Annual Conference on Computer Science Logic, CSL 2015, September 7-10, 2015, Berlin, Germany, volume 41 of LIPIcs, pages 211-228. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: http://www.dagstuhl.de/dagpub/978-3-939897-90-3, URL: http://dx.doi.org/10.4230/LIPIcs.CSL.2015.211.
  3. Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New J. Phys., 13:113036-113075, 2011. URL: http://dx.doi.org/10.1088/1367-2630/13/11/113036.
  4. Samson Abramsky and Jonathan A. Zvesper. From Lawvere to Brandenburger-Keisler: Interactive forms of diagonalization and self-reference. J. Comput. Syst. Sci., 81(5):799-812, 2015. URL: http://dx.doi.org/10.1016/j.jcss.2014.12.001.
  5. Carsten Butz. Regular categories and regular logic. BRICS Lecture Series LS-98-2, BRICS, 1998. Google Scholar
  6. Dov M. Gabbay and Larisa Maksimova. Interpolation and Definability: Modal and Intuitionistic Logics. Oxford Logic Guides. Clarendon Press, Oxford, 2005. Google Scholar
  7. Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber. A general argument against superluminal transmission through the quantum mechanical measurement process. Lettere al Nuovo Cimento, 27:293-298, Mar 1980. URL: http://dx.doi.org/10.1007/BF02817189.
  8. Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger. Going beyond Bell’s theorem. In M. Kafatos, editor, Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, pages 69-72. Kluwer, 1989. Google Scholar
  9. Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett., 71:1665-1668, Sep 1993. URL: http://dx.doi.org/10.1103/PhysRevLett.71.1665.
  10. Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the `magic' for quantum computation. Nature, 510:351-355, June 2014. URL: http://dx.doi.org/10.1038/nature13460.
  11. Chris J. Isham and Jeremy Butterfield. Topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations. Int. J. Theor. Phys., 37:2669-2733, November 1998. URL: http://dx.doi.org/10.1023/A:1026680806775.
  12. Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium, volume 2 of Oxford Logic Guides. Clarendon Press, Oxford, 2002. Google Scholar
  13. Kohei Kishida. Stochastic relational presheaves and dynamic logic for contextuality. In Bob Coecke, Ichiro Hasuo, and Prakash Panangaden, editors, Proceedings of the 11th Workshop on Quantum Physics and Logic (QPL 2014), Kyoto, Japan, 4-6th June 2014, volume 172 of EPTCS, pages 115-132, 2014. URL: http://eptcs.web.cse.unsw.edu.au/content.cgi?QPL2014, URL: http://dx.doi.org/10.4204/EPTCS.172.9.
  14. Jürg Kohlas, Marc Pouly, and Cesar Schneuwly. Generic local computation. J. Comput. Syst. Sci., 78(1):348-369, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.05.012.
  15. F. William Lawvere. Diagonal arguments and cartesian closed categories. Lecture Notes in Mathematics, 92:134-145, 1969. Google Scholar
  16. F. William Lawvere. Quantifiers and sheaves. Actes, Congrès intern. math., 1:329-334, 1970. Google Scholar
  17. Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, 1992. Google Scholar
  18. N. David Mermin. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett., 65:1838-1840, Oct 1990. URL: http://dx.doi.org/10.1103/PhysRevLett.65.1838.
  19. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. Google Scholar
  20. Jaap van Oosten. Basic category theory. BRICS Lecture Series LS-95-1, BRICS, 1995. Google Scholar
  21. Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Found. Phys., 24:379-385, Mar 1994. URL: http://dx.doi.org/10.1007/BF02058098.
  22. Robert Raussendorf. Contextuality in measurement-based quantum computation. Phys. Rev. A, 88:022322, Aug 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.022322.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail