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, Volume 27, Issue 2, pp 177–212 | Cite as

Quantum Logic in Dagger Kernel Categories

  • Chris Heunen
  • Bart Jacobs
Open Access
Article

Abstract

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres.

Keywords

Quantum logic Dagger kernel category Orthomodular lattice Categorical logic 

References

  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Logic in Computer Science, IEEE, pp. 415–425 (2004)Google Scholar
  2. 2.
    Awodey, S.: Category Theory. Oxford Logic Guides, Oxford Univ. Press, Oxford (2006)Google Scholar
  3. 3.
    Barr, M.: Algebraically compact functors. J. Pure Appl. Algebra 82, 211–231 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin. Revised and corrected version available from URL: www.cwru.edu/artsci/math/wells/pub/ttt.html (1985)
  5. 5.
    Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics 50, 51 and 52. Cambridge Univ. Press, Cambridge (1994)Google Scholar
  6. 6.
    Cassinelli, G., De Vito, E.P., Lahti J., Levrero, A.: The Theory of Symmetry Actions in Quantum Mechanics. Number 654 in Lecture Notes in Physics. Springer, Berlin (2004)Google Scholar
  7. 7.
    Coecke, B., Pavlović, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology. Taylor and Francis, Philadelphia. See also: arXiv:/quant-ph/0608035 (2006)
  8. 8.
    Coecke, B., Smets, S.: The Sasaki hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes. Int. J. Theor. Phys. 43(7/8), 1705–1736 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Crown, G.: On some orthomodular posets of vector bundles. J. Nat. Sci. Math. 15(1–2), 11–25 (1975)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Davey, B., Priestley, H.: Introduction to Lattices and Order. Math. Textbooks. Cambridge Univ. Press, Cambridge (1990)Google Scholar
  11. 11.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad., Dordrecht (2000)zbMATHGoogle Scholar
  12. 12.
    Finch, P.D.: Quantum logic as an implication algebra. Bull. Am. Math. Soc. 2, 101–106 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Foulis, D.J., Greechie, R., Bennett, M.: Sums and products of interval algebras. Int. J. Theor. Phys. 33(11), 2119–2136 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Freyd, P.: Abelian Categories: An Introduction to the Theory of Functors. Harper and Row, New York. Available via www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf (1964)
  15. 15.
    Haghverdi, E., Scott, P.: A categorical model for the geometry of interaction. Theor. Comp. Sci. 350, 252–274 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Harding, J.: Orthomodularity of decompositions in a categorical setting. Int. J. Theor. Phys. 45(6), 1117–1128 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Heunen, C.: Compactly accessible categories and quantum key distribution. Logical Methods Comp. Sci. 4(4) (2008). doi: 10.2168/LMCS-4(4:9)2008 zbMATHMathSciNetGoogle Scholar
  18. 18.
    Heunen, C.: Quantifiers for quantum logic. arXiv:0811.1457 (2008)
  19. 19.
    Heunen, C.: An embedding theorem for Hilbert categories. Theory Appl. Categ. 22, 321–344 (2009)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Heunen, C.: Categorical quantum models and logics. Ph.D. thesis, Radboud Univ. Nijmegen (2010)Google Scholar
  21. 21.
    Jacobs, B.: Categorical Logic and Type Theory. North Holland, Amsterdam (1999)zbMATHGoogle Scholar
  22. 22.
    Jacobs, B.: Orthomodular lattices, Foulis semigroups and dagger kernel categories. Logical Methods Comp. Sci. (2009). Available from http://arxiv.org/abs/0905.4090
  23. 23.
    Janowitz, M.F.: Quantifiers and orthomodular lattices. Pac. J. Math. 13, 1241–1249 (1963)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Johnstone, P.: Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge (1982)Google Scholar
  25. 25.
    Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, Oxford (2002)Google Scholar
  26. 26.
    Kalmbach, G.: Orthomodular Lattices. Academic, London (1983)zbMATHGoogle Scholar
  27. 27.
    Kalmbach, G.: Measures and Hilbert Lattices. World Scientific, Singapore (1986)zbMATHGoogle Scholar
  28. 28.
    Kock, A., Reyes, G.: Doctrines in categorical logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 283–313. North-Holland, Amsterdam (1977)CrossRefGoogle Scholar
  29. 29.
    Lane, S.M.: Categories for the Working Mathematician. Springer, Berlin (1971)zbMATHGoogle Scholar
  30. 30.
    Lehmann, D.: A presentation of quantum logic based on an and then connective. J. Log. Comput. 18(1), 59–76 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mac Lane, S.: An algebra of additive relations. Proc. Natl. Acad. sci. 47, 1043–1051 (1961)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Makkai, M., Reyes, G.: First Order Categorical Logic. Number 611 in Lect. Notes Math., Springer, Berlin (1977)Google Scholar
  33. 33.
    Manes, E.: Monads, matrices and generalized dynamic algebra. In: Ehrig, H., Herrlich, H., Kreowski, H.-J., Preuß, G. (eds.) Categorical Methods in Computer Science With Aspects from Topology, number 393 in Lect. Notes Comp. Sci, pp. 66–81 (1989)Google Scholar
  34. 34.
    Piron, C.: Foundations of Quantum Physics. Number 19 in Mathematical Physics Monographs. Benjamin, New York (1976)Google Scholar
  35. 35.
    Puppe, D.: Korrespondenzen in abelschen Kategorien. Math. Ann. 148, 1–30 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Román, L.: A characterization of quantic quantifiers in orthomodular lattices. Theory Appl. Categ. 16, 206–217 (2006)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Román, L., Rumbos, B.: A characterization of nuclei in orthomodular and quantic lattices. J. Pure Appl. Algebra 73, 155–163 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Román, L., Zuazua, R.: On quantic conuclei in orthomodular lattices. Theory Appl. Categ. 2(6), 62–68 (1996)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Rosenthal, K.: Quantales and Their Applications. Number 234 in Pitman Research Notes in Math. Longman Scientific & Technical, Harlow (1990)Google Scholar
  40. 40.
    Selinger, P.: Dagger compact closed categories and completely positive maps (extended abstract). In: Selinger, P. (ed.) Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), number 170 in Elect. Notes in Theor. Comp. Sci., pp. 139–163.  http://dx.doi.org/10.1016/j.entcs.2006.12.018 (2007)
  41. 41.
    Taylor, P.: Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge (1999)Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Oxford University Computing LaboratoryOxfordUK
  2. 2.Institute for Computing and Information Sciences (iCIS)Radboud University NijmegenNijmegenThe Netherlands

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