Foundations of Physics

, Volume 42, Issue 7, pp 856–873 | Cite as

Complementarity in Categorical Quantum Mechanics



We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.


Complementarity Dagger kernel monoidal category Classical structure Orthonormal basis Orthomodular lattice Boolean lattice von Neumann algebra 


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  1. 1.
    Abramsky, S., Coecke, B.: Categorical quantum mechanics. In: Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pp. 261–324. Elsevier, Amsterdam (2009) CrossRefGoogle Scholar
  2. 2.
    Abramsky, S., Heunen, C.: H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. In: Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics (2011) Google Scholar
  3. 3.
    Butterfield, J., Isham, C.J.: A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations. Int. J. Theor. Phys. 37(11), 2669–2733 (1998) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Coecke, B., Duncan, R.: Interacting quantum observables: Categorical algebra and diagrammatics. In: Automata, Languages and Programming, ICALP 2008. Lecture Notes in Computer Science, vol. 5126, pp. 298–310. Springer, Berlin (2008) CrossRefGoogle Scholar
  5. 5.
    Coecke, B., Pavlović, D.: Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Taylor & Francis, London (2007) Google Scholar
  6. 6.
    Coecke, B., Pavlović, D., Vicary, J.: A new description of orthogonal bases. In: Mathematical Structures in Computer Science (2009) Google Scholar
  7. 7.
    Coecke, B., Paquette, É.O., Pavlović, D.: Classical and quantum structuralism. In: Semantic Techniques in Quantum Computation, pp. 29–70. Cambridge University Press, Cambridge (2010) Google Scholar
  8. 8.
    Döring, A., Isham, C.J.: ‘What is a thing?’: Topos theory in the foundations of physics. In: New Structures for Physics. Lecture Notes in Physics. Springer, Berlin (2009) Google Scholar
  9. 9.
    Held, C.: The meaning of complementarity. Stud. Hist. Philos. Sci. Part A 25, 871–893 (1994) CrossRefGoogle Scholar
  10. 10.
    Heunen, C., Jacobs, B.: Quantum logic in dagger kernel categories. Order 27(2), 177–212 (2010) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Heunen, C., Landsman, N.P., Spitters, B.: Bohrification. In: Deep Beauty. Cambridge University Press, Cambridge (2011) Google Scholar
  12. 12.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, New York (1983) MATHGoogle Scholar
  13. 13.
    Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967) MathSciNetMATHGoogle Scholar
  14. 14.
    Kock, J.: Frobenius Algebras and 2-D Topological Quantum Field Theories. London Mathematical Society Student Texts, vol. 59. Cambridge University Press, Cambridge (2003) CrossRefGoogle Scholar
  15. 15.
    Landsman, N.P.: Between classical and quantum. In: Handbook of the Philosophy of Science, vol. 2: Philosophy of Physics, pp. 417–554. North-Holland, Amsterdam (2007) Google Scholar
  16. 16.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  17. 17.
    Parthasarathy, K.R.: On estimating the state of a finite level quantum system. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 607–617 (2006) CrossRefGoogle Scholar
  18. 18.
    Pavlović, D.: Quantum and classical structures in nondeterministic computation. In: Bruza, P., et al. (ed.) Third International Symposium on Quantum Interaction. Lecture Notes in Artificial Intelligence, vol. 5494, pp. 143–157. Springer, Berlin (2009) Google Scholar
  19. 19.
    Petz, D.: Complementarity in quantum systems. Rep. Math. Phys. 59(2), 209–224 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Piron, C.: Foundations of Quantum Physics. Mathematical Physics Monographs, vol. 19. Benjamin, Elmsford (1976) MATHGoogle Scholar
  21. 21.
    Rédei, M.: Quantum Logic in Algebraic Approach. Kluwer, Dordrecht (1998) MATHGoogle Scholar
  22. 22.
    Scheibe, E.: The Logical Analysis of Quantum Mechanics. Pergamon, Elmsford (1973) Google Scholar
  23. 23.
    Selinger, P.: A survey of graphical languages for monoidal categories. In: New Structures for Physics. Lecture Notes in Physics. Springer, Berlin (2010) Google Scholar
  24. 24.
    Strocchi, F.: Elements of Quantum Mechanics of Infinite Systems. World Scientific, Singapore (1985) Google Scholar
  25. 25.
    van den Berg, B., Heunen, C.: Noncommutativity as a colimit. In: Applied Categorical Structures (2010) Google Scholar
  26. 26.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

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