Skip to main content
Springer Nature Link
Log in

Compositional Quantum Logic

  • Chapter
Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7860))

Abstract

Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles’ heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order is extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation. In fact, our construction is entirely compositional, without any additional assumptions on limits or enrichment. Interpreted in the category of finite-dimensional Hilbert spaces, it yields the projection lattices of arbitrary finite-dimensional C*-algebras. Interestingly, there are models that falsify standardly assumed correspondences, most notably the correspondence between noncommutativity of the algebra and nondistributivity of the order.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 415–425. IEEE Computer Society (2004), Extended version: arXiv:quant-ph/0402130

    Google Scholar 

  2. Abramsky, S., Heunen, C.: H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. In: Abramsky, S., Mislove, M. (eds.) Clifford Lectures. Proceedings of Symposia in Applied Mathematics, vol. 71, pp. 1–24. American Mathematical Society (2012)

    Google Scholar 

  3. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boixo, S., Heunen, C.: Entangled and sequential quantum protocols with dephasing. Physical Review Letters 108, 120402 (2012)

    Article  Google Scholar 

  5. Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49, 11–32 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Informational derivation of quantum theory. Physical Review A 84(1), 012311 (2011)

    Article  Google Scholar 

  7. Coecke, B.: Axiomatic description of mixed states from Selinger’s CPM-construction. Electronic Notes in Theoretical Computer Science 210, 3–13 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coecke, B.: The logic of quantum mechanics – take II (2012), arXiv:1204.3458

    Google Scholar 

  9. Coecke, B., Edwards, B., Spekkens, R.W.: Phase groups and the origin of non-locality for qubits. Electronic Notes in Theoretical Computer Science 270(2), 15–36 (2011), arXiv:1003.5005

    Article  MATH  Google Scholar 

  10. Coecke, B., Heunen, C.: Pictures of complete positivity in arbitrary dimension. Quantum Phsyics and Logic, Electronic Proceedings in Theoretical Computer Science 95, 27–35 (2011)

    Article  MATH  Google Scholar 

  11. Coecke, B., Heunen, C., Kissinger, A.: A category of classical and quantum channels. In: QPL 2012 (2012)

    Google Scholar 

  12. Coecke, B., Kissinger, A.: The compositional structure of multipartite quantum entanglement. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 297–308. Springer, Heidelberg (2010), Extended version: arXiv:1002.2540

    Chapter  Google Scholar 

  13. Coecke, B., Paquette, É.O.: Categories for the practicing physicist. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, pp. 167–271. Springer (2011), arXiv:0905.3010

    Google Scholar 

  14. Coecke, B., Paquette, É.O., Pavlović, D.: Classical and quantum structuralism. In: Gay, S., Mackie, I. (eds.) Semantic Techniques in Quantum Computation, pp. 29–69. Cambridge University Press (2010), arXiv:0904.1997

    Google Scholar 

  15. Coecke, B., Pavlović, D., Vicary, J.: A new description of orthogonal bases. Mathematical Structures in Computer Science (2011) (to appear), arXiv:quant-ph/0810.1037

    Google Scholar 

  16. Coecke, B., Sadrzadeh, M., Clark, S.: Mathematical foundations for a compositional distributional model of meaning. Linguistic Analysis (2010)

    Google Scholar 

  17. Coecke, B., Spekkens, R.W.: Picturing classical and quantum Bayesian inference. Synthese 186, 651–696 (2012), arXiv:1102.2368.

    Article  MathSciNet  MATH  Google Scholar 

  18. Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with generalised flow. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 285–296. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  19. Faure, C.-A., Moore, D.J., Piron, C.: Deterministic evolutions and Schrödinger flows. Helvetica Physica Acta 68(2), 150–157 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Foulis, D.J., Randall, C.H.: Operational statistics. I. Basic concepts. Journal of Mathematical Physics 13(11), 1667–1675 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics 6, 885–893 (1957)

    MathSciNet  MATH  Google Scholar 

  22. Harding, J.: A link between quantum logic and categorical quantum mechanics. International Journal of Theoretical Physics 48(3), 769–802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Harding, J.: Daggers, kernels, Baer *-semigroups, and orthomodularity. To appear in Journal of Philosophical Logic (2010)

    Google Scholar 

  24. Hardy, L.: Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012 (2001)

    Google Scholar 

  25. Heunen, C., Contreras, I., Cattaneo, A.S.: Relative frobenius algebras are groupoids. Journal of Pure and Applied Algebra 217, 114–124 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Heunen, C., Jacobs, B.: Quantum logic in dagger kernel categories. Order 27(2), 177–212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Horsman, C.: Quantum picturalism for topological cluster-state computing. New Journal of Physics 13, 095011 (2011), arXiv:1101.4722

    Article  Google Scholar 

  28. Jacobs, B.: Orthomodular lattices, foulis semigroups and dagger kernel categories. Logical Methods in Computer Science 6(2), 1 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ludwig, G.: An Axiomatic Basis of Quantum Mechanics. 1. Derivation of Hilbert Space. Springer (1985)

    Google Scholar 

  31. Mackey, G.W.: The mathematical foundations of quantum mechanics. W. A. Benjamin, New York (1963)

    MATH  Google Scholar 

  32. Ore, O.: Structures and group theory II. Duke Mathematical Journal 4(2), 247–269 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pavlovic, D.: Quantum and classical structures in nondeterminstic computation. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen, K., Klusch, M. (eds.) QI 2009. LNCS, vol. 5494, pp. 143–157. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  34. Piron, C.: Axiomatique quantique. Helvetia Physica Acta 37, 439–468 (1964)

    MathSciNet  MATH  Google Scholar 

  35. Piron, C.: Foundations of quantum physics. W. A. Benjamin (1976)

    Google Scholar 

  36. Rédei, M.: Quantum Logic in Algebraic Approach. Kluwer (1998)

    Google Scholar 

  37. Selinger, P.: Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170, 139–163 (2007)

    Article  MATH  Google Scholar 

  38. Selinger, P.: Idempotents in dagger categories (extended abstract). Electronic Notes in Theoretical Computer Science 210, 107–122 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, pp. 275–337. Springer (2011), arXiv:0908.3347

    Google Scholar 

  40. Solèr, M.P.: Characterization of Hilbert spaces by orthomodular spaces. Communications in Algebra 23(1), 219–243 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Stubbe, I., van Steirteghem, B.: Propositional systems, Hilbert lattices and generalized Hilbert spaces. In: Lehmann, D., Gabbay, D., Engesser, K. (eds.) Handbook Quantum Logic, pp. 477–524. Elsevier Publ. (2007), http://www.mat.uc.pt/~isar/PDF/HilbertLatticesELSEVIER.pdf

  42. Vicary, J.: Categorical formulation of finite-dimensional quantum algebras. Communications in Mathematical Physics 304(3), 765–796 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wigner, E.P.: Gruppentheorie. Friedrich Vieweg und Sohn (1931)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Oxford, UK

    Bob Coecke, Chris Heunen & Aleks Kissinger

Authors
  1. Bob Coecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chris Heunen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aleks Kissinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Parks Road, Wolfson Building, OX1 3QD, Oxford, UK

    Bob Coecke  & Luke Ong  & 

  2. Department of Computer Science, McGill University, 3480 Rue University, H3A 0E9, Montréal, QC, Canada

    Prakash Panangaden

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Coecke, B., Heunen, C., Kissinger, A. (2013). Compositional Quantum Logic. In: Coecke, B., Ong, L., Panangaden, P. (eds) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science, vol 7860. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38164-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-38164-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38163-8

  • Online ISBN: 978-3-642-38164-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation