Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Quantum Interaction

QI 2009: Quantum Interaction pp 143–157Cite as

Quantum and Classical Structures in Nondeterminstic Computation

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5494))

Abstract

In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of non-standard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an ontic-epistemic gap, as it provides no interface to these nonstandard quantum structures.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramsky, S.: No-cloning in categorical quantum mechanics. In: Gay, S., Mackie, I. (eds.) Semantical Techniques in Quantum Computation, 32 p. Cambridge University Press, Cambridge (2008) (to appear)

    Google Scholar 

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

    Chapter  Google Scholar 

  3. Abramsky, S., Gay, S., Nagarajan, R.: Interaction categories and the foundations of typed concurrent programming. In: Broy, M. (ed.) Proceedings of the 1994 Marktoberdorf Summer Sxhool on Deductive Program Design, pp. 35–113. Springer, Heidelberg (1996)

    Google Scholar 

  4. Barr, M., Grillet, P., van Osdol, D. (eds.): Exact Categories and Categories of Sheaves. Lecture Notes in Mathematics, vol. 236. Springer, Heidelberg (1971)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Church, A.: A formulation of the simple theory of types. The Journal of Symbolic Logic 5(2), 56–68 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coecke, B., Duncan, R.: Interacting quantum observables. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 298–310. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases. Math. Structures in Comp. Sci., 13 (2008) (to appear), arXiv:0810.0812

    Google Scholar 

  9. Coecke, B., Edwards, W.: Toy quantum categories. In: Coecke, B., Panangaden, P. (eds.) Proceedings of the 2008 QPL-DCM Workshop, pp. 25–35. Springer, Heidelberg (2008), arXiv:0808.1037

    Google Scholar 

  10. Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology. Taylor and Francis, Abington (2007), arxiv:quant-ph/0608035

    Google Scholar 

  11. Dieks, D.: Communication by EPR devices. Physics Letters A 92(6), 271–272 (1982)

    Article  ADS  Google Scholar 

  12. Coecke, B., Paquette, É.O., Pavlovic, D.: Classical and quantum structuralism. In: Gay, S., Mackie, I. (eds.) Semantical Methods in Quantum Computation, 42 p. Cambridge University Press, Cambridge (2008) (to appear)

    Google Scholar 

  13. Freyd, P.J., Scedrov, A.: Categories, Allegories. North Holland Publishing Company, Amsterdam (1991)

    MATH  Google Scholar 

  14. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatshefte fr Mathematik und Physik 38, 173–198 (1931)

    Article  MATH  Google Scholar 

  15. Joyal, A., Street, R.: An introduction to Tannaka duality and quantum groups. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Category Theory Proceedings, Como 1990. LNM, vol. 1488, pp. 411–492. Springer, Berlin (1991)

    Google Scholar 

  16. Kleene, S.C.: Recursive predicates and quantifiers. Transactions of the American Mathematical Society 53(1), 41–73 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kock, J.: Frobenius Algebras and 2D Topological Quantum Field Theories. London Mathematical Society Student Texts, vol. 59. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  18. William Lawvere, F.: Ordinal sums and equational doctrines. In: Seminar on Triples, Categories and Categorical Homology Theory. Lecture Notes in Mathematics, vol. 80, pp. 141–155. Springer, Heidelberg (1969)

    Chapter  Google Scholar 

  19. Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pati, A.K., Braunstein, S.L.: Impossibility of deleting an unknown quantum state. Nature 404, 164–165 (2000)

    ADS  Google Scholar 

  21. Pavlovic, D.: Geometry of abstraction in quantum computation. In: TANCL 2007 (2007) (manuscript)

    Google Scholar 

  22. Pavlovic, D.: Maps I: relative to a factorisation system. J. Pure Appl. Algebra 99, 9–34 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pavlovic, D.: Maps II: Chasing diagrams in categorical proof theory. J. of the IGPL 4(2), 1–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pavlovic, D.: Categorical logic of names and abstraction in action calculus. Math. Structures in Comp. Sci. 7, 619–637 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pavlovic, D., Abramsky, S.: Specifying interaction categories. In: Moggi, E., Rosolini, G. (eds.) CTCS 1997. LNCS, vol. 1290, pp. 147–158. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  26. Selinger, P.: Dagger compact closed categories and completely positive maps. Electron. Notes Theor. Comput. Sci. 170, 139–163 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Spekkens, R.W.: Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A 75, 30 p. (2007)

    Google Scholar 

  28. Vicary, J.: Categorical formulation of quantum algebras, 37 p. (2008)

    Google Scholar 

  29. Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kestrel Institute and Oxford University, UK

    Dusko Pavlovic

Authors
  1. Dusko Pavlovic
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, Queensland University of Technology, GPO Box 2434, 4001, Brisbane, Australia

    Peter Bruza

  2. Naval Research Laboratory, Navy Center for Applied Research in Artificial Intelligence, 4555 Overlook Avenue S.W, DC 20375, Washington, USA

    Donald Sofge

  3. Paine College, 1235 15th Street, P.O. Box, GA 30901-3182, Augusta, USA

    William Lawless

  4. Department of Computing Science, University of Glasgow, 17 Lilybank Gardens, G12 8QQ, Glasgow, UK

    Keith van Rijsbergen

  5. German Research Center for Artificial Intelligence, Stuhlsatzenhausweg 3, 66123, Saarbrücken, Germany

    Matthias Klusch

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pavlovic, D. (2009). Quantum and Classical Structures in Nondeterminstic Computation. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen, K., Klusch, M. (eds) Quantum Interaction. QI 2009. Lecture Notes in Computer Science(), vol 5494. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00834-4_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00834-4_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00833-7

  • Online ISBN: 978-3-642-00834-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation