Petri Nets, Discrete Physics, and Distributed Quantum Computation

  • Samson Abramsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5065)


We shall describe connections between Petri nets, quantum physics and category theory. The view of Net theory as a kind of discrete physics has been consistently emphasized by Carl-Adam Petri. The connections between Petri nets and monoidal categories were illuminated in pioneering work by Ugo Montanari and José Meseguer. Recent work by the author and Bob Coecke has shown how monoidal categories with certain additional structure (dagger compactness) can be used as the setting for an effective axiomatization of quantum mechanics, with striking applications to quantum information. This additional structure matches the extension of the Montanari-Meseguer approach by Marti-Oliet and Meseguer, motivated by linear logic.


Bell State Monoidal Category Linear Logic Monoidal Structure Symmetric Monoidal Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abramsky, S.: Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 1–31. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    Abramsky, S.: What are the fundamental structures of concurrency? We still don’t know! In: Proceedings of the Workshop Essays on Algebraic Process Calculi (APC 25). Electronic Notes in Theoretical Computer Science, vol. 162, pp. 37–41 (2006)Google Scholar
  3. 3.
    Abramsky, S.: Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics. In: Chen, G., Kauffman, L., Lomonaco, S. (eds.) Mathematics of Quantum Computation and Quantum Technology, pp. 515–558. Taylor and Francis, Abington (2007)Google Scholar
  4. 4.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425 (2004), arXiv:quant-ph/0402130Google Scholar
  5. 5.
    Abramsky, S., Coecke, B.: Abstract physical traces. Theory and Applications of Categories 14, 111–124 (2005)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bell, J.S.: On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics (1966)Google Scholar
  7. 7.
    Bennet, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wooters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters 70, 1895–1899 (1993)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Best, E., Devillers, R.: Sequential and concurrent behaviour in Petri net theory. Theoretical Computer Science (1987)Google Scholar
  9. 9.
    Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D.: Spacetime as a causal set. Phys. Rev. Lett. 59, 521–524 (1987)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Cardelli, L., Gordon, A.D.: Mobile Ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 140–155. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Carnap, R.: Introduction to Symbolic Logic with Applications. Dover Books (1958)Google Scholar
  12. 12.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 3rd edn. Oxford University Press, Oxford (1947)zbMATHGoogle Scholar
  13. 13.
    Gehlot, V., Gunter, C.: Normal process representatives. In: Proceedings LiCS (1990)Google Scholar
  14. 14.
    Girard, J.-Y.: Linear Logic. Theoretical Computer Science 50(1), 1–102 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Joyal, A., Street, R.: The geometry of tensor calculus I. Advances in Mathematics 88, 55–112 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Landauer, R.: Information is physical. Physics Today 44, 23–29 (1991)CrossRefGoogle Scholar
  18. 18.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  19. 19.
    Martin, K., Panangaden, P.: A Domain of spacetime intervals for General Relativity. Communications of Mathematical Physics (November 2006)Google Scholar
  20. 20.
    Marti-Oliet, N., Meseguer, J.: From Petri Nets To Linear Logic Through Categories. IJFCS 2(4), 297–399 (1991)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Meseguer, J., Montanari, U.: Petri Nets Are Monoids. Information and Computation 88, 105–155 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Milner, R.: Calculi for Interaction. Acta Informatica 33 (1996)Google Scholar
  23. 23.
    Milner, R.: Pure bigraphs: Structure and dynamics. Inf. Comput. 204(1), 60–122 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Petri, C.A.: Fundamentals of a Theory of Asynchronous Information Flow. IFIP Congress, 386–390 (1962)Google Scholar
  25. 25.
    Petri, C.-A.: State-Transition Structures in Physics and in Computation. International Journal of Theoretical Physics 21(12), 979–993 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Petri, C.A.: Nets, Time and Space. Theor. Comput. Sci. 153(1–2), 3–48 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Smolin, L.: Three Roads to Quantum Gravity. Phoenix (2000)Google Scholar
  28. 28.
    Sorkin, R.D.: A Specimen of Theory Construction from Quantum Gravity. In: Leplin, J. (ed.) The Creation of Ideas in Physics: Studies for a Methodology of Theory Construction (Proceedings of the Thirteenth Annual Symposium in Philosophy, held Greensboro, North Carolina, March, 1989, pp. 167–179. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  29. 29.
    Winskel, G.: Petri nets, algebras, morphisms and compositionality. Information and Computation 72, 197–238 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Samson Abramsky
    • 1
  1. 1.Oxford University Computing Laboratory 

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