Abstract
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.
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References
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)
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)
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)
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/0402130
Abramsky, S., Coecke, B.: Abstract physical traces. Theory and Applications of Categories 14, 111–124 (2005)
Bell, J.S.: On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics (1966)
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)
Best, E., Devillers, R.: Sequential and concurrent behaviour in Petri net theory. Theoretical Computer Science (1987)
Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D.: Spacetime as a causal set. Phys. Rev. Lett. 59, 521–524 (1987)
Cardelli, L., Gordon, A.D.: Mobile Ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 140–155. Springer, Heidelberg (1998)
Carnap, R.: Introduction to Symbolic Logic with Applications. Dover Books (1958)
Dirac, P.A.M.: The Principles of Quantum Mechanics, 3rd edn. Oxford University Press, Oxford (1947)
Gehlot, V., Gunter, C.: Normal process representatives. In: Proceedings LiCS (1990)
Girard, J.-Y.: Linear Logic. Theoretical Computer Science 50(1), 1–102 (1987)
Joyal, A., Street, R.: The geometry of tensor calculus I. Advances in Mathematics 88, 55–112 (1991)
Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)
Landauer, R.: Information is physical. Physics Today 44, 23–29 (1991)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (1998)
Martin, K., Panangaden, P.: A Domain of spacetime intervals for General Relativity. Communications of Mathematical Physics (November 2006)
Marti-Oliet, N., Meseguer, J.: From Petri Nets To Linear Logic Through Categories. IJFCS 2(4), 297–399 (1991)
Meseguer, J., Montanari, U.: Petri Nets Are Monoids. Information and Computation 88, 105–155 (1990)
Milner, R.: Calculi for Interaction. Acta Informatica 33 (1996)
Milner, R.: Pure bigraphs: Structure and dynamics. Inf. Comput. 204(1), 60–122 (2006)
Petri, C.A.: Fundamentals of a Theory of Asynchronous Information Flow. IFIP Congress, 386–390 (1962)
Petri, C.-A.: State-Transition Structures in Physics and in Computation. International Journal of Theoretical Physics 21(12), 979–993 (1982)
Petri, C.A.: Nets, Time and Space. Theor. Comput. Sci. 153(1–2), 3–48 (1996)
Smolin, L.: Three Roads to Quantum Gravity. Phoenix (2000)
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)
Winskel, G.: Petri nets, algebras, morphisms and compositionality. Information and Computation 72, 197–238 (1987)
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Abramsky, S. (2008). Petri Nets, Discrete Physics, and Distributed Quantum Computation. In: Degano, P., De Nicola, R., Meseguer, J. (eds) Concurrency, Graphs and Models. Lecture Notes in Computer Science, vol 5065. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68679-8_33
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DOI: https://doi.org/10.1007/978-3-540-68679-8_33
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