Abstract
We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical dynamics. We also prove Representation Theorems, showing these axiomatizations to be complete with respect to the natural Hilbert-space semantics. The advantages of this setting are many: (1) it provides a clear and intuitive dynamic-operational meaning to key postulates (e.g. Orthomodularity, Covering Law); (2) it reduces the complexity of the Solèr–Mayet axiomatization by replacing some of their key higher-order concepts (e.g. “automorphisms of the ortholattice”) by first-order objects (“actions”) in our structure; (3) it provides a link between traditional quantum logic and the needs of quantum computation.
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References
Abramsky, S. and Coecke, B. (2004). A categorial semantics of quantum protocols. In Proceedings of the 19th annual IEEE symposium on Logic (LICS) in Computer Science. IEEE Computer Society, 2004. Available at arXiv:quant-ph/0402130.
Amemiya, I. and Araki, H. (1967). A remark on Piron’s paper. Publications of the Research Institute of Mathematical Sciences, Kyoto University, Series A 2, 423–427.
Amira, H., Coecke, B., and Stubbe, I. (1998). How quantales emerge by introducing induction within the operational approach. Helvetica Physica Acta 71, 554–572.
Baltag, A. and Smets, S. (2004). The logic of quantum programs. In Proceedings of the 2nd International Workshop on Quantum Programming Languages (QPL2004), TUCS General Publication, P. Selinger, ed., Turku Center for Computer Science, Finland, vol. 33, pp. 39–56.
Beltrametti, E. G. and Cassinelli, G. (1977). On state transformations induced by yes-no experiments, in the context of quantum logic. Journal of Philosophical Logic 6, 369–379.
Coecke, B. and Smets, S. (to appear). The Sasaki hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes. International Journal of Theoretical Physics. Available at arXiv: quant-ph/0111076.
Coecke, B. and Stubbe, I. (1999). On a duality of quantales emerging from an operational resolution. International Journal of Theoretical Physics 38, 3269–3281.
Coecke, B., Moore, D. J., and Smets, S. (2004). Logic of dynamics & dynamics of logic; some paradigm examples. Logic, Epistemology and the Unity of Science, S. Rahman, J. Symons, D. M. Gabbay, and J. P. Van Bendegem, eds., Kluwer Academic, Dordrecht.
Coecke, B., Moore, D. J., and Stubbe, I. (2001). Quantaloids describing causation and propagation for physical properties. Foundations of Physics Letters 14, 357–367. Available at (arXiv: quant-ph/0009100).
Dalla Chiara, M., Giuntini, R., and Greechie, R. (2004). Reasoning in Quantum Theory, Sharp and Unsharp Quantum Logics, Kluwer Academic, Dordrecht .
Daniel, W. (1982). On the non-unitary evolution of quantum systems. Helvetica Physica Acta 55, 330–338.
Daniel, W. (1989). Axiomatic description of irreversible and reversible evolution of a physical system. Helvetica Physica Acta 62, 941–968.
Faure, C. L.-A., Moore, D. J., and Piron, C. (1995). Deterministic evolutions and schrodinger flows. Helvetica Physica Acta 68, 150–157.
Foulis, D. J. and Randall, C. H. (1971). Lexicographic orthogonality. Journal of Combinatorial Theory 11(2), 157–162.
Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic 3, 19–35.
Goldblatt, R. (1984). Orthomodularity is not elementary. The Journal of Symbolic Logic 49, 401–404.
Hardegree, G. M. (1975). Stalnaker conditional and quantum logic. Journal of Philosophical Logic 4, 399–421.
Hardegree, G. M. (1979). The conditional in abstract and concrete quantum logic. The Logico-Algebraic Approach to Quantum Mechanics, C. A. Hooker, ed., Vol. 2, D. Reidel, Dordrecht.
Harel, D., Kozen, D., and Tiuryn, J. (2000). Dynamic Logic, MIT-Press, Massachusetts.
Jauch, J. M. (1968). Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.
Jauch, J. M. and Piron, C. (1969). On the structure of quantal proposition systems. Helvetica Physica Acta 42, 842–848.
Keller, H. A. (1980). Ein nichtklassischer hilbertscher Raum. Mathematische Zeitschrift 172, 41–49.
Mayet, R. (1998). Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms. International Journal of Theoretical Physics 37(1), 109–114.
Pauli, W. (1980). Handbuch der Physik, Vol. 5, Part 1: Prinzipien der Quantentheorie 1 (1958); English translation by P. Achuthan and K. Venkatsesan: General Principles of Quantum Mechanics, Springer Verlag, Berlin.
Piron, C. (1964). Axiomatique quantique (PhD-Thesis), Helvetica Physica Acta, 37, 439–468, English Translation by M. Cole: “Quantum Axiomatics” RB4 Technical memo 107/106/104, GPO Engineering Department (London).
Piron, C. (1976). Foundations of Quantum Physics, W.A. Benjamin, Massachusetts.
Smets, S. (2001). On Causation and a counterfactual in quantum logic: The Sasaki hook. Logique et Analyse 173–175, 307–325.
Smets, S. (2003). In defense of operational quantum logic. Logic and Logical Philosophy 11, 191–212.
Smets, S. (to appear). From intuitionistic logic to dynamic operational quantum logic. Poznan Studies in Philosophy and the Humanities.
Solèr, M. P. (1995). Characterization of Hilbert spaces by orthomodular spaces. Communications in Algebra 23(1), 219–243.
van Benthem, J. (1996). Exploring Logical Dynamics, Studies in Logic, Language and Information, CSLI Publications, Stanford.
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PACS: 02.10.-v Logic; set theory and algebra; 03.65.-w Quantum mechanics; 03.65.Fd Algebraic methods; 03.67.-a Quantum information.
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Baltag, A., Smets, S. Complete Axiomatizations for Quantum Actions. Int J Theor Phys 44, 2267–2282 (2005). https://doi.org/10.1007/s10773-005-8022-2
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DOI: https://doi.org/10.1007/s10773-005-8022-2