International Journal of Theoretical Physics

, Volume 44, Issue 12, pp 2267–2282 | Cite as

Complete Axiomatizations for Quantum Actions

Article

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.

Keywords

dynamic quantum logic quantum frames quantum dynamic algebra quantum transition systems quantales Piron lattices 

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Oxford University Computing LaboratoryOxford UniversityOxfordUnited Kingdom
  2. 2.Vrije Universiteit BrusselBrusselBelgium
  3. 3.Vrije Universiteit Brussel, Flanders’ Fund for Scientific Research Post-DocBelgium

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