QS7 Complete Axiomatizations for Quantum Actions A. Baltag ∗ and S. Smets † Abstract We present two equivalent axiomatizations for a logic of quantum ac- tions: 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 postu- lates (e.g. Orthomodularity, Covering Law); 2) it reduces the complexity of the Sol` er-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 tradi- tional quantum logic and the needs of quantum computation. KEY WORDS: Dynamic Quantum Logic, Quantum Frames, Quantum Dynamic Algebra, Quantum Transition Systems, Quantales, Piron lat- tices. PACS: 02.10.-v Logic, set theory and algebra, 03.65.-w Quantum mechan- ics, 03.65.Fd Algebraic methods, 03.67.-a Quantum information 1 Introduction Our research is connected to the recent trend towards a “dynamification” of logic, development pursued (mainly, but not exclusively) by the “Dutch school” in modal logic, see e.g. [31]: looking at various “propositional” logics as being about actions, rather than propositions. This is also connected to the older work (originating in Computer Science) on Propositional Dynamic Logic (PDL) and its relatives (such as Hoare logic). More generally, there is already a whole tra- dition in Computer Science of thinking about information systems in a dynamic manner: a “state” of a system is, in this view, identified only by the actions that can be (successfuly) performed on the state. This view is embodied in ∗ Oxford University Computing Laboratory, baltag@comlab.ox.ac.uk † Vrije Universiteit Brussel, Flanders’ Fund for Scientific Research Post-Doc, sons- mets@vub.ac.be 1