Complete axiomatizations for quantum actions

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 Soler-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.