Duality for the Logic of Quantum Actions

In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” (Baltag et al. in Int J Theor Phys, 44(12):2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a (generalized) Hilbert space. The other approach considers a “Piron lattice” (Piron in Foundations of Quantum Physics, 1976), which characterizes the algebra of closed linear subspaces of a (generalized) Hilbert space. We define categories of these two sorts of structures and show a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice.