Discrete quantum structures II: Examples

Abstract

Part I of this paper established the basic properties of quantum predicate logic as the internal logic of discrete quantum structures. We now show that a majority of the established quantum generalizations of discrete structures are naturally axiomatizable within this internal logic. In particular, we axiomatize the quantum graphs of Duan, Severini, and Winter, the quantum metric spaces of Kuperberg and Weaver, the quantum isomorphisms of Atserias, Mančinska, Roberson, Šámal, Severini, and Varvitsiotis, and the quantum groups of Woronowicz. In each instance, we consider only those structures that are discrete in the sense that the underlying von Neumann algebra is hereditarily atomic.