A Graph Theoretic Perspective on CPM(Rel)

Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger’s CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched †-compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.