The way I interpret your question is: deterministic = pure state on an abelian C${}^*$-algebra, random = arbitrary state on an abelian C${}^*$-algebra, quantum = pure state on an arbitrary C${}^*$-algebra. There's one further level of generality, arbitrary state on an arbitrary C${}^*$-algebra, which gives you statistical ensembles of quantum states.
(Note that under this interpretation you do not have "random $\subset$ quantum", unless your "quantum" includes ensembles.)