On real-valued classical probability as a special case of complex-valued quantum probability. More generally (?), non-commutative probability.
Chaitanya Rao, Notes on von Neumann’s algebra formulation of Quantum Mechanics has a side-by-side comparison table, and a reading list for getting up to speed fast.
David Ellerman On Classical and Quantum Logical Entropy: The analysis of measurement advertises an article on this theme.
Terry Tao, maximally general: Notes on Free probability.
Peter Woit has just made the final draft (dated 10/25/16) of his new textbook Quantum Theory, Groups and Representations: An Introduction available for free download.