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.
Allen, John-Mark A., Jonathan Barrett, Dominic C. Horsman, Ciarán M. Lee, and Robert W. Spekkens. 2017. “Quantum Common Causes and Quantum Causal Models.” Physical Review X 7 (3): 031021. https://doi.org/10.1103/PhysRevX.7.031021.
Baez, John, and Jacob Biamonte. 2012. “A Course on Quantum Techniques for Stochastic Mechanics.” arXiv:1209.3632, September. http://arxiv.org/abs/1209.3632.
Carmi, Avishy Y., and Daniel Moskovich. 2015. “Statistics Limits Nonlocality,” July. http://arxiv.org/abs/1507.07514.