Hoisted from the comments, Robin asks:

So, with that in mind, here’s a question. What do you think about teaching quantum mechanics as noncommutative probability theory? In other words, by starting with probability theory and alluding to probabilistic mechanics (e.g., distributions on phase space), and then introducing quantum theory as a generalization of probability.

This is how I think of quantum theory all the time now — and it makes tremendous sense to me. I think it’s how I want to teach it. And I’m curious what y’all think.

This is roughly how I like to introduce quantum theory, although I’m not sure that non-commutative probability theory is the right description. I mean, classical probability theory is not commutative: the order of dynamics certainly matters in both deterministic and classic theories. I mean pressing the accelerator and turning the wheel is different than turning the wheel and pressing the accelerator. In one you hit grandma, and in the other you miss her. But this got me thinking: what is the proper words to describe this approach: where one begins by introducing probabilistic theories and then moves to quantum theory. Probability theory with the wrong norm?