Reference request: “mathematical systems for probability”

This question is in response to an answer here on Physics.SE, but is essentially about math.

There are basically two kinds of mathematical systems that can yield a nontrivial formalism for probability. One is the kind we're familiar with from everyday life... But there's another system of probability, very different from what you and I are used to. It's a system where each event has an associated vector (or complex number), and the sum of the squared magnitudes of those vectors (complex numbers) is 1... There are only these two systems. It is mathematically proven that you couldn't have, say, an amplitude that must be raised to the 4th power. There is only classical probability as we know it and the quantum kind.

For me, this raises at least the following two questions:

1. Are the quoted assertions correct?

(It seems to be referencing the fact that the only two division algebras over the real numbers which are fields are $\mathbb{R}$ itself and $\mathbb{C}$, but I'm not sure how that result applies to a discussion of formalisms of probability.)

2. Does any one have a reference for this alternative formulation of probability, which is ideally as mathematical as possible?

(I.e. avoiding or not focusing on "physical interpretation" or quantum mechanics, since I am not familiar with quantum mechanics, rather something like Durrett's book, Probability: Theory and Examples, but for $\mathbb{C}$-valued "probability" instead of the typical/original theory.)

• The term you want to look up is "quantum probability" or maybe "noncommutative probability" depending on taste. For a relatively sophisticated discussion see qchu.wordpress.com/2012/08/18/noncommutative-probability. As a simple toy case, over a finite state space probability distributions are not given by vectors of nonnegative real numbers summing up to $1$, but by vectors of complex numbers with norm $1$. From here you can write down "quantum Markov chains" given by unitary instead of stochastic matrices, and then the Born rule describes observations. – Qiaochu Yuan Dec 15 '16 at 17:50