What conditions distinguish noncommutative probability from general GPTs?

I know in quantum foundations, a lot of research studies physical theories via the concept of a generalized probabilistic theories (GPTs). In a GPT, states and outcomes are represented in essentially the most general possible way that could still conceivably be called a "theory of physics". GPTs are nice because they encompass classical mechanics and quantum mechanics (along with many other possibilities).

Another structure that can describe both classical and quantum mechanics is the notion of noncommutative probability. In this context observables are a subset of a von Neumann algebra, and states are specific functionals on that algebra.

Every noncommutative probability space has an associated GPT, but not every GPT can be represented by noncommutative probability.

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In some sense, noncommutative probability is enough to describe all "reasonable" physics theories (quantum mechanics, classical mechanics, probabilistic restrictions of those).

As such, I'm curious about whether there's a set of reasonable, or physically motivated conditions you could add to the theory of GPTs to get noncommutative probability.

Does anyone have any answers about this, or have any papers/books they'd recommend looking at to learn more about this?