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?

As you have said, every GPT is defined by a positive cone in Banach space and a positive cone in its dual, by Ludwig’s embedding theorem. Von Neumann algebras are also Banach spaces, and the space of states is (up to normalisation) the positive cone of the predual.

In finite dimensions, Koecher–Vinberg gives conditions for the cone of effects to be the cone of a Jordan algebra. To restrict the case to von Neumann algebras, I am not sure.

Fun fact, its known that demanding no "higher-order Sorkin interference", i.e. that the 3-slit experiment results can be derived from all the combinations of 1 and 2 slits, is equivalent to picking our a Jordan algebra plus a few other natural ideas about functional calculus of observables. So if you want the physical answer, I believe it would be something along the lines of the usual GPT axioms + https://arxiv.org/pdf/0912.0203

Thanks!

Does the Koecher-Vinberg theorem not work in infinite dimensions? If not, do you know if there's a more general theorem that works in infinite dimensional systems?

Well hopefully someone else can give the restriction to von Neumann algebras part, or maybe I can do some googling for theorems making Jordan algebras into von Neumann algebras.

Fun fact, its known that demanding no "higher-order Sorkin interference", i.e. that the 3-slit experiment results can be derived from all the combinations of 1 and 2 slits, is equivalent to picking our a Jordan algebra

Sorry, I don't quite understand this. Are you saying that no higher order Sorkin interference is necessary and sufficient to enforce the GPT being a Jordan algebra?

Koecher's original paper was "Positivitatsbereiche im R^(n)" so it would have to be a later extension. I'm not so familiar with the existing proofs, and most GPT work is in finite dimensions so I don't know if there is a canonical reference that is good for infinite dimensions.

I believe the higher order interference papers (which start with finite dims) end up constraining to "Euclidean" Jordan algebras, which include the symmetric/self-adjoint subalgebra of M\_n(\\mathbb F) (recall that the Jordan algebras have the Jordan product, which makes the the symmetric/self-adjoint matrices into an algebra) where the field can be real, complex or quaternion, and then two strange cases. Certainly the last of the strange cases (called the exceptional Jordan algebra) is not associative, and the quarternions are not either, so associativity restricts you a lot already, possibly completely to the real/complex matrices.

\> Sorry, I don't quite understand this. Are you saying that no higher order Sorkin interference is necessary and sufficient to enforce the GPT being a Jordan algebra?

No higher order interference is sufficient when coupled with the axioms in the paper above, or similarly those in this one https://arxiv.org/abs/1704.05106. Necessary also seems likely but I don't know of any proof off the top of my head.