During my regular* sit down with Nature magazine today I stumbled upon an intriguing quote with regards to attempts to reformulate quantum mechanics as a theory of observation under uncertainty:
“A lot of features we think of as uniquely quantum are generic to many probabilistic theories.”
Apparently various quantum phenomena, such as entanglement, can be shown to arise ‘naturally’ as the consequence of applying a few simple axioms formulated to describe the preparation and observation of a physical system. These so-called General Probablistic Theories (GPTs)—e.g. Hardy—are certainly at lot more physics-y than stats-y (for one thing, they allow only linear transformations on finite-dimensional vector spaces and their topological dual) but they do seem sufficiently powerful to recover a lot of observable quantum effects. I’ll need to read more of the GPT literature but at face value it seems like there could be some low hanging fruit for a more mathematical statistics based reformulation of these quantum theoretical reformulations!
* I’m a vegetarian!