My initial question below appears to be unclear so I am rewording in more succinctly here. The pre-edit question remains below.
There exists two types of probability theory: 1-norm (classic stochastic theory) and 2-norm (quantum mechanics). 1-norm has been shown to emerge from the lack of information about a deterministic process (statistical mechanics). Have we proven that a 2-norm probability theory cannot possibly emerge from a lack of information about a deterministic process?
From my understanding a ‘hidden variable theory’ is just saying that the current quantum theory is not a complete description of nature, and just like thermodynamics can be reduced to deterministic rules (statistical mechanics), the probabilistic nature of the wave function can be reduced to deterministic rules. Hidden variable theory simply says that it is possible traditional quantum theory is an emergent theory that results from lack of information about the specific state of this new 'hidden variable'.
One crucial difference to note here is that ‘probability’ in the quantum sense is very different than traditional probability. The probabilistic state evolves unitarily instead of stochastically and the traditional probability is recovered by squaring the wave function. This leads to interference and most of the oddities associated with quantum theory.
Scott Aaronson in his paper here reduces the possibility of hidden variable theory to whether we can map unitary to stochastic matrices in a way that has certain desirable properties. Isn’t this just proving that quantum theory cannot be explained with ‘traditional’ stochastic probability? Couldn’t there still be a way to reduce this ‘new’ type of probability down to deterministic rules with missing information (again just like statistical mechanics did for stochastic probability). Is stochastic probability the only possible probability theory that can result from a lack of information? If so, why?