A prolegomena to future posts (on economics)

Before i start muttering about Walrasian auctioneers and such i want to ask a question to the mathematically inclined. In quantum mechanics the claim is made that we effectively can interpret the operation bra Operator ket as a probability. This is all well and good and seems to line up with experimentation. But, from the point of view of mathematics there must be a justification of the form "i'm using this theory of probability, and we can interpret this operation in said theory thusly". So, on the basis of which theory of probability can we interpret the operation as giving up a probability? Is it frequentist? Is it Bayesian?

i'm asking this question in a serious spirit of inquiry. i don't want a hand-wavy argument. i want to see the paper where this is argued out in enough detail that i can do the calculations myself. So, if you, theoretical physicist or mathematician or logician, are reading this and know the answer, or know someone who might know, please post a comment with a reference.