In particular, has anyone tries to apply the bayesian framework to the whole "the wave collapses upon measurement" shebang?
Gelman, Yao - Holes in Bayesian Statistics
> Every philosophy has holes, and it is the responsibility of proponents of a philosophy to point out these problems. Here are a few holes in Bayesian data analysis: (1) the usual rules of conditional probability fail in the quantum realm, (2) flat or weak priors lead to terrible inferences about things we care about, (3) subjective priors are incoherent, (4) Bayes factors fail in the presence of flat or weak priors, (5) for Cantorian reasons we need to check our models, but this destroys the coherence of Bayesian inference.
In particular, you may appreciate the example and discussion in section 2:
> Probability theory isn’t true (quantum physics)
http://www.stat.columbia.edu/~gelman/research/unpublished/ba...
from Quantum mechanics: The Bayesian theory generalised to the space of Hermitian matrices by Alessio Benavoli, Alessandro Facchini, Marco Zaffalon
There are two caveats somewhat unique to quantum mechanics.
First, measurement has a known kickback effect on the state. So you always have to be careful to distinguish inferences of what the value of the state was before measurement vs inferences of the state after measurement. In general, after the observation has been made and the back-action accounted for, you know more about the post-measurement state than the pre-measurement state.
Second, many apparently different probability distributions of wavefunctions are observationally indistinguishable. For example, a qubit with a 50% chance of being definitely ON and a 50% chance of being definitely OFF is observationally indistinguishable from a qubit known to be in a state of the form 1/sqrt(2)|ON> + e^(i theta)/sqrt(2) |OFF> where theta is a uniform random variable ranging from 0 to 2pi. Those two hypotheses give identical predictions for all possible measurements. So people usually work with equivalence classes of probability distributions, represented by mathematical objects called density matrices, instead of individual probability distributions. Density matrices massively simplify everything, to the point where you might not even recognize the Bayes bits happening underneath.
https://www.quantamagazine.org/quantum-bayesianism-explained...
"Quantum measurement is nothing more, and nothing less, than a refinement and a readjustment of one’s initial state of belief. [...] Let us look at two limiting cases of efficient measurements. In the first, we imagine an observer whose initial belief structure ρ = |ψ⟩⟨ψ| is a maximally sharp state of belief. By this account, no measurement whatsoever can refine it. [...] The only state change that can come about from a measurement must be purely of the mental-readjustment sort: We learn nothing new; we just change what we can predict as a consequence of the side effects of our experimental intervention. That is to say, there is a sense in which the measurement is solely disturbance."
https://www.perimeterinstitute.ca/personal/cfuchs/Oviedo.pdf
By the way, the "collapse of the wave function" describes as well the transition from one state to another state. I don't understand what do you mean by "multiple simultaneous states".
He wrote a monograph on the approach: "Entropic Inference and the Foundations of Physics" which should be available at https://www.albany.edu/physics/ACaticha-EIFP-book.pdf but that link doesn't work now. You can find it here: https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsd...