I'm sure this question is a bit gauche for this site, but I'm just a mathematician trying to piece together some physical intuition.
*Question:*Is the statistical interpretation of Quantum Mechanics still, in any sense, viable? Namely, is it completely ridiculous to regard the theory as follows: Every system corresponds to a Hilbert space, to each class of preparations of a system corresponds to a state functional and to every class of measurement procedure there is a self-adjoint operator, and finally, a state functional evaluated at one of these self-adjoint operators yields the expected value of numerical outcomes of measurements from the class of measurement procedures, taken over the preparations represented by the state?
I am aware of Bell's inequalities and the fact that the statistical interpretation can survive in the absence of locality, and I am aware of the recent work (2012) which establishes that the psi-epistemic picture of quantum mechanics is inconsistent with quantum predictions (so the quantum state must describe an actual underlying physical state and not just information about nature). Nevertheless, I would really like a short summary of the state of the art with regard to the statistical interpretation of QM, against the agnostic (Copenhagen interpretation) of QM, at present.
Is the statistical interpretation dead, and if it isn't...where precisely does it stand?
An expert word on this from a physicist would be very, very much appreciated. Thanks, in advance.
EDIT: I have changed the word "mean" to "expected" above, and have linked to the papers that spurred this question. Note, in particular, that the basic thing in question here is whether the statistical properties prescribed by QM can be applied to an individual quantum state, or necessarily to an ensemble of preparations. As an outsider, it seems silly to attach statistical properties to an individual state, as is discussed in my first link. Does the physics community share this opinion?
EDIT: Emilio has further suggested that I replace the word "statistical" by "operational" in this question. Feel free to answer this question with such a substitution assumed (please indicate that you have done this, though).