What can be a scientific solution to the Q-measurement problem (other than many worlds idea)? Can it be somehow verified through experiment?
The quantum measurement problem cannot be satisfied by an experiment really. Suppose I say that the God can do anything (I do not believe in God). This hypothesis cannot be disproved by any experiment. The same is the problem of QM.
A solution to the quantum measurement problem can only be experimentally meaningful if it conflicts with ordinary QM, since the existence of Everett interpretations shows that it is consistent to view quantum mechanics by itself as mathematically complete.
An example of a resolution to the quantum measurement problem is a hidden variables theory that reproduces QM approximately for a few particles, but fails to reproduce QM for many highly entangled particles. Such hidden variables schemes can plausibly allow a quantum computer to fail without allowing measurable deviations for less highly entangled quantum systems. Nobody has an agreed example of such a scheme of hidden variables today.
For me, the main question is whether quantum computation actually works. If it does, then QM is right, and you might as well believe Everett. If it doesn't work when factoring 10,000 digit numbers, because of fundamental decoherence, then you expect that the theory is really hidden variables on the cosmological horizon reproducing approximate QM. Those are the two real options as I see it, everything else is philosophy.
I don't think Everett Hypothesis or the Bohemian Interpretation of quantum mechanics is correct. They are devised to are escape from real issues.
If one assumes measuring apparatus exist, without asking for how they have been made and how they work, then there is no doubt that Quantum Mechanics as formulated by Heisenberg, Schrodinger or Feynman is complete and has nothing out of place. Niels Bohr understood the meaning behind Quantum mechanics. He asked what does it mean to observe?
The Issue with measuring apparatus, in my opinion has not been completely understood. Decoherence offers a partial solution that can be applied in some situations.
I think what the key missing ingredient is What does it mean to make an irreversible record on the measurement apparatus? I think this question is closely related to Thermodynamics. It appears that Free energy must be expended during measurement,(or vise verse free energy must be expended to reset the state of an apparatus) and inevitably Entropy of the measurement apparatus is increased after the experiment has been performed.
Feynman seems to have understood clearly, what the right questions are. In my opinion as far as I have spoken to people, These questions have been not been fully answered.
“We and our measuring instruments are part of nature and so are, in principle, described by an amplitude function satisfying a deterministic equation. Why can we only predict the probability that a given experiment will lead to a definite result? From what does the uncertainty arise? Almost without a doubt it arises from the need to amplify the effects of single atomic events to such a level that they may be readily observed by large systems.” -Feynman
“. . . In what way is only the probability of a future event accessible to us, whereas the certainty of a past event can often apparently be asserted?. . . Obviously, we are again involved in the consequences of the large size of ouselves and of our measuring equipment. The usual separation of observer and observed which is now needed in analyzing measurements in quantum mechanics should not really be necessary, or at least should be even more thoroughly analyzed. What seems to be needed is the statistical mechanics of amplifying apparatus.” -Feynman