I would like to know what physicists think about the propensity viewpoint.If this latter one is in line with physics and especially Quantum Mechanics. Otherwise, what is the most coherent philosophical perspective on probability with Quantum Mechanics?
Most physicists don't care about the interpretation of probability. Nor do they care about the controversy over the "interpretation" of quantum mechanics.
The philosophy of probability is not in a good state. It is mostly divided between frequentists who think that probability is defined by long run frequency and Bayesians who think that probability has something or other to do with credence. No real experiment involves measuring an infinite set of systems, so the frequency interpretation is irrelevant to all real experiments. It's also difficult to see why credence should play any role at all in physics.
The propensity interpretation of probability has the merit of saying that a theory of probability has to be relevant to real experiments, but falls down when it comes to explaining that relevance.
There are various attempts to explain probability in quantum mechanics:
but most physicists don't seem to think there is any problem here worth considering. I think this is much to their discredit since having a theory where one of the most important concepts is unexplained makes properly understanding, testing, criticising and replacing that theory impossible. David Deutsch has made some interesting comments on these issues:
According to Richard Feynman this is exactly where the central mystery in QM lies.
Probability was implicated in QM right from the start. Recall, that probability in its usual sense - as frequentist is the technology that allowed Boltzmann to think of thermodynamics as the statistics of a large number of atoms and one of the central problems here was to work out the spectrum of black body radiation. A solution based on Boltzmanns theory was found by Wien & Raleigh but both were inaccurate in some domain, one in the low energy regime or the infrared and the other in the high energy or the ultraviolet.
It was Plancks achievement to come up with an accurate formula and he did this by postulating - as purely a mathematical fix - a discrete and indivisible amount of energy; this was reified by Einstein into quanta, the 'atoms' of energy.
It was Born that came up with the interpretation of the square of the quantum wave function as probability, but unlike Boltzmanns probability, which was epistemological, this was ontological.
The mystery is that physically speaking, quantum amplitudes are the square root of probabilities. There is no good explanation of this within the frequentist interpretation of why we can take the square root of probabilities and why this should make sense.
This suggests an analogy with the square root of -1 which was for centuries seen as a notion useful in helping to solve equations but had no ontological standing as some definite number until the 19th C when a geometric rather than arithmetic understanding was developed and then this became reified into a duality in mathematics between geometry and algebra, two perspectives on the same mathematical reality.
It may take likewise a similar amount of time before we can fully understand how probability is to be properly understood in QM.