I finally got as far as your main calculation (part IV in the paper). You have a two-state quantum system, a "qubit", and another two-state quantum system, a "photon". You make some assumptions about how the photon scatters from the qubit. Then you show that, given those assumptions, if the coefficients of the photon state are randomly distributed, then applying the Born rule to the eigenvalues of the old "objective state" (density operator) of the qubit, gives the probabilities for what the "dominant eigenstate" of the new objective state of the qubit will be (i.e. after the scattering).
My initial thoughts are 1) it's still not clear that this has anything to do with real physical processes 2) it's not surprising that an algebraic combination of quantum coefficients with random variables is capable of yielding new random variables with a Born-rule distribution 3) if you try to make this work in detail, you will end up with a new modification of quantum mechanics - perhaps a stochastic, piecewise-linear Bohmian mechanics, or just a new form of "objective collapse" theory - and not a derivation of the Born rule from within quantum mechanics.
Are you saying that actual physical systems contain populations of photons with randomly distributed coefficients such as you describe? edit Or perhaps just that this is a feature of electromagnetically mediated measurement interactions? It sounds like a thermal state, and I suppose it's plausible that localized thermal states are generically involved in measurement interactions, but these details have to be addressed if anyone is to understand how this is related to actual observation.
There must be something that you have fundamentally misunderstood. I will try to clear up some aspects that I think may cause this confusion.
First of all, the scattering processes presented in the paper are very generic to demonstrate the range of possible processes. The blog contains a specific realization which you may find closer to known physical processes.
Let me explain in detail again what this section is about, maybe this will help to overcome our misunderstanding. A photon scatters on a single qubit. The photon and the qubit each bring in a two dim...
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