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I don't see where the problem is. I use a magnet and choose one of the two ensembles by beam-dumping the other. Then I've a new ensemble. It's a preparation procedure in two steps.A. Neumaier said:The main difference is that mathematicians are trained to use precise concepts and smell easily when something informal cannot be made precise. This is usually the case where informal mathematical arguments go wrong, hence our sensitivity to lack of conceptual precision. Physicists are much more liberal in this respect and aim for precision only when their informal thinking led them completely astray. Thus they tend not to notice the many subtleties inherent in the quest for good conceptual foundations.
But I asked for the precise physical definition of what you call an ensemble (as contrasted to state). In your description the term didn't occur: you only explained the meaning of states and observables.
This is only mock-mathematical, as preparation procedures are no mathematical entities, and the equivalence relation in question is not even specified.
According to which notion of ensemble? That's the question of interest in the present context.
If ''ensemble of colliding proton beams'' is just another phrase for ''several colliding proton beams'' then it is plain wrong to later claim subensembles by filtering according to spin, say. The magnet creates two beams from one, hence two ensembles from one.
So what else did you mean?
Of course, a complete random experiment is only defined if also the measurement on the prepared state is given, and only then the Kolmogorov axioms make sense.
I also have nothing against mathematical regidity and refinements of physical definitions, but it's impossible to make sense of a mathematical abstract prescription like the POVM, if there's not a single example, where it is applied to a real-world experiment. It would be great, if there'd be a simple example, like the one I tried somewhere in this Forum about a position measurement with a detector. So far I've only seen very abstract descriptions with no reference to a real-world measurement.