Hi Alan,
Sorry it took me so long to get back here…
On the joint-probability issue, I see two ways to go. In my first response, way above, I mentioned that one also needs the rule that tells one how to figure out which allowed outcomes correspond to which measurement settings/geometries. I would think that this rule would solve the problem you mention, in that it would be this rule that told you the outcomes with non-zero probability, and then P(A,B) that would allow you to work out the conditional probabilities for the allowed outcomes.
But the other, better, way to go would be to have a function P(A,B) that went to zero (or nearly zero) whenever B was a non-allowed (non-eigenstate) outcome. That’s what I was trying to accomplish in http://arxiv.org/abs/1301.7012 , and I should try to get back to that style of story at some point (I’ve put it aside for now). From an experimental-verification perspective, I find this a very promising path forward, because if there was some tiny-but-nonzero probability of a non-eigenstate outcome, this might be detectable with current technology — and better yet, also plausible that most experimentalists would have thrown out any prior indications of such outcomes as experimental error (since there’s no framework in standard QM that would predict such things).
As far as the hidden variable space goes, I absolutely agree that the global phase should be part of that space (it’s by far the most natural piece!), but you might like to see where this starting point leads for the case of spin-1/2 systems (where the global phase is hard to define without hitting coordinate singularities). The result is a much larger hidden variable space. (Also, such larger spaces are generally needed if we’re going to expand the sort of hidden-variable models that Nathan Argaman and I are developing to partially-entangled states.) If you’re interested, you can take a look at the recent JPA paper: http://iopscience.iop.org/1751-8121/48/23/235302. Another motivation for such a large hidden variable space comes from relativity, where second-order-in-time equations are far more natural than the first order ones (standard quantum theory knows how to handle the latter).
Best, Ken
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