DarMM
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Of course, but there is a set ##A## that is the intersection of the supports of all preparation states. This intersection has to overlap with at least one of the measurement states, or else an ontic state drawn from that sample could not lead to any of the outcomes. So if it overlaps with at least one, just consider that outcome state (regardless of which one it is).zonde said:Overlap assumption just says that there is overlap, it does not say where it is. If the support of ##|0\rangle\otimes|0\rangle## is not exactly the same as support of measurement state (i.e. outcome probability is less than 1) then overlap could be in the part that is excluded by the measurement state.
Then there will be one of the preparation states incompatible with that measurement state. The only way to avoid this would be if ##\left(\lambda,\lambda\right)## did "jump" upon measurement out of this measurement state's support. However this would only be possible if it knew about the epistemic state it was incompatible with.