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Nice paper! Yes, I think you got the SEPRB example just about right… As you note, we didn’t really commit to an intermediate ontology, so that’s where things get tricky, but even without committing to a particular ontology I think you’re perfectly justified in making your conclusions.
Still, since I’m quite interested in the pros and cons of possible intermediate ontologies, and you’re proposing something a bit different starting on Page 8, I’d like to delve into the intermediate details, a bit…
First of all, before you start dealing with the L/R asymmetry on Page 9, you of course have a past-future asymmetry in SEPRB. One strange way that this manifests itself, that you might not have considered, is what happens in a series of consecutive measurements. You’re right in footnote 5 that it’s *possible* to assign the correct probabilities to all of these hidden properties. But as soon as one of them is measured, the rest of the properties have to shift their probabilities in a strange and time-asymmetric manner (in order to account for the distribution that would be observed at a future measurement).
Okay, then we get into the time-symmetric version where you note that all these w- and z- probabilities are conditional on each other. I think this part would be greatly enhanced by explicitly noting the unconditional (joint!) probabilities P(w,z), from which your conditional probabilities can be derived. You could even list the 8 equal-joint-probability cases:
w | z
If each of these 8 cases have an equal chance of jointly occurring, you can easily show that you get the correct marginal probabilities for w and z, as well as the correct conditional probabilities for P(w|z) and P(z|w). As it is now, you only really tell the conditional-probability account, leaving it somewhat unclear what the unconditional story you’re telling really looks like. I think having this explicit account would strongly support the epistemic/ontic distinction you’re making in the last paragraph of section 3.
Now, to the question of your ontology (having a built-in outcome associated with each possible measurement direction), and how it constrasts with the “real polarization” ontology that Huw and I have been using. The downside of our approach is that we either need *two* polarizations (one controlled from each side), or else an anomalous rotation from one to the other. You don’t have this problem, which seems like a plus.
But another issue is whether one needs a discontinuity at a given measurement. These Stern-Gerlach experiments can easily be staged, one after the other, simply by blocking one of the two output ports, and sending the open port directly into another experiment. If the blocked port doesn’t detect a particle, then you have “measured” the intermediate spin of the particle without an entropy-increasing interaction.
As I noted above, all of your beables would generally have to shift at such an intermediate measurement. Maybe this is okay, but I strongly lean toward stories where one doesn’t need any sudden change at all in such non-invasive scenarios. That’s why I don’t particularly like even the two-polarization stories (for which one polarization is unchanged, but the other one changes). I much prefer the anomalous rotation account, where there’s only one “beable”, and it doesn’t need to do anything at all when it passes through such an intermediate measurement.
But that point aside, I actually think your approach here is easier to analyze than our polarization examples… and I may be stealing it for general-audience summaries! 🙂
One last nitpick: I thought the statement “there is NO sense in which [preparation and measurement] are the time-reverse of each other” was too strong… I think there are lots of senses in which they are, even taking our human asymmetries into account.
Thanks for a thought-provoking paper!