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In case I'm going from a wrong assumption here's how I understood the experiment:

The singlet electron-positron pair has spin zero, so if we measure one electron as +x (i.e. +1 on x axis) then the positron must have -x. If we measure the electron again then it still has +x (unless we did a measurement on another axis in the meantime). Since the electron can come up as +x or -x with 50% probability each but the positron always precisely mirrors the outcome, the appearance of superluminal interaction arises, violating locality.

This disturbed Einstein, hence he postulated hidden variables, i.e. that all possible measurements (i.e. whether along x, y or z axis) are already present at creation of the singlet (a position called "realism"). But Bell's theorem shows that local realism does not hold, i.e. the distribution of probabilities when measuring at various angles (and hence various superpositions of the axes) do not follow what would be expected from hidden variables.

Now the claim seems to be that since local realism doesn't work one has to sacrifice either locality or realism to keep the other. But I can't see how giving up realism does anything to ensure locality since realism was invented by Einstein precisely to get around apparent non-locality, so if we drop realism we're still going to have non-locality.

Hence I would think that Bell's theorem only disproves hidden variables (i.e. realism) but doesn't really say anything about (non-)locality - rather, the appearance of non-locality is only due to the specific interpretation of quantum mechanics (since non-locality isn't a problem with photons and e.g. considering a -x positron as the time-reversed +x electron makes it also go away for electrons).

Does that make sense?

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  • $\begingroup$ Giving up realism simply doesn't necessarily preserve locality, you have to work a bit to get a local description even if you're non-realist. Conversely, giving up locality doesn't automatically make you realist. $\endgroup$
    – ACuriousMind ♦
    Feb 25, 2015 at 18:23
  • $\begingroup$ Yes, "non-realism implies local" and "non-local implies realism" are logically equivalent and false. But what I'm arguing against is the claim that Bell's inequality forces the choice between "realism implies non-local" and the logically equivalent "locality implies non-realism" since Bell's inequality doesn't just disprove that realism cannot be local but disproves realism altogether. $\endgroup$
    – user66554
    Feb 25, 2015 at 18:42
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    $\begingroup$ Bohmian mechanics is explicitly constructed to be non-local and realist and to agree with all QM predictions, so locality is necessary for Bell's theorem to rule out realism. $\endgroup$
    – ACuriousMind ♦
    Feb 25, 2015 at 18:47
  • $\begingroup$ Yeah, I forgot that one, thinking more about the reverse case. So Bell's theorem says that one cannot have local realism - but local non-realism, non-local realism and non-local non-realism are all possible. But I knew that going in. So WTF is my question? How about: does anyone have a chart sorting the interpretations by whether they are local and whether they are realist? (Except for Copenhagen and Bohm I'm not sure where each fits.) $\endgroup$
    – user66554
    Feb 25, 2015 at 19:24
  • $\begingroup$ Wiki has a chart, where counterfactual definiteness corresponds to realism, I think. $\endgroup$
    – ACuriousMind ♦
    Feb 25, 2015 at 20:06

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definition of locality :

$$Prob[a]_{A , B} = Prob[a]_A, \ \text {and} \ Prob[b]_{A , B} = Prob[b]_B \tag{i}$$

where a and b are the results obtained by respectively Alice and Bob each one on her/his particle, where $A$ and $B$ are the respective types of measurements.

Now, you say, "the appearance of non-locality is only due to the specific interpretation of quantum mechanics (since non-locality isn't a problem with photons".

No, the appearance of non-locality is recognized by the standard QM - no interpretation, as one doesn't get the equalities (i). About photons, the Aspect's experiments on the polarization singlet were with polarized photons,

$$ |S\rangle = \frac {|x\rangle |x \rangle + |y\rangle |y \rangle}{\sqrt{2}} \tag{ii}$$

A very convincing argument against locality is the GHZ experiment. The greatness of the GHZ experiment is that it doesn't even work with probabilities, it works per single trials.

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