bohm2 said:
This is the part that confuses me. It isn't only DrC who uses sees contextuality as implying non-realism. There are a number of other authors like Nieuwenhuizen, Hess, Krennikov, Accardi, Pitowsky, Rastal , Kupczynski, de Raedt, etc. who see contextuality in a somewhat similar light but draw different conclusions:
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From the basics, Bell's inequality can be written as ##P(a, c) - P(b, a) - P(b, c) \le 1## Which we compare with QM/experiment. Experimentally, we are measuring three corresponding averages ## \langle AC\rangle, \langle BA \rangle, \langle BC \rangle## for which we assume that ##P(a, c) ≈ \langle AC\rangle, P(b, a) ≈ \langle BA \rangle, P(b, c) ≈ \langle BC \rangle## for a large number of measurements.
According to Bell's local realistic prescription (equation 2), we then have three quantities:
P(a,b) = \int_{[\lambda _{1..n}]} A(a,\lambda )B(b,\lambda ) \rho (\lambda ) d\lambda
P(b,c) = \int_{[\lambda _{n+1..m}]} A(b,\lambda )B(c,\lambda ) \rho (\lambda ) d\lambda
P(a,c) = \int_{[\lambda _{m+1..l]}]} A(a,\lambda )B(c,\lambda ) \rho (\lambda ) d\lambda
Reflecting the fact that experimentally, we can only measure each expectation value on a different set of particle pairs. Why is this relevant to contextuality, realism, locality, loophole etc. In a simple way:
Realism: Bell's inequality can not be violated if ## [\lambda _{1..n}] = [\lambda _{n+1..m}] = [\lambda _{m+1..l]}]##. This is equivalent to measuring each particle pair at three angles simultaneously (a practical impossibility). No spreadsheet of three columns of outcomes, one for each pair of particles can ever violate Bell's inequality. On the other hand, it is very easy to violate the inequality if ## [\lambda _{1..n}] \neq [\lambda _{n+1..m}] \neq [\lambda _{m+1..l]}]## (if you are interested I can show you examples). Therefore the derivation of the inequality must include (even implicitly) the assumption that 3 outcomes exist simultaneously for each particle pair. There is one problem though, you can't then rely on experimental data measured on different particle pairs to rule out realism. You can't claim 3 values do not exist simultaneously based on experiments which can never measure the 3 values simultaneously even if they existed.
Locality: If you assume that the same process is generating the particles ## [\lambda _{1..n}], [\lambda _{n+1..m}], [\lambda _{m+1..l]}]## used for each correlation, for large enough particle pairs, even if the sets are not the same, the probability distributions may be so similar that you should still obtain the same expectation values as you would have obtained from a single set as in the realism case. Then you can violate the inequality is if there is some communication between the sides. Another way to see this is to create a new "non-local" variable and assign it to the sets. Then you end up with ## [\lambda _{1..n}, \eta_1], [\lambda _{n+1..m}, \eta_2], [\lambda _{m+1..l]}, \eta_3]## Then you end up with different probability distributions which can violate the inequality. While we do not assume that the particles are the same set, we still assume that we should have had the same distribution, unless there is a non-local influence which changes the effective distribution after emission of the particle pairs. Note, if the non-local variables are the same value, you still won't be able to violate the inequality. You need non-local variables which change value in an angle dependent way.
Contextuality: This is very similar to the Locality case in that, we could encapsulate the context in another variable such that we now have sets ## [\lambda _{1..n}, \beta_1], [\lambda _{n+1..m}, \beta_2], [\lambda _{m+1..l]}, \beta_3]## If the contexts are different, then we have a way to introduce differences between the distributions and the inequality can be violated. Hess, De Raedt, Accardi and others argue that when measuring on different sets of particles at different times, it is natural to expect differences in context that are angle dependent. This is what loopholes are about, they are just ways of introducing differences in context and thus different distributions. For example:
- detection loophole: particle pairs are less likely to be detected at certain angles than others
- coincidence loophole: the likelihood of matching a pair varies with angle difference.
- "Superdeterminism": Same thing. Alice and Bob do not have the free to control the experiment such that ## [\lambda _{1..n}, \beta_1] = [\lambda _{n+1..m}, \beta_2] = [\lambda _{m+1..l]}, \beta_3]##.
There isn't a whole lot of difference between them.
