minkwe wrote:Let lambda be a non-local hidden variable, please write down the integral for the expectation value of the paired-product of outcomes at Alice and Bob.
When lambda is non-local hidden variable, What is wrong with
)
)
 = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda)
Please show exactly why the above three expressions are as you call it "undefined" or "meaningless" for a non-local hidden variable lambda. I've been asking you for a while, maybe one of your Bell-believer friends can help you explain what exactly in the above expression restricts it to LHV.
There are many kinds of non-local hidden variables models! Perhaps most general would be that Alice's setting is somehow available both at the source and at Bob's measurement station, and vice versa. If both settings are available at the source, then the probability distribution of the hidden variable could depend on the settings. If they are available at both measurement stations, then the measurement functions could depend on both. How about
 = \int A(a, b, \lambda) B(a, b, \lambda) \rho(\lambda | a, b) d\lambda)
for the most general case?
Christian's model (which is Pearle's model interpreted as a non-local model rather than as a detection loophole model) has the distribution of lambda depending on both the settings as one can easily see from his R code where "bad" values of lambda are rejected depending on criteria which depend on a and on b. ie he has
 = \int A(a, \lambda) B(b, \lambda) \rho(\lambda | a, b) d\lambda)
Whether one calls a model like this non-local or conspiratorial is a matter of interpretation, a matter of direction of causality.
Were a and b known in advance and the hidden variable's distribution depending on a and b; the experimenter having no choice at all? (conspiracy or super-determinism)? Or were a and b chosen by the experimenter at the last moment but their values instanteously somehow infecting the hidden variable at the source? (violation of locality)?
Here is Pearle's model:
The hidden variable lambda is the pair (X, Y) where:
X ~ uniform on S^2 … = unit vectors in R^3
Y ~ uniform on (1, 4), independent of X
C := (2 – √Y) / √Y
a and b are Alice, Bob’s settings, in S^2
A := sign(a . X) if |a . X | > C , otherwise 0 (Pearle) or no particle pair (Christian)
B := sign(– b . X) if |b . X | > C , otherwise 0 (Pearle) or no particle pair (Christian)
The hidden variable probability distribution rho used by Christian is the distribution of (X, Y) conditional on |a . X | > C and |b . X | > C where C = (2 – √Y) / √Y
Michel should note that this model is better than his epr-simple in that (a) settings are now direction in space, not just in the plane, (b) the singlet correlation is reproduced exactly, not only roughly. The model violates CHSH but not Larsson modified CHSH, nor CH, nor CHSH with outcomes "0" not discarded.
All these years the experimenters have been testing the wrong inequalities, as Caroline Thompson kept on emphasizing. Till last year, in fact.
meant to ask: