Minkwe talks about pairs of particles. However we should forget all about particles and pairs of particles. Listen to Bell (1981) in "Bertlmann’s socks and the nature of reality", Chapter 16 of "Speakable and Unspeakable":
You might suspect that there is something specially peculiar about spin-1/2 particles. In fact there are many other ways of creating the troublesome correlations. So the following argument makes no reference to spin-1/2 particles, or any other particular particles.
Finally you might suspect that the very notion of particle, and particle orbit, freely used above in introducing the problem, has somehow led us astray. Indeed did not Einstein think that fields rather than particles are at the bottom of everything? So the following argument will not mention particles, nor indeed fields, nor any other particular picture of what goes on at the microscopic level. Nor will it involve any use of the words ‘quantum mechanical system’, which can have an unfortunate effect on the discussion. The difficulty is not created by any such picture or any such terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups.
Consider the general experimental set-up of Fig. 7. To avoid inessential details it is represented just as a long box of unspecified equipment, with three inputs and three outputs.
Here is the figure in question:
The text continues:
Consider the general experimental set-up of Fig. 7. To avoid inessential details it is represented just as a long box of unspecified equipment, with three inputs and three outputs. The outputs, above in the figure, can be three pieces of paper, each with either ‘yes’ or ‘no’ printed on it. The central input is just a ‘go’ signal which sets the experiment off at time t_1. Shortly after that the central output says ‘yes’ or ‘no’. We are only interested in the ‘yes’s, which confirm that everything has got off to a good start (e.g., there are no ‘particles’ going in the wrong directions, and so on). At time t_1 + T the other outputs appear, each with ‘yes’ or ‘no’ (depending for example on whether or not a signal has appeared on the ‘up’ side of a detecting screen behind a local Stern–Gerlach magnet). The apparatus then rests and recovers internally in preparation for a subsequent repetition of the experiment. But just before time t_1 + T, say at time t_1 + T – δ, signals a and b are injected at the two ends. (They might for example dictate that Stern–Gerlach magnets be rotated by angles a and b away from some standard position). We can arrange that cδ << L, where c is the velocity of light and L the length of the box; we would not then expect the signal at one end to have any influence on the output at the other, for lack of time, whatever hidden connections there might be between the two ends.
Sufficiently many repetitions of the experiment will allow tests of hypotheses about the joint conditional probability distribution
for results A and B at the two ends for given signals a and b.
Yablon asks: "where is the evidence"? Evidence can be of two kinds: experimental, theoretical.
On the experimental side the answer is that there is no proof, the good experiment still hasn't been done. Possibly it might be done within a year from now, the experimentalists are getting close.
On the theoretical side the answer is just the old QM story which everyone here knows. According to QM it seems that that good experiment ought to be feasible. The experimentalists have been trying hard for 50 years and still haven't managed to do it.
Yablon talks about superluminal signalling as being what is at issue. However in the just mentioned paper by Bell, the author lists four possible metaphysical conclusions. Only one of them involves effects spreading faster than the speed of light, and this only happens in a "hidden" layer not directly accessible to observations. Since this hidden layer is purely theoretical, saying that it "exists" is pretty meaningless.
Bell's list of four positions to take is not exhaustive. Some writers consider the 50 years of failure to perform the desired experiment as strong experimental proof that it can never be done, because quantum mechanics itself prevents it from being done: uncertainty relations preventing the required state to be created and measured under the needed spatial and temporal constraints.
Quantum mechanics itself prevents
superluminal signalling between observable experimental inputs and outputs. Whether or not superluminal signalling is needed to create the cosine correlations, then, even if they could be created in the interesting context of a loophole-free experiment, you couldn't use them to signal.
So whatever the Bell discussion is all about, it is not about
superluminal signalling.
Michel asks, how on earth can a spin-half pair of particles produce a cosine correlation? Well one pair just produces one outcome pair, but the outcomes of many pairs create what we call a correlation. But how can that be done?
The answer is, nobody knows. It can't be done in a local realistic way, it can't be done in a simulation of a local hidden variables model by a network of computers in the arrangement of a loophole-free experiment. (Fast, rapid, local, random selection of measurement settings; no non-detections; rigorous enforced time-space separations as in Bell's story). Bell's theorem says it can't be done that way, and Bell's theorem is so incredibly elementary there can be no doubt about it. But if you don't trust elementary calculus and logic, then there is the experimental evidence: there have been 50 years for inventors of local hidden variables theories to come up with a computer simulation model which is a counterexample to Bell's theorem ... they haven't done it (and they won't, ever). And plenty have tried!