Ben6993 wrote:Michel, I have done more calculations following your reply.
CHSH statistic = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′). The settings a, a′, b and b′ are 0°, 90°, 45° and 135°, respectively
For the trivial case of a CHSH run of one pair of particles, CHSH = AB - AB' + A'B + A'B'
Ben you are confused. In one run (one pair of particles) of a CHSH experiment only one of A and A' is observed, only one of B and B'. The other two are counterfactual. You don't observe them. And it is only under a local hidden variables theory, that they can be said to "exist".
You said the "independence case" was that the quadruple (A, A', B and B') took on any of 16 possible combinations of values. But this is already counterfactual. Factually, only one of A and A' exists, and only one of B and B' exists.
Then you talked about the "counterfactual case" and said that A would equal -A' and B would equal -B'. But we are talking about spin half particles and there is no deterministic relation between any of the pairs which you can form (under counterfactual definiteness). There is no pair of angles differing by 0 or by 180 degrees.
However your observation that all 16 combinations in what you incorrectly called the "independence case" were either equal to -2 or +2. This means that if you would repeat a calculation of AB - AB' + A'B + A'B' (in the counterfactual world which only exists if we believe in local hidden variables) many many times, independently, and go to the infinite N limit, you would (in the limit) find the value of E(AB - AB' + A'B + A'B') where E stands for expectation value. So this limit has to lie between -2 and +2. Read a book about probability theory if you don't know what all this means.
Now the expectation operator is linear, so we find E(AB) - E(AB') + E(A'B) + E(A'B') lies between -2 and 2.
All this in the limit of infinitely many repetitions and in the imaginary world where local hidden variables exist so that it makes sense to talk about the value of AB - AB' + A'B + A'B' for one particle pair, even though this value is "hidden".
Now we change gears and think of an experiment. If we do N repetitions and measure every time A and B, we can average the products of A and B. Call this average ave(AB). Note that it is random - if we do the same thing again, we can expect to get a different answer. It certainly will never (or almost never) equal E(AB) defined earlier. However it should tend to be only about 1/sqrt N off target, sometimes more, sometimes less, sometimes in one direction, sometimes in the other direction.
Do another finite number of repetitions, I don't mind if its the same number N or another number N'. For simplicity assume it's the same number. But this time measure A and B'. Calculate the average of the products and call it ave(AB').
Similarly for A'B and A'B'.
Putting everything together, we expect that ave(AB) - ave(AB') +ave(A'B) +ave(A'B') lies within +/- a few multiples of 1/sqrt N away from E(AB) - E(AB') + E(A'B) + E(A'B'), with large probability.
So that's the rational of a CHSH experiment. Choose N large enough that 1/sqrt N is of order of size, say, 0.01.
Measure 4N particle pairs, N with each pair of settings, calculate ave(AB) - ave(AB') +ave(A'B) +ave(A'B')
It should be close to E(AB) - E(AB') + E(A'B) + E(A'B')
Which is not larger than 2 if local hidden variables well describe the mathematical physics of what is going on.
But can equal 2 sqrt 2 = 2.8 if quantum mechanics of the singlet state with the "optimal" set of measurement directions well describes the mathematical physics of what is going on.
So if ave(AB) - ave(AB') +ave(A'B) +ave(A'B') is larger than 2.4 (half way between 2 and 2.8) we would be advised to trash local hidden variables, because if local hidden variables theory had been true, something very unlikely has happened. While under QM predictions, what we have seen is quite plausible.
Of course if we saw ave(AB) - ave(AB') +ave(A'B) +ave(A'B') larger than 3.2 we should start worrying about QM because under QM it's not possible for E(AB) - E(AB') + E(A'B) + E(A'B') to be larger than 2.828...