Guest wrote:There is dependency between the terms: it is *assumed* that Phi = phi_1 & phi_2 & ... & phi_N is impossible.
Yes, since it is "supposed" that you have P = 0 earlier in that section. However, the dependency should be better expressed mathematically in eq. (1).
Guest wrote:To give an example: suppose a local hidden variables model is true in the Bell-CHSH set-up. Define X_1 and X_2 to be the outcomes which Alice would have observed, had she used either of her two settings 1 or 2 (outcomes +/- 1 in the conventional scenario), define Y_1 and Y_2 to be the outcomes which Bob would have observed, had he used either of his two settings 1 or 2 (outcomes +/- 1 in the conventional scenario).
Notice that (since these four variables are restricted to being equal to +/- 1) it is impossible that X_1 = Y_1 & X_1 != Y_2 & Y_2 != X_2 & X_2 != Y_1. For instance if X_1 = Y_1 = +1, then Y_2 would have to be -1, X_2 would have to be +1, and Y_1 would have to be -1 ... a logical contradiction. Similarly in the other case X_1 = Y_1 = -1.
It follows that Prob(X_1 = Y_1) + Prob(X_1 != Y_2) + Prob(Y_2 != X_2) + Prob(X_2 != Y_1) <= 3.
The point is that these variables X_1, X_2, Y_1, Y_2 exist according to a local hidden variables theory, but quantum mechanics says nothing about them at all. In QM we don't assume they exist at all.
That is fine. No problem there.
Guest wrote:Quantum mechanics does predict the value of Prob(Alice's outcome = Bob's outcome | Alice uses setting i and Bob uses setting j) for each i, j = 1, 2. And experiment allows us to measure the value of Prob(Alice's outcome = Bob's outcome | Alice uses setting i and Bob uses setting j).
So in an experiment with many trials with each of the four pairs of settings we can get to see (up to statistical error) the values of Prob(Alice's outcome = Bob's outcome | Alice uses setting 1 and Bob uses setting 1), Prob(Alice's outcome != Bob's outcome | Alice uses setting 1 and Bob uses setting 2), Prob(Alice's outcome != Bob's outcome | Alice uses setting 2 and Bob uses setting 2), Prob(Alice's outcome != Bob's outcome | Alice uses setting 2 and Bob uses setting 1).
And QM allows the sum of these four probabilities to exceed 3. In fact, QM allows 1 + 2 sqrt 3 = 3.4 approx
What you fail to realize is that the QM experiments use the inequality,
(independent terms)
Instead of,
(with dependent terms)
So they don't "violate" the Bell-CHSH inequality for LHV. It is mathematically impossible for anything to violate it.