Sadly there is a major mistake here. Let Ω = 1/√2. On page 116 the book "Introduction to quantum mechanics" by Neilsen and Chuang gives actual conditions for Tsirelson's Bound to be attained. Alice has detectors Q and R, Bob has detectors S and T. They are associated with the expectations E1, E2, E3, E4 used in the Bell test by the formulae E1 = ⟨QS⟩, E2 = ⟨QT⟩, E3 = ⟨RS⟩, E4 = ⟨RT⟩. The book's equation (2.226) gives a quantum configuration |ψ⟩ = Ω(|01⟩ − |10⟩) prepared by Charlie. It then demonstrates in equation (2.229) that ⟨QS⟩ = Ω, ⟨RS⟩ = Ω, ⟨RT⟩ = Ω, ⟨QT⟩ = −Ω leading straight to the conclusion in (2.230) that ⟨QS⟩ + ⟨RS⟩ + ⟨RT⟩ − ⟨QT⟩ = 4Ω which is equal to the Bound.
Expressing this equation in Bell's terms gives E1 − E2 + E3 + E4 = 4Ω. Let's suppose that Alice's and Bob's detectors record 100 qubits each, and start working in round figures using 0.7 as a close approximation to Ω. In accordance with Bell's specification E1 is calculated by comparing the recordings of Q and S. It is the agreements less the disagreements divided by 100, thus for it to be 0.7 requires 85 agreements and 15 disagreements.
- Similarly E3 = 0.7 requires 15 disagreements between R and S, so there are at most 30 disagreements between Q and R;
- Moreover E4 = 0.7 requires 15 disagreements between R and T, so there are at most 45 disagreements between Q and T;
- But E2 = − 0.7 indicates there are actually 85 disagreements between Q and T.
Therefore 85 ≤ 45 !!!
But there's far worse to come. The article clearly states, "For comparison in the classical (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of +1, −1 is allowed, it is 4". It isn't to prove (though I shan't do so here) that with an arbitrary assignment the upper bound is 2, not 4. Returning to page 116, this means that those experiments that have yielded results "resoundingly in favor of the quantum mechanical prediction", simply haven't followed Bell's instructions. This puts local realism back on the table, and now it's quantum theory that has a case to answer.