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Treatment of the parameters in CHSH and Bell inequalities is different.
In CHSH inequality, there are two settings on Alice side, and two settings on Bob side. Nothing special (like perfect anticorrelation) is assumed; thus, it is pointless to ask whether etc. The parameters are just (labels of) settings of measuring devices (possibly, of quite different type), not necessarily angles; thus, one should not try to rotate them. CHSH is purely informational. Accordingly, for proving it we introduce observables such that etc; assume they all commute (according to the local realism); and get .
Labels are assigned to the two Alice settings arbitrarily. Equally well we may swap them, getting . There are four such inequalities. In addition, labels are assigned to the two possible outcomes arbitrarily, and may be swapped for (say) Alice but not Bob, which changes the sign: (and therefore ). These are symmetries of the CHSH inequality.
In Bell inequality, there are three angles (not just "settings") ; two of them, , are used by Alice, and two of them, , are used by Bob; note that is used by both. Denoting the two corresponding observables of Alice by we see that the first observable of Bob is due to the perfect anticorrelation. The second observable of Bob may be denoted by or by ; both options are legitimate. Let us choose the former one. Then , , , thus, assuming commutativity (as before) we get .
In order to reduce Bell inequality to CHSH inequality, we subtract , thus rewriting Bell inequality as , that is, . Taking into account the symmetries of CHSH we see that the latter is its special case.
In the presence of such august editors, I should perhaps not say too much. But in response to Editor Tsirel's comment "assume they all commute (according to the local realism)", I feel it could be mentioned that 'local realism' is an item of Bellspeak. At stake here are two things: a metaphysical principle, and a religious principle. The metaphysical principle is about causality. The religious principle is that Albert Einstein is a silly old goat, not nearly as clever as his critics. The causality thing is translated by his Bellspeaking critics into a matter that is exactly stated by classical mechanics, which is not a metaphysical principle, but something downstream from there, a physical theory. We don't need Bellspeakers to tell us that classical mechanics is unable to account for the existence and behaviour of atoms. That was apparent by 1920. The Bellspeakers muddle the metaphysical principle with the physical theory, and crow that because the physical theory is inadequate, they have overthrown the metaphysical principle and can thereby see in the dark and walk on water; well, at least they will be able to do so given one more research grant. But anyway, they are cleverer than Albert Einstein; the religious principle stands.Chjoaygame (talk) 22:47, 20 January 2016 (UTC)
I may add that the actual discussion on the formulas offered here by Editor Tsirel is helpful and illuminating.11:51, 21 January 2016 (UTC)
Thanks for the compliment. If your "august" was toward me, you are (to the best of my knowledge) the first to call me this way. Another very generous compliment "great magician" is here. What a collection! Boris Tsirelson (talk) 13:21, 21 January 2016 (UTC)