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In this section of the Scholarpedia article on Bell's theorem, the first paragraph comments that Bell's original inequality is not ideal for experimental verification because it requires perfect anti-correlation of the variables, and for that reason the CHSH inequality is better. I don't really understand what this means, since as far as I'm aware experiments with e.g. the CHSH game also involve Bell pairs which are perfectly correlated. They allude to some continuity in the correlations, but I don't know what this means. Could anyone elucidate?

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The difference is as follows:

  • the original Bell inequality requires that outcomes from the same setting are always perfectly anti-correlated. It says nothing about the case where they are even marginally different.

  • by contrast, in CHSH, the ideal (giving maximum violation) is that outcomes from the same setting would be anti-correlated, but it is not required.

Thus, in an experiment with imperfections, CHSH still applies, but the original may not.

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  • $\begingroup$ I understand that anti-correlation is required to show $P(Z_a^1 \neq Z_b^2) + P(Z_b^1 \neq Z_c^2) + P(Z_c^1 \neq Z_a^2) \geq 1$. But if they were not perfectly anti-correlated, just nearly so, wouldn't this quantity still be close to 1, and still quite distinct from the $3/4$ of the quantum case? $\endgroup$
    – Pedro
    Nov 5 '20 at 17:39
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    $\begingroup$ You might anticipate that, but to build it into the proof it essentially leads you in the direction of CHSH (I believe. It's not a distinction I've studied in detail). $\endgroup$
    – DaftWullie
    Nov 6 '20 at 7:49
  • $\begingroup$ Yeah I've come to believe that as well. $\endgroup$
    – Pedro
    Nov 6 '20 at 15:57

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