Editing CHSH inequality (section)

== Statement ==
The usual form of the CHSH inequality is
{{NumBlk||<math display="block">|S|\leq 2,</math>|{{EquationRef|1}}}}
where
{{NumBlk||<math display="block">S = E(a, b) - E\left(a, b'\right) + E\left(a', b\right) + E\left(a', b'\right).</math>|{{EquationRef|2}}}}

<math>a</math> and <math>a'</math> are detector settings on side <math>A</math>, <math>b</math> and <math>b'</math> on side <math>B</math>, the four combinations being tested in separate subexperiments. The terms <math>E(a,b)</math> etc. are the [[Quantum correlation|quantum correlations]] of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of <math>A(a) \times B(b)</math>, where <math>A,B</math> are the separate outcomes, using the coding +1 for the '+' channel and &minus;1 for the '&minus;' channel. Clauser et al.'s 1969<ref name="Clauser-1969">{{citation |author1=J.F. Clauser |author2=M.A. Horne |author3=A. Shimony |author4=R.A. Holt |year=1969 |title=Proposed experiment to test local hidden-variable theories |journal=Phys. Rev. Lett. |volume=23 |issue=15 |pages=880–4 |doi=10.1103/PhysRevLett.23.880 |bibcode=1969PhRvL..23..880C|doi-access=free }}</ref> derivation was oriented towards the use of "two-channel" detectors, and indeed it is for these that it is generally used, but under their method the only possible outcomes were +1 and &minus;1.  In order to adapt to real situations, which at the time meant the use of polarised light and single-channel polarisers, they had to interpret '&minus;' as meaning "non-detection in the '+' channel", i.e. either '&minus;' or nothing. They did not in the original article discuss how the two-channel inequality could be applied in real experiments with real imperfect detectors, though it was later proved<ref name="Bell-1971">J. S. Bell, in ''Foundations of Quantum Mechanics'', Proceedings of the International School of Physics “Enrico Fermi”, Course XLIX, B. d'Espagnat (ed.) (Academic, New York, 1971), p. 171 and Appendix B.  Pages 171-81 are reproduced as Ch. 4 of J. S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'' (Cambridge University Press 1987)</ref> that the inequality itself was equally valid.  The occurrence of zero outcomes, though, means it is no longer so obvious how the values of ''E'' are to be estimated from the experimental data.

The mathematical formalism of quantum mechanics predicts that the value of <math>S</math> exceeds 2 for systems prepared in suitable entangled states and the appropriate choice of measurement settings (see below). The maximum violation predicted by quantum mechanics is <math>2 \sqrt{2}</math> ([[Tsirelson's bound]])<ref>{{cite journal|last1=Cirel'son|first1=B. S.|title=Quantum generalizations of Bell's inequality|journal=Letters in Mathematical Physics | date=March 1980|volume=4|issue=2|pages=93–100|doi=10.1007/BF00417500|bibcode = 1980LMaPh...4...93C |s2cid=120680226 }}</ref> and can be obtained from a maximal entangled [[Bell state]].<ref>{{cite book |first=Asher |last=Peres |title=Quantum Theory: Concepts and Methods |publisher=Kluwer Academic |year=2002 |isbn=0-792-33632-1 |pages=164–165}}</ref>