The CHSH inequality, named for Clauser, Horne, Shimony, and Holt (1969), is a Bell inequality involving two measurement choices
at each of two locations. Let ,,, and be random variables taking
values in
on one probability space, and put ,
where the brackets denote expectation values.
Then
The common probability space is essential: the four pairwise correlations must be compatible with one joint
distribution. Quantum measurement correlations need not satisfy this requirement
and can violate the bound.
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J. L. "Identification of All Hardy-Type Correlations for Two Photons or
Particles with Spin 1/2." Found. Phys. Lett.14, 401-424, 2001.
https://doi.org/10.1023/A:1015520603468.Clauser,
J. F.; Horne, M. A.; Shimony, A.; and Holt, R. A. "Proposed Experiment
to Test Local Hidden-Variable Theories." Phys. Rev. Lett.23,
880-884, 1969. https://doi.org/10.1103/PhysRevLett.23.880.Clauser,
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, 
, 
, and 
be random variables taking
values in 
on one probability space, and put 
,
where the brackets denote expectation values.
Then
and 
is zero. Taking its expectation
value proves the bound.
and 
, the Clauser-Horne
inequality has expression 
. Thus the probability
form and the form in expectation values are
equivalent under this change of variables.
