A Bell inequality is a constraint on correlations that follows from a classical joint distribution of measurement outcomes. Such inequalities arise in models in which outcomes at two separated locations depend on local measurement choices and shared random variables, but not on the measurement choice at the other location (Bell 1964).
For example, let 
,
, and 
be random variables taking
values in 
on one probability space. Then
 |
where 
denotes the expectation value of 
. The result follows by taking expectation
values in the pointwise identity
 |
The CHSH inequality is a related bound involving two measurement choices at each location. Quantum correlations can violate these bounds because the different measurement choices need not admit the assumed joint classical distribution; this does not invalidate the mathematical inequalities under their stated hypotheses. The Greenberger-Horne-Zeilinger theorem instead gives a contradiction with predetermined local outcomes using perfect correlations, without an inequality.
