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 random variables taking
values in probability space . Then
where expectation value of 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 .
See also CHSH Inequality ,
Clauser-Horne Inequality ,
Correlation Polytope ,
Expectation Value ,
Greenberger-Horne-Zeilinger
Theorem ,
Grothendieck's Constant ,
Random Variable
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D. M.; Horne, M. A.; Shimony, A.; and Zeilinger, A. "Bell's Theorem
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R. F. and Wolf, M. M. "All-Multipartite Bell-Correlation Inequalities
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Cite this as:
Weisstein, Eric W. "Bell Inequality."
From MathWorld https://mathworld.wolfram.com/BellInequality.html
Subject classifications
, 
, and 
be random variables taking
values in 
on one probability space. Then
denotes the expectation value of 
. The result follows by taking expectation
values in the pointwise identity
