The Clauser-Horne inequality is a Bell inequality for jointly distributed indicator functions ,
written in terms of single-event and pair-event probabilities
(Clauser and Horne 1974). Let probability space .
Writing expectation values ,
put
The assumption of a common joint distribution
is essential.
For deterministic values
Taking expectation values proves the bound. For CHSH inequality uses correlation
polytope expresses these consistency constraints geometrically (Filipp and Svozil
2003, Trott 2006).
See also Bell Inequality ,
CHSH Inequality ,
Correlation Polytope ,
Indicator Function ,
Joint
Distribution
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References Clauser, J. F. and Horne, M. A. "Experimental Consequences of Objective Local Theories." Phys. Rev. D 10 , 526-535,
1974. https://doi.org/10.1103/PhysRevD.10.526 . Filipp,
S. and Svozil, K. "Boole-Bell-Type Inequalities in Mathematica." In Challenging
the Boundaries of Symbolic Computation (Ed. P. Mitic, P. Ramsden, and
J. Carne). London: Imperial College Press, pp. 215-222, 2003. https://doi.org/10.1142/9781848161313_0028 . Trott,
M. "Polynomials in Inequalities." §1.2.3 in The
Mathematica GuideBook for Symbolics. https://doi.org/10.1007/0-387-28815-5 .
https://www.mathematicaguidebooks.org/ .
Cite this as:
Weisstein, Eric W. "Clauser-Horne Inequality."
From MathWorld https://mathworld.wolfram.com/Clauser-HorneInequality.html
Subject classifications
, 
, 
, and 
take values in 
on one probability space.
Writing 
,
,
and 
,
where the brackets denote expectation values,
put 
.
Then
, 
, 
, and 
in 
,
and 
,
the CHSH inequality uses 
.
Expanding gives 
, so 
is equivalent to 
. The correlation
polytope expresses these consistency constraints geometrically (Filipp and Svozil
2003, Trott 2006).
