The correlation polytope 
is the convex hull of the
vectors 
for 
, where 
is a positive integer.
Equivalently,
It has 
vertices and dimension 
(Pitowsky 1991).
A vector 
belongs to 
iff there are events 
, ..., 
on one probability space
with 
and 
.
Indeed, the coefficients 
are the probabilities
of the possible joint assignments of the indicator
functions of those events. They therefore define a
joint distribution, and conversely any such
joint distribution gives a convex
combination of the vertices.
For 
,
is the tetrahedron with vertices 
, 
, 
, and 
. Its inequalities can
be written as
The four joint probabilities are 
, 
, 
, and 
(Filipp and Svozil 2003).
Keeping only selected pair coordinates gives a projection of 
. Bell inequalities, including
the Clauser-Horne inequality, constrain
such projected polytopes (Pitowsky and Svozil 2001,
Trott 2006).
See also
Bell Inequality,
Clauser-Horne Inequality,
Convex Hull,
Joint
Distribution,
Polytope
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References
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.Pitowsky,
I. Quantum
Probability--Quantum Logic. Berlin, Germany: Springer-Verlag, 1989. https://doi.org/10.1007/BFb0021186.Pitowsky,
I. "Correlation Polytopes: Their Geometry and Complexity." Math. Programming 50,
395-414, 1991. https://doi.org/10.1007/BF01594946.Pitowsky,
I. and Svozil, K. "Optimal Tests of Quantum Nonlocality." Phys. Rev.
A 64, 014102, 2001. https://doi.org/10.1103/PhysRevA.64.014102.Trott,
M. "Polynomials in Inequalities." §1.2.3 in The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5.
https://www.mathematicaguidebooks.org/.Werner,
R. F. and Wolf, M. M. "Bell Inequalities and Entanglement." Quantum
Inform. Comput. 1, 1-25, 2001. https://doi.org/10.26421/QIC1.3-1.
Cite this as:
Weisstein, Eric W. "Correlation Polytope."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CorrelationPolytope.html
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