Bell inequalities demonstration

[microsansfil]

TL;DR

I try to understand why the correlator boundaries is −1 ≤ C(x y) ≤ 1

Hello,

In this thesis https://tel.archives-ouvertes.fr/tel-01743877/document at "1.2.2 Bell inequalities" page 7-8 it's define a correlation function :

C(x y) = P(+ + |x y) + P(− − |x y) − P(+ − |x y) − P(− + |x y), with −1 ≤ C(x y) ≤ 1.

How do one get to this relationship −1 ≤ C(x y) ≤ 1 ?

Thanks
Patrick

 

[wle]

Summary:: I try to understand why the correlator boundaries is −1 ≤ C(x y) ≤ 1

Hello,

In this thesis https://tel.archives-ouvertes.fr/tel-01743877/document at "1.2.2 Bell inequalities" page 7-8 it's define a correlation function :

C(x y) = P(+ + |x y) + P(− − |x y) − P(+ − |x y) − P(− + |x y), with −1 ≤ C(x y) ≤ 1.

How do one get to this relationship −1 ≤ C(x y) ≤ 1 ?

Thanks
Patrick

It's just a consequence of positivity $$P(\pm \pm' | xy) \geq 0$$ and normalisation $$P(++|xy) + P(+-|xy) + P(-+|xy) + P(--|xy) = 1$$ of the probabilities.