Bell inequality

A Bell inequality is a constraint on observable correlations satisfied by the specified class of local hidden-variable models. In finite-setting scenarios such constraints bound linear combinations of probabilities or expectation values. The CHSH inequality is one example, following from Bell factorization, measurement independence and bounded outcomes. A violation rules out the conjunction of the assumptions defining that model class; it does not alone identify a unique false premise.

A bound on the correlations attainable by measurement in quantum measurements for a specified Bell expression. The CHSH Tsirelson bound is $2\sqrt{2}$; the number depends on which Bell expression is being optimized.

For dichotomic observables with $A_i^2=B_j^2=I$, let $\mathcal{B}=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$. Expanding gives ${\mathcal{B}}^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, so $\Vert \mathcal{B}{\Vert }^2\le 8$ and $|⟨\mathcal{B}⟩|\le 2\sqrt{2}$. Appropriate spin measurements of a Bell state attain equality. The local hidden-variable bound of the corresponding CHSH inequality is two.