= CHSH inequality
{c}
{wiki}
For local outcomes $A_0,A_1,B_0,B_1\in\{-1,1\}$, the Clauser–Horne–Shimony–Holt inequality is
$$
|E(A_0B_0)+E(A_0B_1)+E(A_1B_0)-E(A_1B_1)|\leq2.
$$
An arbitrary no-signalling correlation can reach the algebraic maximum $4$, while quantum correlations are bounded by $2\sqrt2$.
For a short proof of the local bound, fix a <deterministic local hidden-variable model> assignment. The combination is $A_0(B_0+B_1)+A_1(B_0-B_1)$. Exactly one bracket vanishes and the other is $\pm2$, so every assignment contributes $\pm2$. Averaging with a single setting-independent <probability distribution> preserves the interval $[-2,2]$. Mixtures also cover stochastic <local hidden-variable theories>. Rejecting a different sample at each setting would invalidate precisely that common-average step.
The quantum bound follows just as economically. For <Hermitian operators> with $A_i^2=B_j^2=I$, the <CHSH inequality> operator $C=A_0\otimes(B_0+B_1)+A_1\otimes(B_0-B_1)$ obeys $C^2=4I-[A_0,A_1]\otimes[B_0,B_1]$. Each <commutator> has <operator norm> at most two, giving $\|C\|\leq2\sqrt2$. The <Polarization CHSH count model> attains this bound without allowing either station to control the distant marginal <probability distribution>.
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