CHSH inequality

Remarks

For each $\lambda \in \Lambda$, the integrand is $A_0(B_0 + B_1) + A_1(B_0 - B_1)$, evaluated at $\lambda$. Its absolute value is at most $\abs{B_0 + B_1} + \abs{B_0 - B_1} = 2\max(\abs{B_0}, \abs{B_1}) \le 2$, and integrating gives the inequality. In a local hidden-variable theory, $\lambda$ is the hidden variable and $A_i(\lambda)$, $B_j(\lambda)$ are the outcomes of measurement $i$ on one side and $j$ on the other. The essential hypotheses are that one measure $\mu$ serves for all four pairs and that $A_i$ does not depend on $j$, nor $B_j$ on $i$. Without them the bound fails: four independent pairs with correlations $1, 1, 1, -1$ give the value $4$. Example: for the two-qubit state $(\lvert 00 \rangle + \lvert 11 \rangle)/\sqrt{2}$, suitable spin measurements give the value $2\sqrt{2}$, so the quantum correlations have no such model.

Going deeper

For a student

The inequality was published by Clauser, Horne, Shimony and Holt in 1969, as a version of Bell's inequality suited to real experiments. Its proof needs only boundedness of the outcomes, not perfect correlations. The quantum maximum $2\sqrt{2}$ is Tsirelson's bound, and the algebraic maximum $4$ is reached by hypothetical nonlocal boxes. The CHSH inequality is the key step in the usual proof of Bell's theorem, and it is the basis of device-independent quantum cryptography.