 Solution

For a deterministic local hidden variable, write the four possible measurement results as $q,r,s,t\in \{\pm 1\}$. Its Bell combination isIf $s=t$, the first bracket vanishes and the second is $\pm 2$; if $s=-t$, the second vanishes and the first is $\pm 2$. Thus $b=\pm 2$ for every hidden variable. Averaging over any local hidden-variable distribution gives $|\mathcal{B}|\le 2$, and in particular the requested upper bound.

For the specified settings, the correlation formula in part b givesConsequentlyThis violates the local bound precisely whenThe maximum is $2\sqrt{2}$ at $\phi =0$ modulo $2\pi$. The state remains maximally entangled at other phases, even when this fixed choice of real-plane analyser settings gives no violation. Local phase compensation or adapted measurement settings can recover the optimum.