Controlled-basis measurement causality obstruction ($p_B(1\mid A=1)=\frac{1}{2}{\sin }^2(2\theta )$)

A rank-one projective measurement whose Bob basis is computational when Alice is zero and rotated by $\theta$ when Alice is one can signal. Bob starts in zero. Alice zero leaves Bob zero; Alice one dephases Bob in the rotated basis, giving the displayed nonzero flip probability for $0<\theta \le \pi /4$. This violates quantum no-signalling. The conclusion applies to a measurement distinguishing the four product-basis states, rather than a degenerate observable that does not resolve them.