= Controlled-basis measurement causality obstruction
{title2=$p_B(1\mid A=1)=\tfrac12\sin^2(2\theta)$}
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 $0<\theta\leq\pi/4$. 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.
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