= Solution
The <quantum no-signalling> principle says Alice's uncommunicated local choice cannot alter Bob's reduced state. Expanding the two tensor products cancels the cross terms and gives
$$
|\mu\mu\rangle+|\nu\nu\rangle=|00\rangle+|11\rangle.
$$
Alice obtains either result with probability $1/2$; conditionally Bob has probability $\cos^2\theta$ or $\sin^2\theta$ of $0$, respectively. Averaging,
$$
P_B(0)=\tfrac12(\cos^2\theta+\sin^2\theta)=\tfrac12,\qquad P_B(1)=\tfrac12,
$$
independent of $\theta$, exactly as no-signalling requires.
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