 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 givesAlice obtains either result with probability $1/2$; conditionally Bob has probability ${\cos }^2\theta$ or ${\sin }^2\theta$ of $0$, respectively. Averaging,independent of $\theta$, exactly as no-signalling requires.