= Solution
Use the rotation convention $R_z(\phi)=e^{-i\phi Z/2}=\operatorname{diag}(e^{-i\phi/2},e^{i\phi/2})$. For uniform $\phi$ modulo $2\pi$,
$$
\boxed{\rho'=\frac1{2\pi}\int_0^{2\pi}R_z(\phi)\rho R_z(\phi)^\dagger\,d\phi
=\operatorname{diag}(|\alpha|^2,|\beta|^2),}
$$
because the off-diagonal entry acquires $e^{-i\phi}$ with zero average. Each trial retains a <pure state> with a randomly rotated relative phase, reversible if $\phi$ is known. All three procedures therefore give identical statistics for every measurement on the unconditioned system, but their records and conditional ensembles differ. A <density operator> does not identify its preparation ensemble uniquely; access to a record changes which information is available.
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