 Solution

Use the rotation convention $R_z(\varphi )=e^{-i\varphi Z/2}=\mathrm{diag}(e^{-i\varphi /2},e^{i\varphi /2})$. For uniform $\varphi$ modulo $2\pi$,because the off-diagonal entry acquires $e^{-i\varphi }$ with zero average. Each trial retains a pure state with a randomly rotated relative phase, reversible if $\varphi$ 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.