Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 5 / a

The normalized input density operator isA quantum measurement in the computational basis has projectors $P_j=|j⟩⟨j|$. Discarding its recorded outcome gives the dephasing channelThe original off-diagonal coherence disappears. If the outcome is retained instead, the conditional state is $|0⟩$ or $|1⟩$ and the record carries information about which occurred. Thus discarding the record is an essential part of this channel specification.

A random Pauli Z gate with equal application and nonapplication probabilities givesThis is exactly the same dephasing channel as the unrecorded measurement, for every input density operator, including one entangled with an auxiliary system. Individual trials differ: conditioned on the classical random choice, the state is the pure state $|\psi ⟩$ or $Z|\psi ⟩$, generally neither a computational basis vector. Knowledge of that choice permits recovery by the same Pauli Z gate; a measurement outcome does not restore an unknown destroyed superposition.

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.