Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 3 / a / ii

Conjugation by ${\sigma }_z$ maps $(r_x,r_y,r_z)$ to $(-r_x,-r_y,r_z)$. The phase-flip channel therefore acts on the Bloch vector as

In the computational basis,At $p=1/2$, the two opposite off-diagonal contributions cancel, giving

A computational-basis projective measurement has projections $P_0=|0⟩⟨0|$ and $P_1=|1⟩⟨1|$. Discarding its result produces the nonselective projective measurementThusComplete phase randomization and unread computational-basis measurement implement the same dephasing on this qubit.

The four index pairs giveThe last global phase cancels under conjugation. Hence the Pauli channel probabilities are

Conjugation by one Pauli matrix preserves its parallel Bloch component and reverses the other two. Therefore