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
$$
\boxed{(r_x,r_y,r_z)\longmapsto
((1-2p)r_x,(1-2p)r_y,r_z)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 3 / a / ii / 1 / Solution

The phase-flip channel is $\Lambda_p(\rho)=(1-p)\rho+p\sigma_z\rho\sigma_z$. It multiplies the $x$ and $y$ components of the Bloch vector by $1-2p$ and leaves the $z$ component fixed.


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