Pauli channel

A one-qubit Pauli channel applies $I,{\sigma }_x,{\sigma }_y,{\sigma }_z$ with probabilities $p_0,p_x,p_y,p_z$. It maps a Bloch vector diagonally by independently contracting or reversing its three components.

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.

A dephasing channel suppresses off-diagonal density-matrix entries in a preferred basis while preserving populations. Complete dephasing is identical to a projective measurement in that basis with its outcome discarded.