The four index pairs give
$$
\sigma_z^0\sigma_x^0=I,\quad
\sigma_z^0\sigma_x^1=\sigma_x,\quad
\sigma_z^1\sigma_x^0=\sigma_z,\quad
\sigma_z\sigma_x=i\sigma_y.
$$
The last global phase cancels under conjugation. Hence the <Pauli channel> probabilities are
$$
\boxed{p_0=p_{00},\qquad
p_x=p_{01},\qquad
p_y=p_{11},\qquad
p_z=p_{10}}.
$$

Solution

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

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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.


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 Incoming links: Pauli channel