= Pauli decomposition of a qubit-environment isometry
{c}
{title2=$V|\psi\rangle=\sum_{k=0}^3\sigma_k|\psi\rangle\otimes|e_k\rangle$}
Every linear map from a <qubit> into the <tensor product> of that <qubit> and an environment can be expanded in the <Pauli matrix> <orthonormal basis> of operators. If $V|0\rangle=|0\rangle|e_{00}\rangle+|1\rangle|e_{01}\rangle$ and $V|1\rangle=|0\rangle|e_{10}\rangle+|1\rangle|e_{11}\rangle$, then the four coefficient vectors are
$$
|e_0\rangle=\frac{|e_{00}\rangle+|e_{11}\rangle}{2},\quad
|e_1\rangle=\frac{|e_{01}\rangle+|e_{10}\rangle}{2},\quad
|e_2\rangle=\frac{i(|e_{10}\rangle-|e_{01}\rangle)}{2},\quad
|e_3\rangle=\frac{|e_{00}\rangle-|e_{11}\rangle}{2}.
$$
These environment vectors need not be orthogonal, so this is an operator expansion rather than a claim that the <quantum channel> is a classical <Pauli channel>. Physical <isometries> additionally preserve the <inner products> of the two input basis states.
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