= Pauli error criterion for a stabilizer code
{c}
Let $S$ exclude $-I$, let $C(S)$ be its Pauli centralizer, and let $\mathcal Z=\{\pm I,\pm iI\}$. A set of <Pauli errors> is jointly correctable exactly when every product $E_a^\dagger E_b$ belongs to $\mathcal ZS$ or lies outside $C(S)$. Outside the centralizer its code compression is zero by anticommutation; in $\mathcal ZS$ it acts as a scalar. A centralizing operator outside $\mathcal ZS$ has zero trace on the code but is unitary there, so cannot be scalar. This gives precisely the <Knill--Laflamme condition>. Scalar phases must be retained when formulating the group criterion.
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