The <centralizer of a stabilizer group> is
$$
C(\mathcal S)=\{P\in\mathcal P_n:PS=SP\text{ for every }S\in\mathcal S\}.
$$
The nontrivial logical Pauli operators are represented by
$$
\operatorname{LO}_{\mathcal S}=C(\mathcal S)\setminus\mathcal S,
$$
or more invariantly by the quotient $C(\mathcal S)/\mathcal S$ after phases are removed.

For $P=E_a^\dagger E_b$, there are three cases. If $P\in\mathcal S$, it acts as a scalar on the code. If $P\notin C(\mathcal S)$, it anticommutes with a stabilizer and $\Pi_{\mathcal L}P\Pi_{\mathcal L}=0$. If $P\in C(\mathcal S)\setminus\mathcal S$, it acts as a nontrivial logical operator and is not proportional to the identity. Therefore
$$
\boxed{
\text{the KL conditions hold}
\iff E_a^\dagger E_b\notin\operatorname{LO}_{\mathcal S}
\quad\text{for every }a,b.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 2 / b / Solution

The Pauli centralizer $C(S)$ consists of Pauli operators commuting with every stabilizer. Elements of $C(S)\setminus S$, modulo stabilizers and phases, are the nontrivial logical Pauli operators.


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 Incoming links: Centralizer of a stabilizer group