C (S) = {P \in P_{n} : PS = SP for every S \in S} .

The nontrivial logical Pauli operators are represented by

LO_{S} = C (S) ∖ S,

or more invariantly by the quotient

C (S) / S

after phases are removed.

For

P = E_{a}^{†} E_{b}

, there are three cases. If

P \in S

, it acts as a scalar on the code. If

P \in / C (S)

, it anticommutes with a stabilizer and

Π_{L} P Π_{L} = 0

. If

P \in C (S) ∖ S

, it acts as a nontrivial logical operator and is not proportional to the identity. Therefore

the KL conditions hold ⟺ E_{a}^{†} E_{b} \in / LO_{S} for every a, b .

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