4 Stabilizer groups and the codespace
A state \(\psi \) is stabilized by \(g \in \mathcal{P}_n\) when \(g\psi = \psi \), i.e. \(\psi \) is a \(+1\) eigenvector of the unitary definition 5.
A *stabilizer group* is an abelian subgroup \(\mathcal{S} \le \mathcal{P}_n\) that does not contain \(-I\). The two conditions are exactly what is needed for the common \(+1\) eigenspace to be nonzero: commutativity makes the eigenspace projectors compatible, and excluding \(-I\) rules out the contradiction \(\psi = -\psi \).
\(-I \notin \mathcal{S}\) for any stabilizer group \(\mathcal{S}\).
Immediate from the defining field of definition 21. It is recorded separately because it is the obligation that concrete codes must discharge by hand, and it is the step that fails for a would-be "stabilizer group" generated by anticommuting operators.
The codespace of \(\mathcal{S}\) is the set of states stabilized by every element of \(\mathcal{S}\) — the simultaneous \(+1\) eigenspace. Logical information is stored here.
The two conditions in definition 21 — abelian, and not containing \(-I\) — are exactly what is needed for the codespace to be nonzero. Commutativity makes the eigenspace projectors compatible; excluding \(-I\) rules out the contradiction \(\psi = -\psi \). The proof that the codespace is genuinely nonempty runs through the group sum.
The operator \(\sum _{g \in \mathcal{S}} g\), which is \(|\mathcal{S}|\) times the orthogonal projector onto the codespace. Summing over the group is the standard route to showing the codespace is nonzero.
Every stabilizer group has a state in its codespace. A stabilizer code is therefore never vacuous.
The stabilizer sum definition 24 has trace \(2^n \neq 0\): the identity contributes \(2^n\) and every other element of \(\mathcal{S}\) is a non-identity Pauli, hence traceless. A nonzero operator has a nonzero column, and rescaling that column by \(|\mathcal{S}|\) gives a state fixed by every \(g \in \mathcal{S}\), because \(g\) permutes the summands of definition 24.
4.1 The centralizer
Operators that preserve the codespace are those commuting with every stabilizer. For the Pauli group — and this is special to it — normalizing and centralizing coincide, which is why the literature uses the two words interchangeably in this setting.
The centralizer \(\mathcal{C}(\mathcal{S})\) of a stabilizer group inside \(\mathcal{P}_n\): the Pauli operators commuting with every element of \(\mathcal{S}\). These are exactly the operators that preserve the codespace, so they are the candidates for logical operators.
For a stabilizer group, the Pauli normalizer and the centralizer coincide. This is special to the Pauli group and is the reason the literature uses the two words interchangeably here.
One inclusion is general. Conversely, if \(g\) normalizes \(\mathcal{S}\) then for \(s \in \mathcal{S}\) we have \(gsg^{-1} \in \mathcal{S}\), and by theorem 13 that conjugate is \(\pm s\). The value \(-s\) is impossible: it would put \(-I = (-s)s^{-1}\) in \(\mathcal{S}\), contradicting theorem 22. So \(gsg^{-1} = s\).
\(\mathcal{S} \le \mathcal{C}(\mathcal{S})\).
Restatement of commutativity of definition 21. The quotient \(\mathcal{C}(\mathcal{S})/\mathcal{S}\) is what carries the logical operators.