Stabilizer code space

Let $n\ge 1$ and let $S$ be an abelian subgroup of the Hermitian elements of the phase-aware Pauli group ${\mathcal{P}}_n$ on the ordered $n$-qubit register. Assume $-I\mathrm{\notin }S$. The stabilizer code space is the common $+1$ eigenspace

$\mathcal{C}(S):=\{\psi \in ({\mathbb{C}}^2{)}^{\otimes n}:g\psi =\psi \text{ for every }g\in S\}\mathrm{.}$

If $g_1,\dots ,g_r$ are independent generators of $S$, the associated stabilizer rank is $r$; independence means no nonempty product of distinct generators is $I$. The code-space projector and its nonzero dimension are established by Stabilizer projector and code dimension ↗.

The subgroup elements here are Hermitian Paulis in the phase-aware $i^k{X}^x{Z}^z$ convention. The binary label records commutation but does not replace the phase needed to determine whether a generator fixes the code with eigenvalue $+1$.