Stabilizer state

Let $S$ be a rank-$n$ stabilizer subgroup on an ordered $n$-qubit register $H_n=({\mathbb{C}}^2{)}^{\otimes n}$. A pure stabilizer state is the unique ray in the common $+1$ fixed space

$\mathcal{C}(S):=\{\psi \in H_n:g\psi =\psi \text{ for every }g\in S\}\mathrm{.}$

The cited projector-rank theorem proves that this common space is nonzero and one-dimensional, so the ray is well defined. A normalized representative may be used for probabilities, but the state itself is the ray, not a chosen vector.

De Wolf §20.1, PDF p. 185, and Watrous Lesson 14 §14.3, printed pp. 415–416 (PDF pp. 421–422), define a stabilizer state from a full-rank commuting Pauli subgroup and its common $+1$ eigenspace: https://arxiv.org/pdf/1907.09415; https://jhwatrous.github.io/UQIC.pdf.