Binary symplectic space of Pauli labels ($\omega ((a,b),(a^{\prime },b^{\prime }))=a\cdot b^{\prime }+b\cdot a^{\prime }$)

Modulo scalar phases, the n-qubit Pauli group is the additive space ${\mathbb{F}}_2^{2n}$, with label $(a,b)$ for $X^a{Z}^b$. Multiplication adds labels, and commutation contributes the sign $(-1{)}^{\omega }$. The displayed alternating form is nondegenerate. An isotropic subspace $L$ satisfies $L\subseteq L^{\perp }$, hence $\dim L\le n$; it labels commuting stabilizers.

A subspace $L$ is isotropic when the symplectic pairing vanishes between every pair of its vectors. Its binary Pauli labels therefore commute. Nondegeneracy identifies the ambient space with its dual, so the map to $L^{\ast }$ is surjective and $\dim L^{\perp }=2n-\dim L$. Since $L\subseteq L^{\perp }$, it follows that $\dim L\le n$. This explains the maximum number of independent generators of a stabilizer group.