Lemma: Stabilizer reductions have flat nonzero spectra

Every reduced density matrix of a pure qubit stabilizer state is proportional to a projector. Its nonzero eigenvalues are equal, and its rank is a power of two.

Proof. For an $n$-qubit pure stabilizer state with stabilizer group $\mathcal{S}$,

Tracing the complementary subsystem leaves only Paulis supported on the retained subsystem $A$:

If ${\mathcal{S}}_A$ has $2^r$ elements, its group average $2^{-r}{\sum }_{P_A\in {\mathcal{S}}_A}{P}_A$ is a projector. Therefore ${\rho }_A$ has flat nonzero eigenvalues $2^{r-n_A}$ and rank $2^{n_A-r}$.