Purity bound for generalized Bloch vectors ($\Vert s{\Vert }^2=\mathrm{Tr}({\rho }^2)-1/N\le 1-1/N$)

For a Hilbert-Schmidt orthonormal traceless Hermitian basis, the Generalized Bloch representation gives the displayed identity by orthogonality. Nonnegative eigenvalues of a density operator sum to one, so their squared sum is at most one, with equality precisely for a rank-one projector. For $N>2$, the radius bound alone does not guarantee positivity: $(2/N)I-P$ for a rank-one projector $P$ has trace one and squared norm one, but one negative eigenvalue.