= Purity bound for generalized Bloch vectors
{title2=$\|s\|^2=\operatorname{Tr}(\rho^2)-1/N\leq1-1/N$}
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>.
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