= Solution
Take any rank-one <orthogonal projection> $P$ and its generalized <Bloch vector> $s$. The antipodal vector $-s$ is on the same <sphere>, but its corresponding trace-one <Hermitian matrix> is
$$
\rho_- =\frac IN-\left(P-\frac IN\right)=\frac2N I-P.
$$
On the range of $P$, this <matrix> has <eigenvalue> $2/N-1$, which is negative for $N>2$. It is therefore not a <positive semidefinite matrix> and cannot be a <density operator>. Hence \b[the pure states do not cover the sphere when $N>2$]. This also shows directly why the <purity bound for generalized Bloch vectors> does not by itself guarantee a physical <quantum state>.
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