 Solution

A normalized pure state has $\rho =|\psi ⟩⟨\psi |$, and thereforeConversely, diagonalize a permissible density operator. Its eigenvalues obey $0\le {\lambda }_j\le 1$ and ${\sum }_j{\lambda }_j=1$, soEquality requires every eigenvalue to be zero or one; exactly one is one. Thus the idempotent matrix condition and purity one both characterize a pure state. A mixed state has at least two positive eigenvalues, ${\rho }^2\ne \rho$ and $\mathrm{Tr}{\rho }^2<1$. In dimension $d$ its purity is at least $1/d$.