Show that the density matrix \(\hat{ho}(t)\) has the properties \(\operatorname{Tr} \hat{ho}(t)=1\) and \(\operatorname{Tr} \hat{ho}^{2}(t) \leq 1\), and that the equality is manifested only for the case in which the ensemble used to construct the density matrix is composed of a single pure state. Show that \(0 \leq\langle\phi|\hat{ho}| \phiangle \leq 1\) for any normalized vector \(|\phiangle\), and use this to determine properties of the eigenvalues for mixed states and pure states. Use the properties of the eigenvalues to show that the von Neumann entropy \(S[\hat{ho}]\) is zero for pure states and positive for mixed states.