Extreme points of the density-operator state space ($\rho \text{ extreme}⟺\mathrm{rank}\rho =1$)

The density operators on a finite-dimensional Hilbert space form a convex set. Its extreme points are exactly the rank-one projectors, hence exactly the pure states. To prove this, suppose $|\psi ⟩⟨\psi |={\sum }_i{a}_i{\rho }_i$ with $a_i>0$ and ${\rho }_i$ positive semidefinite. For $w\perp \psi$, $0={\sum }_i{a}_i⟨w|{\rho }_i|w⟩$ is a sum of nonnegative numbers, so every term vanishes. Diagonalizing ${\rho }_i$ shows that $⟨w|{\rho }_i|w⟩=0$ implies ${\rho }_{i}w=0$. Thus each ${\rho }_i$ is supported on the line spanned by $\psi$, and its unit trace forces ${\rho }_i=|\psi ⟩⟨\psi |$.

Conversely, a density operator of rank at least two has a spectral decomposition with at least two positive eigenvalues. If $\lambda$ is one of them, $0<\lambda <1$ and $\rho =\lambda |e⟩⟨e|+(1-\lambda )\sigma$, where $\sigma$ is the normalized remainder. These are distinct density operators, proving that $\rho$ is not an extreme point.