Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 323 / 3 / iii / b / Solution

A density operator is a pure state exactly when its matrix rank is one. Within the admissible interval from part (a), one of $(1+\alpha )/2$ and $(1-\alpha )/2$ must vanish. Therefore the pure-state parameters and vectors arewhere the Bell states are $|{\Phi }^{\pm }⟩=(|00⟩\pm |11⟩)/\sqrt{2}$. Equivalently, the purity of a density operator is $\mathrm{Tr}{\rho }_{AB}^2=(1+{\alpha }^2)/2$, reaching one exactly at these two endpoints.