Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 2 / b / ii / Solution

Yes. Uhlmann's theorem says that, for the fixed purification $|{\Psi }^{\rho }{⟩}_{AR}$,Choose a maximizing purification of ${\sigma }_A$ on the same reference space. In the unsquared fidelity convention,