Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 4 / e / Solution

For outcome probabilities $p_k=\mathrm{Tr}({\Lambda }_k\rho )$ and $q_k=\mathrm{Tr}({\Lambda }_k\sigma )$, the hint givesThe left side is at most $\Vert \rho -\sigma {\Vert }_1/2$ by part d. Choose a POVM attaining the stated minimum characterization of quantum fidelity; then