Solution
= Solution
For outcome probabilities $p_k=\operatorname{Tr}(\Lambda_k\rho)$ and $q_k=\operatorname{Tr}(\Lambda_k\sigma)$, the hint gives
$$
\frac12\sum_k|p_k-q_k|
\geq\frac12\sum_k(\sqrt{p_k}-\sqrt{q_k})^2
=1-\sum_k\sqrt{p_kq_k}.
$$
The left side is at most $\|\rho-\sigma\|_1/2$ by part d. Choose a POVM attaining the stated minimum characterization of <quantum fidelity>; then
$$
\boxed{\frac12\|\rho-\sigma\|_1\geq1-F(\rho,\sigma)}.
$$