For any POVM, part c followed by trace-distance contraction gives
$$
\sum_k|\operatorname{Tr}(\Lambda_k\rho)-\operatorname{Tr}(\Lambda_k\sigma)|
\leq\|\rho-\sigma\|_1.
$$
For the binary POVM $\{P,I-P\}$ from part b, equality holds. Consequently
$$
\boxed{\|\rho-\sigma\|_1
=\max_{\{\Lambda_k\}}\sum_k
|\operatorname{Tr}(\Lambda_k\rho)-\operatorname{Tr}(\Lambda_k\sigma)|}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-323/4/d/solution