Choose a basis $|a\rangle$ of $H$ and define Kraus operators from $H$ to $\mathbb C^m$ by
$$
K_{ka}=|k\rangle\langle a|\sqrt{\Lambda_k}.
$$
Then
$$
\sum_{k,a}K_{ka}\rho K_{ka}^\dagger
=\sum_k|k\rangle\langle k|\operatorname{Tr}(\Lambda_k\rho),
$$
and
$$
\sum_{k,a}K_{ka}^\dagger K_{ka}
=\sum_k\Lambda_k=I.
$$
The displayed map is therefore a completely positive trace-preserving <measure-and-prepare channel>.

Solution

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

 Solution

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