For a purification $|\Psi_\rho\rangle_{RA}$ of $\rho$, the <entanglement fidelity> is
$$
F_e(\rho,\Lambda)=
\langle\Psi_\rho|
(\operatorname{id}_R\otimes\Lambda)
(|\Psi_\rho\rangle\langle\Psi_\rho|)
|\Psi_\rho\rangle.
$$
Using a <Kraus representation> $\Lambda(X)=\sum_kA_kXA_k^\dagger$ gives
$$
F_e(\rho,\Lambda)
=\sum_k\left|
\langle\Psi_\rho|I\otimes A_k|\Psi_\rho\rangle
\right|^2.
$$
In a <Schmidt decomposition> of the purification,
$$
\langle\Psi_\rho|I\otimes A_k|\Psi_\rho\rangle
=\sum_j\lambda_j\langle j|A_k|j\rangle
=\operatorname{Tr}(A_k\rho).
$$
Therefore
$$
\boxed{F_e(\rho,\Lambda)
=\sum_k|\operatorname{Tr}(A_k\rho)|^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 1 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-323/1/a/ii/solution