# Question 7

This question asks to to compute the entanglement fidelity $$F_\text{e}(\rho,\Lambda)$$ where $$\rho$$ is a qubit and $$\Lambda$$ is the amplitude damping channel. The entanglement fidelity is $F_\text{e}(\rho,\Lambda) = \langle \psi_{RA}^\rho | \operatorname{id}\otimes \Lambda(\psi^\rho_{RA}) | \psi^\rho_{RA} \rangle$ where $$\psi^\rho_\text{RA}$$ is a purification of $$\rho$$. Is there a clever trick we can use to compute $$F_\text{e}(\rho,\Lambda)$$? I can’t think of one…. Dear reader: if you know a quick way to do this one, please let me know and I’ll update this answer (and credit you if you’re up for that)!
Update: Seth Musser emailed me and pointed out there is a very useful lemma in Notes 12: $F_\text{e}(\rho,\Lambda) = \sum_k |\operatorname{tr}(A_k \rho)|^2$ whenever $$\{A_k\}$$ is a set of Kraus operators for $$\Lambda$$. Since we are given the two Kraus operators for the amplitude damping channel, this seems like a great way to do the calculation.