Question 7

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Question 7.

Regarding Q2 of the 2012 exam: Is there a better way to calculate part iv) entanglement fidelity than doing purification directly?

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.