Estimation of a generalized amplitude-damping channel

The problem of finding the optimal strategy for estimating a generalized amplitude damping channel ${\Gamma }_{\eta }^{\left(p\right)}$ by means of the extension $\mathrm{id}\otimes {\Gamma }_{\eta }^{\left(p\right)}$ is addressed. We first evaluate the quantum Fisher information of output states based on the symmetric logarithmic derivative and specify all pure-state inputs that maximize the quantum Fisher information. We next investigate the ${\nabla }^e$ autoparallelity of output state manifolds and characterize the condition for the existence of an efficient estimator. A comparison of these results concludes that, while there is no uniformly optimal input for all $p$ and $\eta$, a maximally entangled input is an admissible one under a nonasymptotic setting.