Optimal quantum estimation of loss in bosonic channels

We address the estimation of the loss parameter of a bosonic channel probed by Gaussian signals. We derive the ultimate quantum bound with precision and show that no improvement may be obtained by having access to the environmental degrees of freedom. We find that, for small losses, the variance of the optimal estimator is proportional to the loss parameter itself, a result that represents a qualitative improvement over the shot-noise limit. An observable based on the symmetric logarithmic derivative is obtained, which attains the ultimate bound and may be implemented using Gaussian operations and photon counting.