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.

Figure 1

Normalized quantum Fisher information $H(\varphi )/\overline{n}$ as a function of the squeezing fraction of the probe state for different values of the probe energy and two values of the actual loss parameter. (Left): ${tan}^2\varphi =0.1$. (Right): ${tan}^2\varphi =5$. In both plots, from bottom to top in the region $x\simeq 0$, the curves for $\overline{n}=0.5$, $\overline{n}=1$, $\overline{n}=2$, $\overline{n}=5$, $\overline{n}=10$, and $\overline{n}=100$.

Figure 2

(Left): Rescaled optimal variance $H^{-1}(\varphi )$ as a function of the probe energy for different actual values of the loss parameter: $z=0.01$, 0.1, 1, 5, 10. The larger is $z$ ($\varphi$) the closer is the curve to the asymptotic $H^{-1}(\varphi )=(4\overline{n}{)}^{-1}$. (Right): Rescaled optimal variance compared to the variance obtained for coherent probe as a function of the loss parameter for different values of the probe energy: from top to bottom the curves for $\overline{n}=0.1$, 0.3, 0.7, 1, respectively.