Fig. 9: Convergence to the perfect QFI with the photon-number N for different noisy photodetection channels.

\({\gamma }_{{{{{{{{{\mathcal{M}}}}}}}}}^{\otimes N}}\) for different dark-count rates, \({\mathsf{p}}\), given a finite detection efficiency (losses) η = 0.1. The blue (solid) line corresponds to losses without dark counts \({\mathsf{p}}\) = 0, while the orange dashed line also includes a dark-count rate of \({\mathsf{p}}=0.01\). Inset: The FI as a function of the angle φ between \(|N,0 \rangle\) and \(|0,N \rangle\) in Eq. (46) that parametrises the global control unitary \({V}_{\vec{\Phi }}\). While with only losses (solid blue line) any angle is optimal, dark counts (dashed orange line) remove this symmetry. In this illustration, N = 50.