Optimal transmission estimation with dark counts

Transmission measurements are essential from fiber optics to spectroscopy. Quantum theory dictates that the ultimate precision in estimating transmission or loss is achieved using probe states with definite photon number and photon-number-resolving detectors (PNRDs). Can the quantum advantage relative to classical probe light still be maintained when the detectors fire due to dark counts and other spurious events? We demonstrate that the answer to this question is affirmative and show in detail how the quantum advantage depends on dark counts and increases with Fock-state-probe strength. These results are especially pertinent as the present capabilities of PNRDs are being dramatically improved.