  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.
