Kramers-Kronig relations and precision limits in quantum phase estimation

Phase measurements are of paramount importance in quantum optical sensing.However, the promise of a quantum advantage, the celebrated Heisenberg scaling, is severely curtailed in the presence of noise and loss.Herewe investigate systems in which phase and absorption profiles are linked by Kramers-Kronig relations and show that, in the limit of a large photon number, their use connects the uncertainties on the profiles attainable by optimal probes for loss and phase. This underlines a physical motivation for which the Heisenberg scaling for the phase is lost. Our results bear practical implications, revealing the metrological capabilities of absorption measurements in determining phase profiles.