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. Here we 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.