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The ultimate precision in any measurement is dictated by the physical processimplementing the observation. The methods of quantum metrology have nowsucceeded in establishing bounds on the achievable precision for phasemeasurements over noisy channels. In particular, they demonstrate how theHeisenberg scaling of the precision can not be attained in these conditions.Here we discuss how the ultimate bound in presence of loss has a physicalmotivation in the Kramers-Kronig relations and we show how they link theprecision on the phase estimation to that on the loss parameter.

Kramers-Kronig relations and precision limits in quantum phase estimation