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Quantum metrology uses small changes in the output probabilities of a quantummeasurement to estimate the magnitude of a weak interaction with the system.The sensitivity of this procedure depends on the relation between the inputstate, the measurement results, and the generator observable describing theeffect of the weak interaction on the system. This is similar to the situationin weak measurements, where the weak value of an observable exhibits asymmetric dependence on initial and final conditions. In this paper, it isshown that the phase sensitivity of a quantum measurement is in fact given bythe variance of the imaginary parts of the weak values of the generator overthe different measurement outcomes. It is then possible to include thelimitations of a specific quantum measurement in the uncertainty bound forphase estimates by subtracting the variance of the real parts of the weakvalues from the initial generator uncertainty. This uncertainty relation can beinterpreted as the time-symmetric formulation of the uncertainty limit ofquantum metrology, where the real parts of the weak values represent theinformation about the generator observable in the final measurement result.

Uncertainty limits for quantum metrology obtained from the statistics of weak measurements