We provide a general framework for handling the effects of a unitary disturbance on the estimation of the amplitude $lambda$ associated to a unitary dynamics. By computing an analytical and general expression for the quantum Fisher information, we prove that the optimal estimation precision for $lambda$ cannot be outperformed through the addition of such a unitary disturbance. However, if the dynamics of the system is already affected by an external field, increasing its strength does not necessary imply a loss in the optimal estimation precision.
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