Quantum parameter estimation affected by unitary disturbance

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.