Continuous-variable phase-estimation with unitary and random linear disturbance

We address the problem of continuous-variable quantum phase estimation in the presence of linear disturbance at the Hamiltonian level, by means of Gaussian probe states. In particular we discuss both unitary and random disturbance, by considering the parameter which characterizes the unwanted linear term present in the Hamiltonian as fixed (unitary disturbance) or random with a given probability distribution (random disturbance). We derive the optimal input Gaussian states at fixed energy, maximizing the quantum Fisher information over the squeezing angle and the squeezing energy fraction, and we discuss the scaling of the quantum Fisher information in terms of the output number of photons $n_{out}$. We observe that in the case of unitary disturbance the optimal state is a squeezed vacuum state and the quadratic scaling is conserved. As regards the random disturbance, we observe that the optimal squeezing fraction may not be equal to one, and, for any non-zero value of the noise parameter, the quantum Fisher information scales linearly with the average number of photons. We finally discuss the performance of homodyne measurement, comparing the achievable precision with the ultimate limit posed by the quantum Cram\'er-Rao bound.