  We derive the ultimate bounds on the performance of nonlinear measurement
schemes in the presence of noise. In particular, we investigate the precision
of the second-order estimation scheme in the presence of the two most
detrimental types of noise, photon loss and phase diffusion. We find that the
second-order estimation scheme is affected by both types of noise in an
analogous way as the linear one. Moreover, we observe that for both types of
noise the gain in the phase sensitivity with respect to the linear estimation
scheme is given by a multiplicative term $\mathcal{O}(1/N)$. Interestingly, we
also find that under certain circumstances, a careful engineering of the
environment can, in principle, improve the performance of measurement schemes
affected by phase diffusion.
