  We provide the optimal measurement strategy for a class of noisy channels
that reduce to the identity channel for a specific value of a parameter
(spreading channels). We provide an example that is physically relevant: the
estimation of the absolute value of the displacement in the presence of phase
randomizing noise. Surprisingly, this noise does not affect the effectiveness
of the optimal measurement. We show that, for small displacement, a squeezed
vacuum probe field is optimal among strategies with same average energy. A
squeezer followed by photodetection is the optimal detection strategy that
attains the quantum Fisher information, whereas the customarily used homodyne
detection becomes useless in the limit of small displacements, due to the same
effect that gives Rayleigh's curse in optical superresolution. There is a
quantum advantage: a squeezed or a Fock state with $N$ average photons allow to
asymptotically estimate the parameter with a $\sqrt{N}$ better precision than
classical states with same energy.
