  We address the characterization of dissipative bosonic channels and show that
estimation of the loss rate by Gaussian probes (coherent or squeezed) is
improved in the presence of Kerr nonlinearity. In particular, enhancement of
precision may be substantial for short interaction time, i.e. for media of
moderate size, e.g. biological samples. We analyze in detail the behaviour of
the quantum Fisher information (QFI), and determine the values of nonlinearity
maximizing the QFI as a function of the interaction time and of the parameters
of the input signal. We also discuss the precision achievable by photon
counting and quadrature measurement and present additional results for
truncated, few-photon, probe signals. Finally, we discuss the origin of the
precision enhancement, showing that it cannot be linked quantitatively to the
non-Gaussianity of the interacting probe signal.
