  We address characterization of lossy and dephasing channels in the presence
of self-Kerr interaction using coherent probes. In particular, we investigate
the ultimate bounds to precision in the joint estimation of loss and
nonlinearity and of dephasing and nonlinearity. To this aim, we evaluate the
quantum Fisher information matrix (QFIM), and compare the symmetric quantum
Cram\'er-Rao bound (QCR) to the bound obtained with Fisher information matrix
(FIM) of feasible quantum measurements, i.e., homodyne and double-homodyne
detection. For lossy Kerr channels, our results show the loss characterization
is enhanced in the presence of Kerr nonlinearity, especially in the relevant
limit of small losses and low input energy, whereas the estimation of
nonlinearity itself is unavoidably degraded by the presence of loss. In the low
energy regime, homodyne detection of a suitably optimized quadrature represents
a nearly optimal measurement. The Uhlmann curvature does not vanish, therefore
loss and nonlinearity can be jointly estimated only with the addition of
intrinsic quantum noise. For dephasing Kerr channels, the QFIs of the two
parameters are independent of the nonlinearity, and therefore no enhancement is
observed. Homodyne and double-homodyne detection are suboptimal for the
estimation of dephasing and nearly optimal for nonlinearity. Also in this case,
the Uhlmann curvature is nonzero, proving that the parameters cannot be jointly
estimated with maximum precision.
