  We address multiparameter quantum estimation for coherently driven nonlinear
Kerr resonators in the presence of loss. In particular, we consider the
realistic situation in which the parameters of interest are the loss rate and
the nonlinear coupling, whereas the amplitude of the coherent driving is known
and externally tunable. Our results show that this driven-dissipative model is
asymptotically classical, i.e. the Uhlmann curvature vanishes, and the two
parameters may be jointly estimated without any additional noise of quantum
origin. We also find that the ultimate bound to precision, as quantified by the
quantum Fisher information (QFI), increases with the interaction time and the
driving amplitude for both parameters. Finally, we investigate the performance
of quadrature detection, and show that for both parameters the Fisher
information oscillates in time, repeatedly approaching the corresponding QFI.
