  The characterization of entanglement is a central problem for the study of
quantum many-body dynamics. Here, we propose the quantum Fisher information as
a useful tool for the study of multipartite-entanglement dynamics in many-body
systems. We illustrate this by considering the regular-to-ergodic transition in
the Dicke model---a fully-connected spin model showing quantum thermalization
above a critical interaction strength. We show that the QFI has a rich
dynamical behavior which drastically changes across the transition. In
particular, the asymptotic value of the QFI, as well as its characteristic
timescales, witness the transition both through their dependence on the
interaction strength and through the scaling with the system size. Since the
QFI also sets the ultimate bound for the precision of parameter estimation, it
provides a metrological perspective on the characterization of entanglement
dynamics in many-body systems. Here we show that quantum ergodic dynamics
allows for a much faster production of metrologically useful states.
