  We theoretically investigate parameter quantum estimation in quantum chaotic
systems. Our analysis is based on an effective description of non-integrable
quantum systems in terms of a random matrix Hamiltonian. Based on this approach
we derive an analytical expression for the time evolution of the quantum Fisher
information. We test our random matrix theory prediction with the exact
diagonalization of a non-integrable spin system, focusing on the estimation of
a local magnetic field by measurements of the many-body state. Our numerical
calculations agree with the effective random matrix theory approach and show
that the information on the local Hamiltonian parameter is distributed
throughout the quantum system during the quantum thermalization process. Our
analysis shows a first stage in which the initial information spread is
quadratic in time which quickly passes into linear increase with slope
determine by the decay rate of the measured spin observable. When the
information is fully spread among all degrees of freedom a second quadratic
time scale determines the long time behaviour of the quantum Fisher
information.
