  The dynamics of Fisher information for an accelerated system initially
prepared in the $X$-state is discussed. An analytical solution, which consists
of three parts: classical, the average over all pure states and a mixture of
pure states is derived for the general state and for Werner state. It is shown
that, the Unruh acceleration has a depleting effect on the Fisher information.
This depletion depends on the degree of entanglement of the initial state
settings. For the $X$-state, for some intervals of Unruh acceleration, the
Fisher information remains constant, irrespective to the Unruh acceleration. In
general, the possibility of estimating the state's parameters decreases as the
acceleration increases. However, the precision of estimation can be maximized
for certain values of the Unruh acceleration. We also investigate the
contribution of the different parts of the Fisher information on the dynamics
of the total Fisher information.
