  We analyze the problem of how different ground states associated to the same
set of the Hamiltonian parameters evolve after a sudden quench. To realize our
analysis we define a quantitative approach to the local distinguishability
between different ground states of a magnetically ordered phase in terms of the
trace distance between the reduced density matrices obtained projecting two
ground states in the same subset. Before the quench, regardless the particular
choice of the subset, any system in a magnetically ordered phase is
characterized by ground states that are locally distinguishable. On the other
hand, after the quench, the maximum of the distinguishability shows an
exponential decay in time. Hence, in the limit of very large time, all the
informations about the particular initial ground state are lost even if the
systems are integrable. We prove our claims in the framework of the
magnetically ordered phases that characterize both the $XY$ model and
$N$-cluster Ising models. The fact that we find similar behavior in models
within different classes of symmetry makes us confident about the generality of
our results.
