  In this work we investigate the equilibration dynamics after a sudden
Hamiltonian quench of a quantum spin system initially prepared in a thermal
state. To characterize the equilibration we evaluate the Loschmidt echo, a
global measure for the degree of distinguishability between the initial and
time-evolved quenched states. We present general results valid for small
quenches and detailed analysis of the quantum XY chain. The result is that
quantum criticality manifests, even at small but finite temperatures, in a
universal double-peaked form of the echo statistics and poor equilibration for
sufficiently relevant perturbations. In addition, for this model we find a
tight lower bound on the Loschmidt echo in terms of the purity of the initial
state and the more-easily-evaluated Hilbert-Schmidt inner product between
initial and time-evolved quenched states. This bound allows us to relate the
time-averaged Loschmidt echo with the purity of the time-averaged state, a
quantity that has been shown to provide an upper bound on the variance of
observables.
