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.
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