  A local excitation in a quantum many-spin system evolves deterministically. A
time-reversal procedure, involving the inversion of the signs of every energy
and interaction, should produce the excitation revival. This idea,
experimentally coined in NMR, embodies the concept of the Loschmidt echo (LE).
While such an implementation involves a single spin autocorrelation $M_{1,1}$,
i.e. a local LE, theoretical efforts have focused on the study of the recovery
probability of a complete many-body state, referred here as global or many-body
LE $M_{MB}$. Here, we analyze the relation between these magnitudes, in what
concerns to their characteristic time scales and their dependence on the number
of spins $N$. We show that the global LE can be understood, to some extent, as
the simultaneous occurrence of $N$ independent local LEs, i.e. $M_{MB}\sim
\left( M_{1,1}\right) ^{N/4}$. This extensive hypothesis is exact for very
short times and confirmed numerically beyond such a regime. Furthermore, we
discuss a general picture of the decay of $M_{1,1}$ as a consequence of the
interplay between the time scale that characterizes the reversible interactions
($T_{2}$) and that of the perturbation ($\tau _{\Sigma }$). Our analysis
suggests that the short time decay, characterized by the time scale $\tau
_{\Sigma }$, is greatly enhanced by the complex processes that occur beyond
$T_{2}$ . This would ultimately lead to the experimentally observed $T_{3},$
which was found to be roughly independent of $\tau _{\Sigma }$ but closely tied
to $T_{2}$.
