  A simple relationship between recently proposed measures of non-Markovianity
and the Loschmidt echo is established, holding for a two-level system (qubit)
undergoing pure dephasing due to a coupling with a many-body environment. We
show that the Loschmidt echo is intimately related to the information flowing
out from and occasionally back into the system. This, in turn, determines the
non-Markovianity of the reduced dynamics. In particular, we consider a central
qubit coupled to a quantum Ising ring in the transverse field. In this context,
the information flux between system and environment is strongly affected by the
environmental criticality; the qubit dynamics is shown to be Markovian exactly
and only at the critical point. Therefore non-Markovianity is an indicator of
criticality in the model considered here.
