# Almost Markovian maps and entanglement-based bound on corresponding non-Markovianity. (arXiv:1905.06198v1 [quant-ph])

We identify a set of dynamical maps of open quantum system, and refer to them
as "$\epsilon$-Markovian" maps. It is constituted of maps that possibly
violate Markovianity but only a "little". We characterize the "$\epsilon$-nonmarkovianity" of a general dynamical map by the minimum distance
of that map from the set of $\epsilon$-Markovian maps. We analytically derive
an inequality which gives a bound on the $\epsilon$-nonmarkovianity of the
dynamical map, in terms of an entanglement-like resource generated between the
system and its "immediate" environment. In the special case of a vanishing
$\epsilon$, this inequality gives a relation between the non-Markovianity of
the reduced dynamical map on the system and the entanglement generated between
the system and its immediate environment. We investigate the behavior of the
measures for classes of amplitude damping and phase damping channels.