# 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.