  We conclusively show that the entanglement- and the mutual information-based
measures of quantum non-Markovianity are inequivalent. To this aim, we first
analytically solve the optimization problem in the definition of the
entanglement-based measure for a two-level system. We demonstrate that the
optimal initial bipartite state of the open system and the ancillary is always
given by one of the Bell states for any one qubit dynamics. On top of this
result, we present an explicit example dynamics where memory effects emerge
according to the mutual information-based measure, even though the time
evolution remains memoryless with respect to the entanglement-based measure.
Finally, we explain this disagreement between the two measures in terms of the
information dynamics of the open system, exploring the accessible and
inaccessible parts of information.
