  We establish a direct connection of quantum Markovianity of an open quantum
system to its classical counterpart by generalizing the criterion based on the
information flow. Here, the flow is characterized by the time evolution of
Helstrom matrices, given by the weighted difference of statistical operators,
under the action of the quantum dynamical evolution. It turns out that the
introduced criterion is equivalent to P-divisibility of a quantum process,
namely divisibility in terms of positive maps, which provides a direct
connection to classical Markovian stochastic processes. Moreover, it is shown
that similar mathematical representations as those found for the original trace
distance based measure hold true for the associated, generalized measure for
quantum non-Markovianity. That is, we prove orthogonality of optimal states
showing a maximal information backflow and establish a local and universal
representation of the measure. We illustrate some properties of the generalized
criterion by means of examples.
