We establish a direct connection of
quantum Markovianity of an open 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 map. 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 mathematical representations similar to 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.