We construct a dynamical map which is not positive divisible and does not
present information backflow either (as measured by trace norm quantifiers). It
is formulated for a qutrit system undergoing noninvertible dynamics. This
provides an evidence that the two definitions of quantum Markovianity based on
absence of information backflow and positive divisibility are nonequivalent for
general noninvertible dynamical mapsComment: Submission to a special issue of 'Open Systems and Information
Dynamics' devoted to the memory of Prof. Andrzej Kossakowsk