Unidirectional information flow and positive divisibility are nonequivalent notions of quantum Markovianity for noninvertible dynamics

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 maps