  A non-isolated physical system typically loses information to its
environment, and when such loss is irreversible the evolution is said to be
Markovian. Non-Markovian effects are studied by monitoring how information
quantifiers, such as the distance between physical states, evolve in time. Here
we show that the Fisher information metric emerges as a natural object to study
in this context; we fully characterize the relation between its contractivity
properties and Markovianity, both from the mathematical and operational point
of view. We prove, both for classical and quantum dynamics, that Markovianity
is equivalent to the monotonous contraction of the Fisher metric at all points
of the set of states. At the same time, operational witnesses of
non-Markovianity based on the dilation of the Fisher distance cannot, in
general, detect all non-Markovian evolutions, unless specific physical
postprocessing is applied to the dynamics. Finally, we show for the first time
that non-Markovian dilations of Fisher distance between states at any time
correspond to backflow of information about the initial state of the dynamics
at time 0, via Bayesian retrodiction.
