  The quantum backflow effect is a counterintuitive behavior of the probability
current of a free particle, which may be negative even for states with
vanishing negative momentum component. Here we address the notion of
nonclassicality arising from the backflow effect, i.e. from the negativity of
the probability current, and analyze its relationships with the notion of
nonclassicality based on the negativity of the Wigner function. Our results
suggest that backflow is linked to a different, and in fact more restrictive,
notion of nonclassicality, the negativity of the Wigner function being only a
necessary prerequisite for its occurrence. This hierarchical structure may be
confirmed by looking at the addition of thermal noise, which more easily
destroys the negativity of the probability current than the negativity of the
Wigner function itself.
