  The purpose of this article is to exploit the geometric structure of Quantum
Mechanics and of statistical manifolds to study the qualitative effect that the
quantum properties have in the statistical description of a system. We show
that the end points of geodesics in the classical setting coincide with the
probability distributions that minimise Shannon's Entropy, i.e. with
distributions of zero dispersion. In the quantum setting this happens only for
particular initial conditions, which in turn correspond to classical
submanifolds. This result can be interpreted as a geometric manifestation of
the uncertainty principle.
