  The Mandelstam-Tamm and Margolus-Levitin inequalities play an important role
in the study of quantum mechanical processes in Nature, since they provide
general limits on the speed of dynamical evolution. However, to date there has
been only one derivation of the Margolus-Levitin inequality. In this paper,
alternative geometric derivations for both inequalities are obtained from the
statistical distance between quantum states. The inequalities are shown to hold
for unitary evolution of pure and mixed states, and a counterexample to the
inequalities is given for evolution described by completely positive
trace-preserving maps. The counterexample shows that there is no quantum speed
limit for non-unitary evolution.
