  Inequalities of Mandelstam-Tamm and Margolus-Levitin type provide lower
bounds on the time it takes for a quantum system to evolve from one state into
another. Knowledge of such bounds, called quantum speed limits, is of utmost
importance in virtually all areas of physics, where determination of the
minimum time required for a quantum process is of interest. Most
Mandelstam-Tamm and Margolus-Levitin inequalities found in the literature have
been derived from growth estimates for the Bures length, which is a statistical
distance measure. In this paper we derive such inequalities by differential
geometric methods, and we compare the obtained quantum speed limits with those
involving the Bures length. We also characterize the Hamiltonians which
optimize the evolution time for generic finite-level quantum systems.
