  In this paper, we derive sharp lower bounds, also known as quantum speed
limits, for the time it takes to transform a quantum system into a state such
that an observable assumes its lowest average value. We assume that the system
is initially in an incoherent state relative to the observable and that the
state evolves according to a von Neumann equation with a Hamiltonian whose
bandwidth is uniformly bounded. The transformation time depends intricately on
the observable's and the initial state's eigenvalue spectrum and the relative
constellation of the associated eigenspaces. The problem of finding quantum
speed limits consequently divides into different cases requiring different
strategies. We derive quantum speed limits in a large number of cases, and we
simultaneously develop a method to break down complex cases into manageable
ones. The derivations involve both combinatorial and differential geometric
techniques. We also study multipartite systems and show that allowing
correlations between the parts can speed up the transformation time. In a final
section, we use the quantum speed limits to obtain upper bounds on the power
with which energy can be extracted from quantum batteries.
