  The presence of noise or the interaction with an environment can radically
change the dynamics of observables of an otherwise isolated quantum system. We
derive a bound on the speed with which observables of open quantum systems
evolve. This speed limit divides into Mandalestam and Tamm's original
time-energy uncertainty relation and a time-information uncertainty relation
recently derived for classical systems, generalizing both to open quantum
systems. By isolating the coherent and incoherent contributions to the system
dynamics, we derive both lower and upper bounds to the speed of evolution. We
prove that the latter provide tighter limits on the speed of observables than
previously known quantum speed limits, and that a preferred basis of
\emph{speed operators} serves to completely characterize the observables that
saturate the speed limits. We use this construction to bound the effect of
incoherent dynamics on the evolution of an observable and to find the
Hamiltonian that gives the maximum coherent speedup to the evolution of an
observable.
