Unraveling quantum and classical speed limits on observables

We introduce speed limits to the evolution of observables of open quantum systems with arbitrary differentiable dynamics, generalizing the original derivation by Mandelstam and Tamm in 1945. The upper bounds on speed take the form of an uncertainty relation and unify previous bounds known for unitary quantum and for classical stochastic systems. By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds to the speed of evolution, and prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits.