  In the Schr{\"o}dinger picture, the state of a quantum system evolves in time
and the quantum speed limit describes how fast the state of a quantum system
evolves from an initial state to a final state. However, in the Heisenberg
picture the observable evolves in time instead of the state vector. Therefore,
it is natural to ask how fast an observable evolves in time. This can impose a
fundamental bound on the evolution time of the expectation value of quantum
mechanical observables. We obtain the quantum speed limit time-bound for
observable for closed systems, open quantum systems and arbitrary dynamics.
Furthermore, we discuss various applications of these bounds. Our results can
have several applications ranging from setting the speed limit for operator
growth, correlation growth, quantum thermal machines, quantum control and many
body physics.
