Tight upper bound of the maximum speed of evolution of a quantum state

I report a tight upper bound of the maximum speed of evolution from one quantum state rho to another rho' with fidelity F(rho,rho') less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state DE, one may interpret DE as a meaningful measure of the maximum information-processing capability of a system.