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 ρ to another ρ^' with fidelity F(ρ,ρ^') 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.