Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight

How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread Delta E, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if Delta E = E. Yet, the problem remained open when Delta E not equal E. We consider the unified bound that involves both Delta E and E. We prove that there exist no initial states that saturate the bound if Delta E not equal E. However, the bound remains tight: for any values of Delta E and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.