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The question of how fast a quantum state can evolve has attracted aconsiderable attention in connection with quantum measurement, metrology, andinformation processing. Since only orthogonal states can be unambiguouslydistinguished, a transition from a state to an orthogonal one can be taken asthe elementary step of a computational process. Therefore, such a transitioncan be interpreted as the operation of "flipping a qubit", and the number oforthogonal states visited by the system per unit time can be viewed as themaximum rate of operation.  A lower bound on the orthogonalization time, based on the energy spreadDeltaE, was found by Mandelstam and Tamm. Another bound, based on the averageenergy E, was established by Margolus and Levitin. The bounds coincide, and canbe exactly attained by certain initial states if DeltaE=E; however, the problemremained open of what the situation is otherwise.  Here we consider the unified bound that takes into account both DeltaE and E.We prove that there exist no initial states that saturate the bound if DeltaEis not equal to E. However, the bound remains tight: for any given values ofDeltaE and E, there exists a one-parameter family of initial states that canapproach the bound arbitrarily close when the parameter approaches its limitvalue. The relation between the largest energy level, the average energy, andthe orthogonalization time is also discussed. These results establish thefundamental quantum limit on the rate of operation of anyinformation-processing system.

The fundamental limit on the rate of quantum dynamics: the unified bound is tight