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Quantum lower and upper speed limits

We derive generalized quantum speed limit inequalities that represent limitations on the time evolution of quantum states. They are extensions of the original inequality and are applied to the overlap between the time-evolved state and an arbitrary state. We can discuss the lower limit of the overlap, in addition to the upper limit as in the original inequality, which allows us to estimate the minimum time for the evolution toward a target state. The inequalities are written by using an arbitrary reference state and are flexibly used to obtain a tight bound. We demonstrate these properties by using the twisted Landau-Zener model, the Grover Hamiltonian, and a periodically-oscillating Hamiltonian.