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We present a simple proof of the minimum time for the quantum evolutionbetween two arbitrary states. This proof is performed in the absence of anygeometrical arguments. Then, being in the geometric framework of quantumevolutions based upon the geometry of the projective Hilbert space, we discussthe roles played by either minimum-time or maximum-energy uncertainty conceptsin defining a geometric efficiency measure of quantum evolutions between twoarbitrary quantum states. Finally, we provide a quantitative justification ofthe validity of the efficiency inequality even when the system passes onlythrough nonorthogonal quantum states.

Minimum time for the evolution to a nonorthogonal quantum state and upper bound of the geometric efficiency of quantum evolutions