  We present a simple proof of the minimum time for the quantum evolution
between two arbitrary states. This proof is performed in the absence of any
geometrical arguments. Then, being in the geometric framework of quantum
evolutions based upon the geometry of the projective Hilbert space, we discuss
the roles played by either minimum-time or maximum-energy uncertainty concepts
in defining a geometric efficiency measure of quantum evolutions between two
arbitrary quantum states. Finally, we provide a quantitative justification of
the validity of the efficiency inequality even when the system passes only
through nonorthogonal quantum states.
