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

We present a simple proof of the fact that the minimum time T_AB for quantum evolution between two arbitrary states |A〉 and |B〉 equals T_AB = ℏ cos⁻¹ [|〈A|B〉|]/∆E with ∆E being the constant energy uncertainty of the system. 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 ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε ≤ 1 even when the system only passes through nonorthogonal quantum states.