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

TL;DR

This paper establishes a simple proof for the minimum time required for quantum state evolution without geometric arguments, and analyzes the geometric efficiency of such evolutions, especially when passing through nonorthogonal states.

Contribution

It provides a straightforward proof of minimum quantum evolution time and explores the geometric efficiency bounds in quantum state transitions.

Findings

Proof of minimum evolution time without geometric arguments

Analysis of geometric efficiency in quantum evolutions

Validation of efficiency inequality for nonorthogonal states

Abstract

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.