On Hamiltonians Generating Optimal-Speed Evolutions

TL;DR

This paper derives formulas for Hamiltonians that enable the fastest possible quantum state evolutions, including in pseudo-Hermitian frameworks, highlighting bounds on evolution speed and their metric dependencies.

Contribution

It provides a simple derivation of optimal Hamiltonians for fastest evolutions, extending to pseudo-Hermitian quantum mechanics and clarifying metric effects.

Findings

Explicit formulas for optimal Hamiltonians in standard and pseudo-Hermitian quantum mechanics

Analysis of metric dependence of evolution time bounds

Universal upper bound on evolution speed independent of metric

Abstract

We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric- (inner product-) dependence of the lower bound on the travel time and the universality (metric-independence) of the upper bound on the speed of unitary evolutions.