α>(ϵ)=α<(ϵ) for the Margolus-Levitin quantum speed limit bound

Abstract

<p>The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least πα(ϵ)/2(E-E0), where (E-E0) is the expected energy of the system relative to the ground state of the Hamiltonian and α(ϵ) is a function of the fidelity ϵ between the two states. For a long time, only an upper bound α>(ϵ) and a lower bound α<(ϵ) are known, although they agree up to at least seven significant figures. Recently, Hörnedal and Sönnerborn [arXiv:2301.10063] proved an analytical expression for α(ϵ), a fully classified system whose evolution time saturates the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that α>(ϵ) is indeed equal to α<(ϵ). More importantly, I point out a numerical stability issue in computing α>(ϵ) and report a simple way to evaluate it efficiently and accurately.</p>