$\boldsymbol{\alpha_{>}(\epsilon) = \alpha_{<}(\epsilon)}$ For The Margolus-Levitin Quantum Speed Limit Bound

TL;DR

This paper proves that the upper and lower bounds of the Margolus-Levitin quantum speed limit are equal, providing an elementary proof, identifying states that saturate the bound, and addressing numerical stability in calculations.

Contribution

It offers an elementary proof that the bounds are equal, identifies all states saturating the bound, and improves numerical evaluation methods.

Findings

Upper and lower bounds of the ML bound are equal.

Explicit characterization of states saturating the ML bound.

A simple, stable method for numerical evaluation of the bound.

Abstract

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 $\pi \alpha(\epsilon) / (2 \langle E-E_0 \rangle)$, where $\langle E-E_0 \rangle$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\alpha(\epsilon)$ is a function of the fidelity $\epsilon$ between the two state. For a long time, only a upper bound $\alpha_{>}(\epsilon)$ and lower bound $\alpha_{<}(\epsilon)$ are known although they agree up to at least seven significant figures. Lately, H\"{o}rnedal and S\"{o}nnerborn proved an analytical expression for $\alpha(\epsilon)$, fully classified systems whose evolution times saturate 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 $\alpha_{>}(\epsilon)$ is indeed equal to $\alpha_{<}(\epsilon)$. More importantly, I point out a numerical stability issue in computing $\alpha_{>}(\epsilon)$ and report a simple way to evaluate it efficiently and accurately.