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

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.