${\mathbf{\alpha }}_{>}(\mathbf{\epsilon })={\mathbf{\alpha }}_{<}(\mathbf{\epsilon })$ For The Margolus-Levitin Quantum Speed Limit 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 $\pi \alpha (\epsilon )/(2⟨E-E_0⟩)$, where $⟨E-E_0⟩$ 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, Hörnedal and Sönnerborn 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.

Extensively revised for clarity, 8 pages, 2 figures, to appear in PRA