Comment on "Geometric derivation of the quantum speed limit"

TL;DR

This paper critiques a recent geometric derivation of quantum speed limits, corrects the derivation of a key bound, and proposes two new bounds that closely approximate the Margolus-Levitin inequality.

Contribution

The authors identify errors in the previous derivation and introduce two new bounds on quantum evolution speed, refining the understanding of quantum speed limits.

Findings

The derivation of the bound on the rate of change of statistical distance was incorrect.

Two new upper bounds are proposed, expressed in terms of the generator's standard deviation and expectation value.

The new bounds lead to a quantum speed limit nearly as tight as the Margolus-Levitin inequality.

Abstract

Recently, Jones and Kok [P. J. Jones and P. Kok, Phys. Rev. A 82, 022107 (2010)] presented alternative geometric derivations of the Mandelstam-Tamm [L. Mandelstam and I. Tamm, J. Phys. (USSR) 9, 249 (1945)] and Margolus-Levitin [N. Margolus and L. B. Levitin, Phys. D 120, 188 (1998)] inequalities for the quantum speed of dynamical evolution. The Margolus-Levitin inequality followed from an upper bound on the rate of change of the statistical distance between two arbitrary pure quantum states. We show that the derivation of this bound is incorrect. Subsequently, we provide two upper bounds on the rate of change of the statistical distance, expressed in terms of the standard deviation of the generator K and its expectation value above the ground state. The bounds lead to the Mandelstam-Tamm inequality and a quantum speed limit which is only slightly weaker than the Margolus-Levitin inequality.