Geometric derivation of the quantum speed limit

TL;DR

This paper provides geometric derivations of the Mandelstam-Tamm and Margolus-Levitin quantum speed limits, clarifying their applicability to unitary evolution and highlighting limitations for non-unitary processes.

Contribution

It offers new geometric derivations for key quantum speed limits and demonstrates their validity for pure and mixed states under unitary evolution, with a counterexample for non-unitary cases.

Findings

Derivations based on statistical distance for both inequalities

Inequalities hold for unitary evolution of pure and mixed states

Counterexample shows no quantum speed limit for non-unitary evolution

Abstract

The Mandelstam-Tamm and Margolus-Levitin inequalities play an important role in the study of quantum mechanical processes in Nature, since they provide general limits on the speed of dynamical evolution. However, to date there has been only one derivation of the Margolus-Levitin inequality. In this paper, alternative geometric derivations for both inequalities are obtained from the statistical distance between quantum states. The inequalities are shown to hold for unitary evolution of pure and mixed states, and a counterexample to the inequalities is given for evolution described by completely positive trace-preserving maps. The counterexample shows that there is no quantum speed limit for non-unitary evolution.