Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states

TL;DR

This paper extends the Mandelstam-Tamm quantum speed limit to mixed states, analyzes the tightness of these bounds, and shows that optimal evolutions often involve time-varying Hamiltonians.

Contribution

It introduces a new tightest extension of the quantum speed limit for mixed states and explores the geometric basis of these bounds.

Findings

Uhlmann's energy dispersion estimate often yields loose bounds.

A new geometric construction provides the tightest possible extension.

Tight evolutions of mixed states typically involve time-varying Hamiltonians.

Abstract

The Mandelstam-Tamm quantum speed limit puts a bound on how fast a closed system in a pure state can evolve. In this paper, we derive several extensions of this quantum speed limit to closed systems in mixed states. We also compare the strengths of these extensions and examine their tightness. The most widely used extension of the Mandelstam-Tamm quantum speed limit originates in Uhlmann's energy dispersion estimate. We carefully analyze the underlying geometry of this estimate, an analysis that makes apparent that the Bures metric, or equivalently the quantum Fisher information, will rarely give rise to tight extensions. This observation leads us to address whether there is a tightest general extension of the Mandelstam-Tamm quantum speed limit. Using a geometric construction similar to that developed by Uhlmann, we prove that this is indeed the case. In addition, we show that tight evolutions of mixed states are typically generated by time-varying Hamiltonians, which contrasts with the case for systems in pure states.