# Classical and quantum speed limits

## TL;DR

This paper derives a new quantum speed limit bound using the Mandelstam-Tamm approach, connecting quantum and classical speed limits and exploring how state mixing influences the classical limit.

## Contribution

It introduces a novel bound on quantum speed limits based on relative purity, linking quantum dynamics to classical speed limits through the classical limit analysis.

## Key findings

- New quantum speed limit bound derived
- Quantum-classical correspondence established
- Classical limit depends on state mixing

## Abstract

The new bound on quantum speed limit (in terms of relative purity) is derived by applying the original Mandelstam-Tamm one to the evolution in the space of Hilbert-Schmidt operators acting in the initial space of states. It is shown that it provides the quantum counterpart of the classical speed limit derived in Phys. Rev. Lett. 120 (2018), 070402 and the $\hbar\rightarrow 0$ limit of the former yields the latter. The existence of classical limit is related to the degree of mixing of the quantum state.