# Geometric quantum speed limits: a case for Wigner phase space

## TL;DR

This paper introduces a new, computationally simpler way to determine quantum speed limits using Wigner phase space, which is equivalent to traditional methods but easier to calculate, with applications to harmonic oscillators and quantum Brownian motion.

## Contribution

The authors develop a Wigner phase space approach to quantum speed limits that simplifies calculations while maintaining equivalence with density operator methods.

## Key findings

- Wigner space quantum speed limit is equivalent to density operator space expressions.
- The new bound is significantly easier to compute.
- Applications demonstrated on harmonic oscillator and quantum Brownian motion.

## Abstract

The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are universally characterized by the Schatten-$p$-norm of the generator of quantum dynamics. Since computing Schatten-$p$-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.