Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics

TL;DR

This paper introduces two new quantum speed limits in Liouville space for CPTP dynamics, providing exact bounds and physical interpretations, with applications in quantum chaos, information scrambling, and non-equilibrium quantum phenomena.

Contribution

It presents novel quantum speed limits in Liouville space that are attainable and exact for all states and dynamics, with a geometric and physical interpretation.

Findings

Bounds relate to spectral form factor and Krylov complexity.

Speed limits elucidate the Mpemba effect in quantum systems.

Liouvillian eigenmodes determine minimal relaxation times.

Abstract

Traditional quantum speed limits formulated in density matrix space are generally unattainable for a wide class of dynamics and it is difficult to characterize the fastest possible dynamics. To address this, we present two distinct quantum speed limits in Liouville space for Completely Positive and Trace-Preserving (CPTP) dynamics. The first bound saturates for time-optimal CPTP dynamics, while the second bound is exact for all states and all CPTP dynamics. Our bounds have a clear physical and geometric interpretation arising from the uncertainty relations for operators acting on Liouville space, and the geometry of quantum evolution in Liouville space. We also obtain the form of the Liouvillian, which generates the time-optimal CPTP dynamics that connect the given initial and target states. To illustrate our findings, we show that the speed of evolution in Liouville space bounds the growth of the spectral form factor and Krylov complexity of states, which are crucial for studying information scrambling and quantum chaos. In another important application, we show that our results can help us understand the counter-intuitive phenomenon of the Mpemba effect in non-equilibrium open quantum dynamics, as the minimal relaxation time scale obtained by speed limits is dictated by the eigenmodes of the Liouvillian.