Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics (2406.08584v1)

Abstract: Traditional quantum speed limits formulated in density matrix space perform poorly for dynamics beyond unitary, as they are generally unattainable and fail to characterize the fastest possible dynamics. To address this, we derive two distinct quantum speed limits in Liouville space for Completely Positive and Trace-Preserving (CPTP) dynamics that outperform previous bounds. 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 of superoperators and the geometry of quantum evolution in Liouville space. They can be regarded as the generalization of the Mandelstam-Tamm bound, providing uncertainty relations between time, energy, and dissipation for open quantum dynamics. Additionally, our bounds are significantly simpler to estimate and experimentally more feasible as they require to compute or measure the overlap of density matrices and the variance of the Liouvillian. We have also obtained the form of the Liouvillian, which generates the time-optimal (fastest) CPTP dynamics for given initial and final states. We give two important applications of our bounds. First, 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. Second, using our bounds, we explain the Mpemba effect in non-equilibrium open quantum dynamics.