  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.
