  We investigate the generic bound on the minimal evolution time of the open
dynamical quantum system. This quantum speed limit time is applicable to both
mixed and pure initial states. We then apply this result to the damped
Jaynes-Cummings model and the Ohimc-like dephasing model starting from a
general time-evolution state. The bound of this time-dependent state at any
point in time can be found. For the damped Jaynes-Cummings model, the
corresponding bound first decreases and then increases in the Markovian
dynamics. While in the non-Markovian regime, the speed limit time shows an
interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing
model, this bound would be gradually trapped to a fixed value. In addition, the
roles of the relativistic effects on the speed limit time for the observer in
non-inertial frames are discussed.
