  A unified bound on the quantum speed limit is obtained for open quantum
systems with the mixed initial state by utilizing the function of relative
purity proposed in [Phys. Rev. Lett. 120, 060409 (2018)]. As applications, it
is found that the quantum speed limit bound for the damped Jaynes-Cumming model
is determined by the competition among the non-Markovianity, the population of
initial excited state and the initial-state coherence, which shows that the
population of initial excited state and the coherence of initial state can
sharp the quantum speed limit despite that the non-Markovian effects can
accelerate the evolution of open quantum system. For the dephasing model, a
simple factorization law with the initial-state coherence shows that the
quantum speed limit is only governed by the competition between the
non-Markovianity and the coherence of the initial state.
