  We investigate the speed limit of the state transformation in open quantum
systems described by the Lindblad type quantum master equation. We obtain
universal bounds of the total entropy production described by the trace
distance between the initial and final states in the interaction picture. Our
bounds can be tighter than the bound of Vu and Hasegawa [Phys. Rev. Lett. 126,
010601 (2021)] which measures the distance by the eigenvalues of the initial
and final states: This distance is less than or equal to the trace distance.
For this reason, our results can significantly improve Vu-Hasegawa's bound. The
trace distance in the Schr\"{o}dinger picture is bounded by a sum of the trace
distance in the interaction picture and the trace distance for unitary dynamics
described by only the Hamiltonian in the quantum master equation.
