Speed limits of the trace distance for open quantum system

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 (2021 Phys. Rev. Lett. 126 010601) 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 Schrodinger 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.