Long time quantum-classical correspondence for open systems in trace norm

We consider a frictionless system coupled to an external Markovian environment. The quantum and classical evolution of such systems are described by the Lindblad and the Fokker-Planck equation respectively. We show that when such a system is given by an at most quadratically growing Hamiltonian and at most linearly growing real jump functions, the quantum and classical evolutions remain close on time scales much longer than Ehrenfest time. In particular, we show that the evolution of a density matrix by the Lindblad equation is close in trace norm to the quantization of the corresponding evolution by the Fokker-Planck equation. Such agreement improves upon recent results arXiv:2403.09345, arXiv:2306.13717, arXiv:2307.05326, which proved long time agreement in weaker norms.