The threshold for quantum-classical correspondence is D sim hbar^frac43

In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time t_E sim log\(hbar^-1\), even in the limit hbar → 0. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength D show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever D gg hbar^frac43, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D gg hbar² always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D sim hbar^frac43 is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an hbar-independent quantum-classical discrepancy at the Ehrenfest time whenever D ll hbar^frac43, even for hbar-independent "macroscopic" smooth observables.