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Quantum and classical systems evolving under the same formal Hamiltonian $H$may exhibit dramatically different behavior after the Ehrenfest timescale $t_E\sim \log(\hbar^{-1})$, even as $\hbar \to 0$. Coupling the system to aMarkovian environment results in a Lindblad equation for the quantum evolution.Its classical counterpart is given by the Fokker-Planck equation on phasespace, which describes Hamiltonian flow with friction and diffusive noise. Thequantum and classical evolutions may be compared via the Wigner-Weylrepresentation. Due to decoherence, they are conjectured to match closely fortimes far beyond the Ehrenfest timescale as $\hbar \to 0$. We prove a versionof this correspondence, bounding the error between the quantum and classicalevolutions for any sufficiently regular Hamiltonian $H(x,p)$ and Lindbladfunctions $L_k(x,p)$. The error is small when the strength of the diffusion $D$associated to the Lindblad functions satisfies $D \gg \hbar^{4/3}$, inparticular allowing vanishing noise in the classical limit. We use atime-dependent semiclassical mixture of variably squeezed Gaussian statesevolving by a local harmonic approximation to the Lindblad dynamics. Both theexact quantum trajectory and its classical counterpart can be expressed asperturbations of this semiclassical mixture, with the errors bounded usingDuhamel's principle. We present heuristic arguments suggesting the $4/3$exponent is optimal and defines a boundary in the sense that asymptoticallyweaker diffusion permits a breakdown of quantum-classical correspondence at theEhrenfest timescale. Our presentation aims to be comprehensive and accessibleto both mathematicians and physicists. In a shorter companion paper, we treatthe special case of Hamiltonians of the form $H=p^2/2m + V(x)$ and linearLindblad operators, with explicit bounds that can be applied directly tophysical systems.

The $\hbar\to 0$ limit of open quantum systems with general Lindbladians: vanishing noise ensures classicality beyond the Ehrenfest time