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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 a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker-Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via expectation values of observables under the Weyl correspondence. Sufficient decoherence is conjectured to ensure quantum-classical agreement for times far beyond the Ehrenfest timescale as $\hbar \to 0$ -- but for what strength, and under what conditions? We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian $H(x,p)$ and Lindblad functions $L_k(x,p)$.  The error is small when the strength of the diffusion $D$ associated to the Lindblad functions satisfies $D \gg \hbar^{4/3}$, in particular allowing vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of Gaussian states which, crucially, are squeezed in varying directions in phase space.  We present heuristic arguments suggesting the $4/3$ exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale.  Our presentation aims to be comprehensive and accessible to both mathematicians and physicists.