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In closed quantum systems, wavepackets can spread exponentially in time dueto chaos, forming long-range superpositions in just seconds for ordinarymacroscopic systems. A weakly coupled environment is conjectured to decoherethe system and restore the quantum-classical correspondence while necessarilyintroducing diffusive noise -- but for what coupling strength, and under whatconditions? For Markovian open systems with Hamiltonians of the form $H=p^2/2m+ V(x)$ and linear Lindblad operators, we prove the quantum and classicalevolutions are close whenever the strength of the environment-induced diffusionexceeds a threshold $\hbar^{4/3} A_c$, were $A_c$ is a characteristic scale ofthe classical dynamics. (A companion paper treats more general Hamiltonians andLindblad operators.) The bound applies for all observables and for timesexponentially longer than the Ehrenfest timescale, which is when thecorrespondence can break down in closed systems. The strength of the diffusivenoise can vanish in the classical limit to give the appearance of reversibledynamics. The $4/3$ exponent may be optimal, as Toscano et al. have foundevidence that the quantum-classical correspondence breaks down in some systemswhen the diffusion is any weaker.

Decoherence ensures classicality beyond the Ehrenfest time as $\hbar \to 0$