  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 the Wigner-Weyl
representation. Due to decoherence, they are conjectured to match closely for
times far beyond the Ehrenfest timescale as $\hbar \to 0$. 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. We use a
time-dependent semiclassical mixture of variably squeezed Gaussian states
evolving by a local harmonic approximation to the Lindblad dynamics. Both the
exact quantum trajectory and its classical counterpart can be expressed as
perturbations of this semiclassical mixture, with the errors bounded using
Duhamel's principle. 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. In a shorter companion paper, we treat
the special case of Hamiltonians of the form $H=p^2/2m + V(x)$ and linear
Lindblad operators, with explicit bounds that can be applied directly to
physical systems.
