The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$

• URL: http://arxiv.org/abs/2512.17623v1
• Date: Fri, 19 Dec 2025 14:28:41 GMT
• Title: The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$
• Authors: Felipe Hernández, Daniel Ranard, C. Jess Riedel,
• Abstract summary: In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution.<n>We show that the scaling $D sim hbarfrac43$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time.
• Score: 0.3623365995586146
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. 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 $D \gg \hbar^{2}$ 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.

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