Classical mechanics as the high-entropy limit of quantum mechanics

TL;DR

This work reframes the classical limit as a high-entropy limit of quantum mechanics, showing that when entropy is sufficiently large, quantum features become negligible and classical phase-space descriptions become accurate. The authors develop a concrete mechanism—an entropy-increasing stretching map—within a framework that treats the limit as a group contraction, equivalent to $\hbar \to 0$ but interpreted as $S \gg 0$. They demonstrate this both in classical phase-space (via stretching maps) and in quantum mechanics (via a pure quantum stretching map $T_Q$ realized through anti-normal ordering and Lindblad dynamics), showing how the Wigner distribution becomes positive and converges toward the Husimi distribution, while commutators are rescaled toward classical values. By reinterpreting traditional limits (black-body radiation and thermal equilibrium) within this high-entropy perspective, the paper provides a unified, interpretation-agnostic account of why classical mechanics emerges and how quantum dynamics reduces to classical Hamiltonian flow at high entropy. The results have implications for decoherence, coarse-graining, and the thermodynamic understanding of the quantum-classical transition, linking group contractions to entropic considerations in a principled way.

Abstract

We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. That is, the high entropy masks quantum effects, and mixed states of high enough entropy can be approximated with classical distributions. The mathematical limit $\hbar \to 0$ can be reinterpreted as setting the zero entropy of pure states to $-\infty$, in the same way that non-relativistic mechanics can be recovered mathematically with $c \to \infty$. Physically, these limits are more appropriately defined as $S \gg 0$ and $v \ll c$. Both limits can then be understood as approximations independently of what circumstances allow those approximations to be valid. Consequently, the limit presented is independent of possible underlying mechanisms and of what interpretation is chosen for both quantum states and entropy.

Figures (4)

• Figure 1: The amended four-quadrant picture often used to compare the theories. The distinction between classical and quantum physics is not size, but entropy. Small systems in states of high entropy (e.g. a few molecules at high temperature) can be described by classical mechanics; large systems in states of low entropy (e.g. systems entangled over long distances) cannot be described by classical mechanics.
• Figure 2: Entropy $S$ in nats for a Gaussian state as a function of uncertainty $\Sigma$, measured in units of $\hbar$. In solid blue, the classical case $S = \ln(\Sigma) + 1$. In dotted red, the quantum case $S = \left( \Sigma + \frac{1}{2} \right) \ln \left( \Sigma + \frac{1}{2} \right) - \left( \Sigma - \frac{1}{2} \right) \ln \left( \Sigma - \frac{1}{2} \right)$. In dashed green the difference between the two.
• Figure 3: Entropy $S$ in nats for a Gaussian state as a function of uncertainty $\Sigma$, measured in units of $\hbar$. It shows the effect of the transition from classical to quantum by shrinking the pure states by a factor of $\lambda$ instead of stretching the mixed states by the same factor. The rescaled uncertainty for pure states becomes $\hbar/\lambda$ and the modified Gaussian bound becomes $S_{\lambda Q}(\Sigma) = S_Q(\lambda \Sigma) - \ln(\lambda)$. Taking $\lambda$ to infinity corresponds to taking $\hbar/\lambda$ to zero.
• Figure 4: A cat state stretched with different values of $\lambda$. The axes are stretched by $\sqrt{\lambda}$ as well. The color scale is reduced to match the range of the function, though white always corresponds to zero. While the positive and negative peaks have comparable magnitude initially, the stretching map dampens the negative peak much more quickly. Note that, consistently with Fig. \ref{['fig:uncertainty_scaled']}, the negative values are severely dampened with $\lambda=5$.