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  • Open Access

Restoring the second law to classical-quantum dynamics

Isaac Layton1,2,*,† and Harry J. D. Miller3

  • 1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
  • 2Center for Quantum Computing, RIKEN, Wakoshi, Saitama 351-0198, Japan
  • 3Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom

  • *Contact author: layton@noneq.t.u-tokyo.ac.jp
  • †Present address: Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
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Phys. Rev. Research 8, 023238 – Published 1 June, 2026

DOI: https://doi.org/10.1103/kqy2-1vmx

Abstract

All physical theories should obey the second law of thermodynamics. However, existing proposals to describe the dynamics of hybrid classical-quantum systems either violate the second law or lack a proof of its existence. Here, we rectify this by studying classical-quantum dynamics that are (1) linear and completely positive and (2) preserve the thermal state of the classical-quantum system. We first prove that such dynamics necessarily satisfy the second law. We then show how these dynamics may be constructed, proposing dynamics that generalize the standard Langevin and Fokker-Planck equations for classical systems in thermal environments to include backreaction from a quantum degree of freedom. Deriving necessary and sufficient conditions for completely positive, linear, and continuous classical-quantum dynamics to satisfy detailed balance, we find this property satisfied by our dynamics. To illustrate the formalism and its applications, we introduce two models. The first is an analytically solvable model of an overdamped classical system coupled to a quantum two-level system, which we use to study the total entropy production in both quantum system and classical measurement apparatus during a quantum measurement. The second describes an underdamped classical-quantum oscillator system subject to friction, which we numerically demonstrate exhibits thermalization in the adiabatic basis, showing the relevance of our dynamics for the mixed classical-quantum simulation of molecules.

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Phys. Rev. Research 8, 023238– Published 1 June, 2026
  • Received 24 July 2025
  • Accepted 7 April 2026
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