Restoring the second law to classical-quantum dynamics

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 generalise the standard Langevin and Fokker-Planck equations for classical systems in thermal environments to include back-reaction 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 thermalisation in the adiabatic basis, showing the relevance of our dynamics for the mixed classical-quantum simulation of molecules.