Statistical ensembles in Hamiltonian formulation of hybrid quantum-classical systems

TL;DR

This paper develops a Hamiltonian framework for hybrid quantum-classical systems, defining statistical ensembles and deriving dynamical equations for their mixed states, emphasizing the role of probability densities on the hybrid phase space.

Contribution

It introduces a consistent approach to defining statistical operators and dynamical equations in hybrid quantum-classical systems, considering arbitrary probability densities.

Findings

Statistical operators can be consistently defined for hybrid systems.

Dynamical equations depend on the total probability density.

Arbitrary probability densities are necessary for physically distinguishable ensembles.

Abstract

General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.