The two classes of hybrid classical-quantum dynamics

TL;DR

This paper classifies the possible memoryless hybrid classical-quantum dynamics, proving they fall into two classes distinguished by their jump behavior, and derives their most general forms using advanced positivity conditions.

Contribution

It provides a complete classification and general form of memoryless classical-quantum dynamics, extending the Pawula theorem to hybrid systems.

Findings

Two classes of memoryless hybrid dynamics identified

Most general forms of each class derived

Generalized positivity conditions applied to hybrid systems

Abstract

Coupling between quantum and classical systems is consistent, provided the evolution is linear in the state space, preserves the split of systems into quantum and classical degrees of freedom, and preserves probabilities. The evolution law must be a completely positive and norm preserving map. We prove that if the dynamics is memoryless, there are two classes of these dynamics, one which features finite sized jumps in the classical phase space and one which is continuous. We find the most general form of each class of classical-quantum master equation. This is achieved by applying the complete positivity conditions using a generalized Cauchy-Schwartz inequality applicable to classical-quantum systems. The key technical result is a generalisation of the Pawula theorem.