Hybrid quantum-classical systems: Quasi-free Markovian dynamics

TL;DR

This paper characterizes the most general quasi-free dynamical semigroup for finite-dimensional quantum-classical hybrid systems, generalizing Gaussian dynamics and incorporating classical-quantum interactions and measurement theory.

Contribution

It provides a quantum generalization of the Lévy-Khintchine formula for hybrid systems, including classical and quantum dynamics, and analyzes how classical information can be extracted without perturbing the quantum state.

Findings

Derived the structure of the generator of quasi-free quantum-classical semigroups.

Connected multi-time probabilities to quantum measurement concepts.

Demonstrated how classical systems can input noise and extract information from quantum systems.

Abstract

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operators in the Heisenberg description. The result is a quantum generalization of the L\'evy-Khintchine formula; Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov-Fokker-Planck equation are included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time $t$, we show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator valued measure and instrument. The structure of the generator of the dynamical semigroup is analized, in order to understand how to go on to non quasi-free cases and to understand the possible classical-quantum interactions; in particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a classical component can input noise into a quantum one and how the classical system can extract information on the behaviour of the quantum one.