Hybrid quantum-classical systems: Quasi-free Markovian dynamics

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 Lévy-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 analyzed, 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 behavior of the quantum one.