Hybrid quantum-classical systems: Quasi-free Markovian dynamics

Alberto Barchielli

Xinyu Zhao, Wufu Shi, J.Q. You +1 more

Z. Kanokov

Alberto Barchielli, Reinhard F. Werner

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 [Formula: see text], 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.

Siao Xie, Shanping Yu, Zairong Xi

The classical noise is unavoided in the real world. In what way classical part impact quantum dynamics and whether classical part impact controllability of quantum system have become problems. To figure out these two problems, we focus on a specific hybrid system which is both quasi-free and Markov. Because of its quasi-free property, the hybrid system comprising an open quantum system and a classical stochastic process possesses a distinct transformation formula that determines its dynamical semigroup. All these discussions are described by Weyl operators which also constitue the observable space. The controllability of reduced quantum system is first discussed, given its likely Lindblad form. The drift term is consisted of both classical information and quantum information. By adding classical control term in drift term, the controllability is analyzed using LARC(Lie Algebra Rank Condition). We discover that the reduced quantum system is fully controllable under certain controls. When we take classical information as a control, the reduced quantum system is fully controllable without external input. And the same conclusions hold in hybrid system.

Dariusz Chruściński, Andrzej Kossakowski