Quantum Dissipative Systems and Feedback Control Design by Interconnection

Abstract

The purpose of this paper is to extend J.C. Willems' theory of dissipative systems to the quantum domain. This general theory, which combines perspectives from the quantum physics and control engineering communities, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of exosystems in general quantum feedback network terms. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using quantum network models, which implements Willems' "control by interconnection" for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

Coherently tracking the covariance matrix of an open quantum system

“…(27) and (28). In accordance with the physical form of an open harmonic oscillator described by Eqs.…”

“…The dynamics of an open quantum system are uniquely determined by the parametrization (S, L, H) [26][27][28]. The self-adjoint operator H is the Hamiltonian describing the self-energy of the system.…”

Pole placement approach to coherent passive reservoir engineering for storing quantum information

“…According to [5], passivity of a quantum system P is defined as a property of the system with respect to an output generated by an exosystem W and applied to input channels of the given quantum system on one hand, and a performance operator Z of the system on the other hand. To particularize the definition of [5] in relation to the specific class of annihilation only systems, we consider a class of exosystems, i.e., open quantum systems with zero Hamiltonian, an identity scattering matrix and a coupling operator u which couples the exosystem with its input field.…”

“…To particularize the definition of [5] in relation to the specific class of annihilation only systems, we consider a class of exosystems, i.e., open quantum systems with zero Hamiltonian, an identity scattering matrix and a coupling operator u which couples the exosystem with its input field. The exosystem is assumed to be independent of P in the sense that u commutes with any operator from the C * operator algebra generated by X and X † .…”

Principles and applications of quantum control engineering

“…General conditions for stability, passivity and L 2 -gain for quantum feedback networks have been given by James & Gough [43] in a framework that extends the Willems approach [44,45] to control engineering. A general set-up is sketched in figure 16.…”