Feedback control engineers have been interested in multiple-input--multiple-output (MIMO) extensions of single-input--single-output (SISO) results of various kinds due to its rich mathematical structure and practical applications. An outstanding problem in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodyne-mediated feedback theory to allow for multiple inputs, multiple outputs, and arbitrary diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.
Feedback control engineers have been interested in multiple-input--multiple-output (MIMO) extensions of single-input--single-output (SISO) results of various kinds due to its rich mathematical structure and practical applications. An outstanding problem in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodyne-mediated feedback theory to allow for multiple inputs, multiple outputs, and arbitrary diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.
We study multiple-input multiple-output (MIMO) quantum systems and explore quantum decoupling techniques that can be used to facilitate the design of quantum networks. We provide a quantum state feedback control law to change the structure of a quantum system in order to decouple the multiple channels. We show that an MIMO quantum system can be decoupled as an aggregate of n separated single-input single-output (SISO) quantum systems from an input-output point of view. We also discuss the conditions under which quantum decoupling can be achieved. We illustrate our method with several qubit and atomic systems.
The concept of a controlled flow of a dynamical system, especially when the controlling process feeds information back about the system, is of central importance in control engineering. In this paper, we build on the ideas presented by Bouten and van Handel [Quantum Stochastics and Information: Statistics, Filtering and Control (World Scientific, 2008)] and develop a general theory of quantum feedback. We elucidate the relationship between the controlling processes, Z, and the measured processes, Y, and to this end we make a distinction between what we call the input picture and the output picture. We should note that the input-output relations for the noise fields have additional terms not present in the standard theory but that the relationship between the control processes and measured processes themselves is internally consistent—we do this for the two main cases of quadrature measurement and photon-counting measurement. The theory is general enough to include a modulating filter which post-processes the measurement readout Y before returning to the system. This opens up the prospect of applying very general engineering feedback control techniques to open quantum systems in a systematic manner, and we consider a number of specific modulating filter problems. Finally, we give a brief argument as to why most of the rules for making instantaneous feedback connections [J. Gough and M. R. James, Commun. Math. Phys. 287, 1109 (2009)] ought to apply for controlled dynamical networks as well.
This paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators. A quantum optical closed-loop, including a plant and controller, is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view, in order to facilitate the use of control theory in microscopic world described by quantum theory. In particular, an input-output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed. This would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics. This paper is divided into two parts. The first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device, cavity, that can be regarded as a unit of quantum dynamics of bosonic modes. The formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations. The input-output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems. In the follow-up paper, the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented.
We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple-measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.
The mathematical theory of quantum feedback networks has recently been developed [J. Gough and M. R. James, e-print arXiv:0804.3442v2] for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical Markovian systems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.