In many optical and atomic systems it is now possible to monitor an individual quantum system with high signal to noise and feed back by altering the system dynamics in real time. This motivates further development of quantum mechanical descriptions of feedback control. We discuss recent work on closed loop control of open quantum systems, focusing on general linear systems for which the statistics of the problem are Gaussian. Such systems may be realized with linear optics, parametric amplifiers and homodyne detection. This problem allows a direct comparison with the classical linear, quadratic cost, Gaussian noise (LQG) optimal control problem. This highlights the key distinction between the quantum and classical theories for linear systems: that increased measurement sensitivity may run counter to the control objectives since it increases the backaction noise. While in an idealized classical control problem it is always preferable to obtain a better sensor, quantum mechanical problems generically have an optimal sensitivity since quantum mechanical measurements irreducibly disturb the system. The general theory will be illustrated by reference to specific simple examples.
In many optical and atomic systems it is now possible to monitor an individual quantum system with high signal to noise and feed back by altering the system dynamics in real time. This motivates further development of quantum mechanical descriptions of feedback control. We discuss recent work on closed loop control of open quantum systems, focusing on general linear systems for which the statistics of the problem are Gaussian. Such systems may be realized with linear optics, parametric amplifiers and homodyne detection. This problem allows a direct comparison with the classical linear, quadratic cost, Gaussian noise (LQG) optimal control problem. This highlights the key distinction between the quantum and classical theories for linear systems: that increased measurement sensitivity may run counter to the control objectives since it increases the backaction noise. While in an idealized classical control problem it is always preferable to obtain a better sensor, quantum mechanical problems generically have an optimal sensitivity since quantum mechanical measurements irreducibly disturb the system. The general theory will be illustrated by reference to specific simple examples.
Quantum feedback control is the control of the dynamics of a quantum system by feeding back (in real time) the results of monitoring that system. For systems with linear dynamics, the control problem is amenable to exact analysis. In these cases, the quantum system is equivalent to a stochastic system of classical phase-space variables with linear drift and constant diffusion, and with a measured current (e.g. a homodyne photocurrent) linear in the system variables. However, the classical evolution is constrained in order to represent valid quantum evolution. We quantify this in terms of a linear matrix inequality (LMI) relating the drift and diffusion (a sort of zero temperature fluctuation-dissipation theorem), and another LMI for the covariance matrix of the possible conditioned states (i.e. under all possible monitoring schemes consistent with the master equation). For manipulable systems (i.e. where the experimenter has arbitrary control over the parameters in a Hamiltonian linear in the system variables) the covariance of the conditioned state is all that is needed to calculate the effectiveness of the feedback. In this case the double optimization problem reduces to a semidefinite program, which can be solved efficiently in general. We illustrate this with an example drawn from quantum optics.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement. Using a quantum noise model to describe the stochastic evolution of the open quantum system and working in parallel with classical indeterministic control theory, we present the theory of nonlinear optimal quantum feedback control. The resulting quantum Bellman equation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities with the corresponding classical control problem.
In this paper, we analyze classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that it can be described by a closed-loop feedback problem. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be thought of as constraints in a non-commutative space, then the momentum must be dependent on the position for any generic wave function. This implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. For the sake of exposition, we give a short review of control theory, and some familiar examples at the classical and quantum levels are analyzed.
This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels.
This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework.Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs).Such systems commonly arise in the area of quantum optics and related disciplines.Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems.With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory.Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system.This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation.However, the design of coherent quantum feedback controllers remains a major challenge.This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case.An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability.This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics.The paper will cover recent results relating the question of physical realizability to notions occurring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.