Quantum Linear Systems Theory

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.Research supported by the Australian Research Council (ARC)

A Systems Theory Approach to the Synthesis of Minimum Noise Phase-Insensitive Quantum Amplifiers

We present a systems theory approach to the proof of a result bounding the required level of added quantum noise in a phase-insensitive quantum amplifier. We also present a synthesis procedure for constructing a quantum optical phase-insensitive quantum amplifier which adds the minimum level of quantum noise and achieves a required gain and bandwidth. This synthesis procedure is based on a singularly perturbed quantum system and leads to an amplifier involving two squeezers and two beamsplitters.Comment: To appear in the Proceedings of the 2018 European Control Conferenc

Quantum Robust Stability of a Small Josephson Junction in a Resonant Cavity

This paper applies recent results on the robust stability of nonlinear quantum systems to the case of a Josephson junction in a resonant cavity. The Josephson junction is characterized by a Hamiltonian operator which contains a non-quadratic term involving a cosine function. This leads to a sector bounded nonlinearity which enables the previously developed theory to be applied to this system in order to analyze its stability.Comment: A version of this paper appeared in the proceedings of the 2012 IEEE Multi-conference on Systems and Contro

Coherent-Classical Estimation versus Purely-Classical Estimation for Linear Quantum Systems

We consider a coherent-classical estimation scheme for a class of linear quantum systems. It comprises an estimator that is a mixed quantum-classical system without involving coherent feedback. The estimator yields a classical estimate of a variable for the quantum plant. We demonstrate that for a passive plant that can be characterized by annihilation operators only, such coherent-classical estimation provides no improvement over purely-classical estimation. An example is also given which shows that if the plant is not assumed to be an annihilation operator only quantum system, it is possible to get better estimates with such coherent-classical estimation compared with purely-classical estimation.Comment: 7 pages, 5 figures. Minor corrections. Accepted, 2014 Conference on Decision and Contro

Coherent-Classical Estimation for Quantum Linear Systems

This paper introduces a problem of coherent-classical estimation for a class of linear quantum systems. In this problem, the estimator is a mixed quantum-classical system which produces a classical estimate of a system variable. The coherent-classical estimator may also involve coherent feedback. An example involving optical squeezers is given to illustrate the efficacy of this idea.Comment: A version of this paper will appear in the Proceedings of the 2013 Australian Control Conferenc