This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Ito calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

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An Introduction to Quantum Filtering

Introduction to stochastic control

The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory) We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods

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A Discrete Invitation to Quantum Filtering and Feedback Control

A recently emerging approach to the feedback control of linear quantum 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. An important issue which arises both in the synthesis of linear coherent quantum controllers and in the modeling of linear quantum systems, is the issue of physical realizability. This issue relates to the property of whether a given set of linear quantum stochastic differential equations corresponds to a physical quantum system satisfying the laws of quantum mechanics. Under suitable assumptions, the paper shows that the question of physical realizability is equivalent to a frequency domain (J,J) -unitary condition. This is important in controller synthesis since it is the transfer function matrix of the controller which determines the closed loop system behavior.

A Frequency Domain Condition for the Physical Realizability of Linear Quantum Systems

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 sys- tems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or ap- proximated by linear QSDEs include interconnections of optical cavities, beam-spitters, phase-shifters, optical parametric am- plifiers, 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 imple- mentation. 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 occuring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.

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Quantum Linear Systems Theory

Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.

https://iopscience.iop.org/article/10.1088/0953-4075/38/3/002/pdf

Bellman equations for optimal feedback control of qubit states

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Simon Edwards

Papers

Bellman equations for optimal feedback control of qubit states

Bellman equations for optimal feedback control of qubit states

Quantum Filtering and Optimal Control

Bellman equations for optimal feedback control of qubit states