Luc Bouten

An Introduction to Quantum Filtering

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.

A Discrete Invitation to Quantum Filtering and Feedback 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.

ON THE SEPARATION PRINCIPLE IN QUANTUM CONTROL

It is well known that quantum continuous observations and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of Hudson-Parthasarathy. The addition of real-time feedback control has been discussed by many authors, but the foundations of the theory still appear to be relatively undeveloped. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is a memoryless function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on existence and uniqueness of the solutions of controlled quantum filtering equations and on the innovations problem in the quantum setting.

Quantum Risk-Sensitive Estimation and Robustness

This paper studies a quantum risk-sensitive estimation problem and investigates robustness properties of the filter. This is a direct extension to the quantum case of analogous classical results. All investigations are based on a discrete approximation model of the quantum system under consideration. This allows us to study the problem in a simple mathematical setting. We close the paper with some examples that demonstrate the robustness of the risk-sensitive estimator.

Adiabatic Elimination in Quantum Stochastic Models

We consider a physical system with a coupling to bosonic reservoirs via a quantum stochastic differential equation. We study the limit of this model as the coupling strength tends to infinity. We show that in this limit the solution to the quantum stochastic differential equation converges strongly to the solution of a limit quantum stochastic differential equation. In the limiting dynamics the excited states are removed and the ground states couple directly to the reservoirs.

Physical model of continuous two-qubit parity measurement in a cavity-QED network

We propose and analyze a physical implementation of two-qubit parity measurements as required for continuous error correction, assuming a setup in which the individual qubits are strongly coupled to separate optical cavities. A single optical probe beam scatters sequentially from the two cavities, and the continuous parity measurement is realized via fixed quadrature homodyne photodetection. We present models based on quantum stochastic differential equations (QSDEs) for both an ideal continuous parity measurement and our proposed cavity quantum electrodynamics (cavity QED) implementation; a recent adiabatic elimination theorem for QSDEs is used to assert strong convergence of the latter to the former in an appropriate limit of physical parameters. Performance of the cavity QED scheme is studied via numerical simulation with experimentally realistic parameters.

Optimality of Feedback Control Strategies for Qubit Purification

Recently two papers [K. Jacobs, Phys. Rev. A 67, 030301(R) (2003); H. M. Wiseman and J. F. Ralph, New J. Physics 8, 90 (2006)] have derived a number of control strategies for rapid purification of qubits, optimized with respect to various goals. In the former paper the proof of optimality was not mathematically rigorous, while the latter gave only heuristic arguments for optimality. In this paper we provide rigorous proofs of optimality in all cases, by applying simple concepts from optimal control theory, including Bellman equations and verification theorems.

Discrete approximation of quantum stochastic models

We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.

Scattering of polarized laser light by an atomic gas in free space: A quantum stochastic differential equation approach

We propose a model, based on a quantum stochastic differential equation (QSDE), to describe the scattering of polarized laser light by an atomic gas. The gauge terms in the QSDE account for the direct scattering of the laser light into different field channels. Once the model has been set, we can rigorously derive quantum filtering equations for balanced polarimetry and homodyne detection experiments, study the statistics of output processes and investigate a strong driving, weak coupling limit.

Optimal pointers for joint measurement of σx and σz via homodyne detection

We study a model of a qubit in interaction with the electromagnetic field. By means of homodyne detection, the field-quadrature At + A*t is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred to the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of σx and σz. Using variational calculus, we find the pointers that are optimal. These optimal pointers are very close to hitting the bound imposed by Heisenbergs uncertainty relation on joint measurement of two non-commuting observables. We close the paper by giving the probability densities of the pointers.