A Discrete Invitation to Quantum Filtering and Feedback Control

Abstract

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.Key words. discrete quantum filtering, quantum feedback control, quantum probability, conditional expectation, dynamic programming, stochastic Lyapunov functions.AMS subject classifications. 93E11, 93E15, 93E20, 81P15, 81S25, 34F05.

Quantum trajectories for a system interacting with environment in a single-photon state: Counting and diffusive processes

“…It is the main difference between the considered situation and the standard cases when the environment is in the factorisable state [37,42]. The physical interpretation of |Ψ j|η η ηj is very intuitive:…”

“…Its rigorous definition together with discussion of its continuous limit were given in number of publications [30,[34][35][36][37][38][39][40][41][42]. Derivations of discrete version of quantum filtering equations with its continuous limits for the case when the environment is initially in factorized state with the qubits prepared in some mixed state one can find in [43] and for the qubits prepared in the ground state (the environment prepared in the vacuum state) in [37,42].…”

Feedback policies for measurement-based quantum state manipulation

“…We now present the solution to the optimal policy for the considered quantum state manipulation in light of the classical work of quantum feedback control theory derived by Belavkin [1] (also see [2] and [3] for a thorough treatment).…”

Quantum Risk-Sensitive Estimation and Robustness

“…The systems we consider are taken from quantum optics and consist of a quantum system in interaction with the quantized electromagnetic field. The field is described by a discretized model [9] that converges to a quantum stochastic dynamics [23] when the discretization step is taken to zero [2], [3], [11], [20], [30]. The discretized model has the advantage of being very tractable mathematically.…”