arXiv:math-ph/0511021

On the separation principle of quantum control

Luc Bouten, Ramon van Handel

Published Nov 5, 2005• 32 citations

Abstract

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.

DOI: 10.48550/arxiv.math-ph/0511021

Related papers

Hamilton–Jacobi–Bellman equations for quantum optimal feedback control

John Edward Gough, V. P. Belavkin, Oleg Georgievich Smolyanov

We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.

arXiv:quant-ph/0502155Journal of Optics B Quantum and Semiclassical OpticsSep 14, 200562 citations

Eventum Mechanics of Quantum Trajectories: Continual Measurements, Quantum Predictions and Feedback Control

V. P. Belavkin

Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.

arXiv:math-ph/0702079Feb 23, 200710 citations

Dynamical programming of continuously observed quantum systems

V. P. Belavkin, Antonio Negretti, Klaus Mølmer

We develop dynamical programming methods for the purpose of optimal control of quantum states with convex constraints and concave cost and bequest functions of the quantum state. We consider both open loop and feedback control schemes, which correspond, respectively, to deterministic and stochastic master equation dynamics. For the quantum feedback control scheme with continuous nondemolition observations, we exploit the separation theorem of filtering and control aspects for quantum stochastic dynamics to derive a generalized Hamilton-Jacobi-Bellman equation. If the control is restricted to only Hamiltonian terms this is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term. In this work, we consider, in particular, the case when control is restricted only to observation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure state from a mixed state of a quantum two-level system.

arXiv:0805.4741Physical Review AFeb 27, 200924 citations

THEORY OF THE CONTROL OF OBSERVABLE QUANTUM SYSTEMS

V. P. Belavkin

Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.

Jan 1, 200556 citations

Classical and quantum filtering and control based on Hudson-Parthasarathy calculus, and filter design methods

Harish Parthasarathy

In this chapter, the author proposes to solve three kinds of related problems related to quantum filtering and control. The first deals with designing standard finite matrix based MATLAB simulation algorithms for the Hudson-Parthasarathy noisy Schrodinger equation which is today accepted as the standard technique for describing the unitary evolution of system and noisy bath with the system getting noisy inputs from the bath in the form of the three kinds of fundamental quantum noise processes. The second deals with simulating the well known Belavkin quantum filter for obtaining estimates of system states observables on a real time basis. The final problem dealt in this work involves computing the evolution of the Von-Neumann entropy of, the Hudson-Parthasarathy noisy Schrodinger system state after tracing out over the bath variables. The Belavkin filter may be simulated both the observable and the state domain either as ordinary non-random matrix differential equations or equivalently as classical stochastic differential equations.

CRC Press eBooksFeb 3, 20210 citations

Stabilizing feedback controls for quantum systems

Mazyar Mirrahimi, Ramon, Van Handel

Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.

Jan 1, 2005184 citations