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Control of quantum stochastic differential equations

Luigi Accardi, Andreas Boukas

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

We review the basic features of the quantum stochastic calculus. Iteration schemes for the computation of the matrix elements of \nsolutions of unitary quantum stochastic evolutions and associated quantum flows are provided along with a basic error analysis of the convergence of \nthe iteration schemes. The application of quantum stochastic calculus to the solution of the quantum version of the quadratic cost control problem \nis described.

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We review the basic features of the quantum stochastic calculus. Iteration schemes for the computation of the matrix elements of \nsolutions of unitary quantum stochastic evolutions and associated quantum flows are provided along with a basic error analysis of the convergence of \nthe iteration schemes. The application of quantum stochastic calculus to the solution of the quantum version of the quadratic cost control problem \nis described.
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The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.
Within the framework of the Accardi-Fagnola-Quaegebeur (AFQ) representation free calculus of \cite{b}, we consider the problem of controlling the size of a quantum stochastic flow generated by a unitary stochastic evolution affected by quantum noise. In the case when the evolution is driven by first order white noise (which includes quantum Brownian motion) the control is shown to be given in terms of the solution of an algebraic Riccati equation. This is done by first solving the problem of controlling (by minimizing an associated quadratic performance criterion) a stochastic process whose evolution is described by a stochastic differential equation of the type considerd in \cite{b}. The solution is given as a feedback control law in terms of the solution of a stochastic Riccati equation.