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Control of elementary quantum flows

Luigi Accardi, Andreas Boukas

Published Jan 1, 2003• 5 citations
OpenAlex

Abstract

We consider the problem of controlling the size of an elementary quantum stochastic flow generated by a unitary stochastic evolution driven by first order white noise.

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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.
The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.
The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.