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arXiv:0709.2216 · Infinite Dimensional Analysis Quantum Probability and Related Topics

THE STABILITY OF QUANTUM MARKOV FILTERS

Ramon van Handel

Published Mar 1, 2009• 33 citations
arXivHTMLPDF

Abstract

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

DOI
10.1142/s0219025709003549

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Stability of quantum Markov systems is investigated in terms of stability of invariant states. Evolutions of a quantum system in the Heisenberg picture are considered which are modeled in terms of a quantum stochastic differential equation. Using a Markov operator semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. The conditions are formulated in terms of algebraic constraints suitable for engineering quantum systems to be used in coherent feedback networks. To derive these conditions, we use quantum analogues of the stochastic Lyapunov stability theory.
Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.
Abstract We address the question of asymptotic stability of quantum trajectories, also referred as quantum filters. We determine the limit of the quantum fidelity between the true quantum trajectory and the estimated one. Under a purification and absolute continuity of initial conditions assumptions we show that this limit equals to one meaning that quantum filters are asymptotically stable. In the general case, under an identifiability, a spectral assumption and absolute continuity of initial conditions assumption we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one.
Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. Furthermore, this paper proves the quantum invariance principle, which extends the LaSalle invariance principle to quantum systems in the Heisenberg picture. These results are formulated in terms of algebraic constraints suitable for engineering quantum systems that are used in coherent feedback networks.
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.