On stability of continuous-time quantum filters

Abstract

International audienceWe prove that the fidelity between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters

Coupling rotational and translational motion via a continuous measurement in an optomechanical sphere

“…The stochastic master equation that describes the evolution of the system under the continuous measurement was solved numerically using a method proposed by Rouchon and collaborators that was specially designed for these types of equations [26,27]. In doing so we have confirmed, for systems much larger than those previously considered, that Rouchon's method provides significant computational advantages compared to the standard Euler-Milstein method.…”

“…The Euler-Milstein increment neither guarantees the positivity nor the Hermiticity of the conditional density matrix. In practice, we have found that Rouchon's method [26,27] provides more accurate state estimates with fewer time steps per period of time, and involves fewer calculations per time step than the EulerMilstein method. For the cases modeled in this paper, typical savings in computational time are a factor of four or five for the increment (14) over the increment (16).…”

“…the Euler-Maruyama method which is weakly convergent to first order or the Euler-Milstein method which is strongly convergent to first order [28]. Here we adopt instead a numerical integration method specifically developed for SMEs [26,27] that we will refer to as Rouchon's method.…”

“…The present paper has two principal objectives: the introduction of a novel measurement interaction for small optically trapped objects, and the demonstration of the advantages of a numerical integration method proposed by Rouchon et al for stochastic master equations [26,27]. The measurement relies on the fact that the object is spatially extended, which allows one part of it to be localized without necessarily localizing the position of the whole object (that is, without localizing the center of mass).…”

“…The second objective of the present work is to apply a numerical stochastic integration method proposed by Rouchon et al [26,27] to a system which is significantly larger, in terms of its quantum state-space, than that studied previously [27]. The integration method is specific to stochastic master equations and we confirm here that it offers significant advantages: i) it requires far fewer time increments to obtain an accurate solution when compared to more general numerical integration methods; ii) it is simpler in that each time increment requires fewer operator calculations than comparable methods -both i) and ii) reduce the simulation time; and, iii) it is much more stable in that, unlike previous methods, it does not produce unphysical density matrices in which the purity is greater than unity.…”