The stability of quantum Markov filters

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.Comment: Final versio

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces

Stabilizing feedback controls for quantum systems

No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is neccessarily 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

Uniform Time Average Consistency of Monte Carlo Particle Filters

We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions.Comment: 21 pages, 1 figur

Equivalent Conditions for Weak Continuity of Nonlinear Filters

This paper studies weak continuity of nonlinear filters. It is well-known that Borel measurability of transition probabilities for problems with incomplete state observations is preserved when the original discrete-time process is replaced with the process whose states are belief probabilities. It is also known that the similar preservation may not hold for weak continuity of transition probabilities. In this paper we show that the sufficient condition for weak continuity of transition probabilities for beliefs introduced by Kara, Saldi, and Yuksel (2019) is a necessary and sufficient condition for semi-uniform Feller continuity of transition probabilities. The property of semi-uniform Feller continuity was introduced in Feinberg, Kasyanov, and Zgurovsky (2021), and, if the original transition probability has this property, then the transition probability of the process, whose state is a pair consisting of the belief probability and observation, also has this property. Thus, this property implies weak continuity of nonlinear filters. This paper also reviews several necessary and sufficient conditions for semi-uniform Feller continuity.Comment: arXiv admin note: substantial text overlap with arXiv:2108.09232; text overlap with arXiv:2107.02207, arXiv:2103.1325

Implementation of robust image artifact removal in SWarp through clipped mean stacking

We implement an algorithm for detecting and removing artifacts from astronomical images by means of outlier rejection during stacking. Our method is capable of addressing both small, highly significant artifacts such as cosmic rays and, by applying a filtering technique to generate single frame masks, larger area but lower surface brightness features such as secondary (ghost) images of bright stars. In contrast to the common method of building a median stack, the clipped or outlier-filtered mean stacked point-spread function (PSF) is a linear combination of the single frame PSFs as long as the latter are moderately homogeneous, a property of great importance for weak lensing shape measurement or model fitting photometry. In addition, it has superior noise properties, allowing a significant reduction in exposure time compared to median stacking. We make publicly available a modified version of SWarp that implements clipped mean stacking and software to generate single frame masks from the list of outlier pixels.Comment: PASP accepted; software for download at http://www.usm.uni-muenchen.de/~dgruen