Iterated stochastic measurements

We describe a measurement device principle based on discrete iterations of Bayesian updating of system-state probability distributions. Although purely classical by nature, these measurements are accompanied with a progressive collapse of the system-state probability distribution during each complete system measurement. This measurement scheme finds applications in analysing repeated non-demolition indirect quantum measurements. We also analyse the continuous time limit of these processes, either in the Brownian diffusive limit or in the Poissonian jumpy limit. In the quantum mechanical framework, this continuous time limit leads to Belavkin's equations which describe quantum systems under continuous measurements. This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday. (paper)