Quantum Filtering with Counting Measurements

Continuous time quantum filtering problems are formulated based on ideas from quantum stochastic processes. This formulation extends previous work in that it allows modelling of the interaction between the measurement process and the state of the quantum field. However, the abstract quantum measurements are restricted to those that can be implemented by counting measurements. The type and properties of the resulting point process are discussed. Following ideas of Davies on Quantum stochastic processes the optimal filtering problem, including optimal measurement selection (from the above type) is formulated. Methods of solutions are discussed and examples illustrate the results. The operator differential equation satisfied by the density operator is analyzed and relations with stochastic partial differential equations of a specific type are illustrated.