Selective quantum evolution of a qubit state due to continuous measurement

We consider a two-level quantum system (qubit) that is continuously measured by a detector. The information provided by the detector is taken into account to describe the evolution during a particular realization of the measurement process. We discuss the Bayesian formalism for such ``selective'' evolution of an individual qubit and apply it to several solid-state setups. In particular, we show how to suppress qubit decoherence using continuous measurement and a feedback loop.