Non-Markovian quantum trajectories, instruments and time-continuous measurements

The linear and nonlinear non-Markovian quantum state diffusion (NMQSD) equations are well-known tools for the description of certain non-Markovian open quantum systems. In this work, we systematically investigate whether the normalized linear NMQSD or the nonlinear NMQSD solutions can be generated by means of a time-continuous measurement. By considering any conceivable measurement scheme, we derive a necessary criterion for the measurement interpretation of both equations. Concrete examples show that the normalized linear NMQSD solutions are realizable only in the Markovian limit in general. The application of the presented criterion to the nonlinear NMQSD remains an open issue. Furthermore, this criterion is a necessary one not only in the context of the NMQSDs but also for a much broader class of stochastic Schrödinger equations.