  The linear and the nonlinear non-Markovian quantum state diffusion equation
(NMQSD) 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 in the framework of instruments, we derive a necessary
criterion for a 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.
