  A $D$-dimensional Markovian open quantum system will undergo stochastic
evolution which preserves pure states, if one monitors without loss of
information the bath to which it is coupled. If a finite ensemble of pure
states satisfies a particular set of constraint equations then it is possible
to perform the monitoring in such a way that the (discontinuous) trajectory of
the conditioned system state is, at all long times, restricted to those pure
states. Finding these physically realizable ensembles (PREs) is typically very
difficult, even numerically, when the system dimension is larger than 2. In
this paper, we develop symmetry-based techniques that potentially greatly
reduce the difficulty of finding a subset of all possible PREs. The two
dynamical symmetries considered are an invariant subspace and a Wigner
symmetry. An analysis of previously known PREs using the developed techniques
provides us with new insights and lays the foundation for future studies of
higher dimensional systems.
