  We investigate quantum tomography in scenarios where prior information
restricts the state space to a smooth manifold of lower dimensionality. By
considering stability we provide a general framework that relates the topology
of the manifold to the minimal number of binary measurement settings that is
necessary to discriminate any two states on the manifold. We apply these
findings to cases where the subset of states under consideration is given by
states with bounded rank, fixed spectrum, given unitary symmetry or taken from
a unitary orbit. For all these cases we provide both upper and lower bounds on
the minimal number of binary measurement settings necessary to discriminate any
two states of these subsets.
