  The purpose of quantum tomography is to determine an unknown quantum state
from measurement outcome statistics. There are two obvious ways to generalize
this setting. First, our task need not be the determination of any possible
input state but only some input states, for instance pure states. Second, we
may have some prior information, or premise, which guarantees that the input
state belongs to some subset of states, for instance the set of states with
rank less than half of the dimension of the Hilbert space. We investigate state
determination under these two supplemental features, concentrating on the cases
where the task and the premise are statements about the rank of the unknown
state. We characterize the structure of quantum observables (POVMs) that are
capable of fulfilling these type of determination tasks. After the general
treatment we focus on the class of covariant phase space observables, thus
providing physically relevant examples of observables both capable and
incapable of performing these tasks. In this context, the effect of noise is
discussed.
