  Determining the state of a quantum system is a consuming procedure. For this
reason, whenever one is interested only in some particular property of a state,
it would be desirable to design a measurement setup that reveals this property
with as little effort as possible. Here we investigate whether, in order to
successfully complete a given task of this kind, one needs an informationally
complete measurement, or if something less demanding would suffice. The first
alternative means that in order to complete the task, one needs a measurement
which fully determines the state. We formulate the task as a membership problem
related to a partitioning of the quantum state space and, in doing so, connect
it to the geometry of the state space. For a general membership problem we
prove various sufficient criteria that force informational completeness, and we
explicitly treat several physically relevant examples. For the specific cases
that do not require informational completeness, we also determine bounds on the
minimal number of measurement outcomes needed to ensure success in the task.
