  A symmetry witness is a suitable subset of the space of selfadjoint trace
class operators that allows one to determine whether a linear map is a symmetry
transformation, in the sense of Wigner. More precisely, such a set is invariant
with respect to an injective densely defined linear operator in the Banach
space of selfadjoint trace class operators (if and) only if this operator is a
symmetry transformation. According to a linear version of Wigner's theorem, the
set of pure states, the rank-one projections, is a symmetry witness. We show
that an analogous result holds for the set of projections with a fixed rank
(with some mild constraint on this rank, in the finite-dimensional case). It
turns out that this result provides a complete classification of the set of
projections with a fixed rank that are symmetry witnesses. These particular
symmetry witnesses are projectable; i.e., reasoning in terms of quantum states,
the sets of uniform density operators of corresponding fixed rank are symmetry
witnesses too.
