  The question of whether given density operators for subsystems of a
multipartite quantum system are compatible to one common total density operator
is known as the quantum marginal problem. We briefly review the solution of a
subclass of such problems found just recently. In particular, this provides the
solution of the $1$-body $N$-representability problem. Its solution, the
so-called generalized Pauli constraints, restrict the set of mathematically
possible fermionic occupation numbers significantly, and strengthens Pauli's
exclusion principle. Moreover, we review the study of a concrete physical model
of interacting fermions confined to a harmonic trap. There, we found occupation
numbers close, but not exactly on the boundary of the allowed region. This new
effect of quasipinning is physically relevant since it corresponds to a
simplified structure of the corresponding $N$-fermion quantum state.
