  The possible spectra of one-particle reduced density matrices that are
compatible with a pure multipartite quantum system of finite dimension form a
convex polytope. We introduce a new construction of inner- and outer-bounding
polytopes that constrain the polytope for the entire quantum system. The outer
bound is sharp. The inner polytope stems only from doubly excited states. We
find all quantum systems, where the bounds coincide giving the entire polytope.
We show, that those systems are: i) any system of two particles ii) $L$ qubits,
iii) three fermions on $N\leq 7$ levels, iv) any number of bosons on any number
of levels and v) fermionic Fock space on $N\leq 5$ levels. The methods we use
come from symplectic geometry and representation theory of compact Lie groups.
In particular, we study the images of proper momentum maps, where our method
describes momentum images for all representations that are spherical.
