*reduced density matrix*, or quantum marginal, of a few particles. For instance, the binding energy of a molecule is given as a minimization over two-electron reduced density matrices arising from $N$-electron wavefunctions. Mathematically, the reduced density matrix is given as the contraction of (or trace over) the indices of the projection operator onto the wavefunction over the remaining particles (see for the precise definition). The problem of characterizing the set of possible reduced density matrices, known as the *quantum marginal problem* in quantum information theory and as the $N$-representability problem in quantum chemistry [@ruskai69; @colemanyukalov00], has therefore been considered one of the most fundamental problems in quantum theory [@stillinger95]. The general problem is computationally intractable, even on a quantum computer; more precisely it is QMA-complete and NP-hard [@liu06; @liuchristandlverstraete07]. However, the characterization of the one-body reduced density matrices of a pure global quantum state [@klyachko04; @daftuarhayden04; @klyachko06] admits a very elegant mathematical interpretation: it amounts to a description of the possible eigenvalues of the reduced density matrices; and this, just as for Horn’s problem, requires the computation of moment polytopes for coadjoint orbits of unitary groups, as first observed in [@christandlmitchison06; @daftuarhayden04; @klyachko04]. An interesting consequence is that defining linear 