the possible eigenvalues of the sum $A + B$ of two Hermitian matrices $A$, $B$ with fixed eigenvalues [@helmkerosenthal95; @klyachko98; @knutsontao99; @fulton00; @knutsontao01; @knutsontaowoodward03]. The corresponding eigenvalue distribution for randomly-chosen matrices $A$ and $B$ has also been studied in the literature, and our methods allow us to recover the main results of [@dooleyrepkawildberger93] (); see also [@frumkingoldberger06] for a more concrete approach.

However, one striking difference between existing approaches to the one-body quantum marginal problem and the corollary to our approach is that the subtleties associated with the non-Abelian nature of their solutions can be completely bypassed. For instance, their reliance upon cohomology of Schubert cycles, and the interplay of different Weyl groups and sub-tori that feature because of repeated use of the Hilbert–Mumford criterion, may be seen as incidental. The problem, even in the full generality of the symplectic setting, is at heart an Abelian one whose essential combinatorics is encoded in the maximal torus action together with taking finitely many explicit derivatives. Remarkably, this is more than philosophy, as it has real import for computation.

The first mathematical contribution of this paper is an effective technique for providing explicit Duistermaat–Heckman measures under rather weaker assumptions than appear in 