@ressayre10] for the moment polytope.

We emphasize that the computed eigenvalue distributions can be directly used to infer distributions of Rényi and von Neumann entropies, both of which play a fundamental role in statistical physics and quantum information theory and are functions of the eigenvalues only. In particular, one can recover the average entropy of a subsystem [@lubkin78; @page93], which featured in an analysis of the black hole entropy paradox [@page94; @HaydenPreskill]. For applications to the study of quantum entanglement, see .

From a mathematical perspective, the eigenvalue distributions that we compute are *Duistermaat–Heckman measures*, which are defined using the push-forward of the Liouville measure on a symplectic manifold along the moment map (see for the precise definition which we use in this article) [@heckman82; @guilleminsternberg82b; @guilleminsternberg84; @guilleminlermansternberg88; @guilleminlermansternberg96; @guilleminprato90]. The support of such a measure is a moment polytope, which in our physical context is the solution to the one-body quantum marginal problem. We will work and establish our results in this more general symplectic setting, in similar spirit to the existing solution to the one-body quantum marginal problem [@berensteinsjamaar00; @klyachko04]. We remark that, in particular, the one-body quantum marginal problem subsumes the well-known Horn’s problem, which asks for 