[@lloydpagels88], i.e., in the simplest case of two large “particles”, one representing the system and one the bath. In this paper we are concerned with computing exact eigenvalue distributions for any number of particles, rather than with asymptotic results or with simple model systems of two particles. More precisely, we are concerned with the following question:

> *What is the probability distribution of the eigenvalues of the one-body reduced density matrices of a pure many-particle quantum state drawn at random from the unitarily invariant distribution?*

In this article, we answer this question completely by describing an explicit algorithm to compute the joint eigenvalue distribution of the reduced density matrices for an arbitrary number of particles of any statistics (distinguishable, Bose, Fermi). As a special case we easily recover the Lloyd–Pagels result for two distinguishable particles. As a corollary, our work naturally leads to a solution of the one-body quantum marginal problem in terms of a finite union of polyhedral chambers, thereby providing a complementary perspective on the work of Klyachko and of Berenstein–Sjamaar, and more recently of Ressayre, who each instead provided procedures based on geometric invariant theory (in particular the Hilbert–Mumford criterion) to list characterizing linear inequalities [@berensteinsjamaar00; @klyachko04; 