  Given a random quantum state of multiple distinguishable or indistinguishable
particles, we provide an effective method, rooted in symplectic geometry, to
compute the joint probability distribution of the eigenvalues of its one-body
reduced density matrices. As a corollary, by taking the distribution's support,
which is a convex moment polytope, we recover a complete solution to the
one-body quantum marginal problem. We obtain the probability distribution by
reducing to the corresponding distribution of diagonal entries (i.e., to the
quantitative version of a classical marginal problem), which is then determined
algorithmically. This reduction applies more generally to symplectic geometry,
relating invariant measures for the coadjoint action of a compact Lie group to
their projections onto a Cartan subalgebra, and can also be quantized to
provide an efficient algorithm for computing bounded height Kronecker and
plethysm coefficients.
