  The randomized quantum marginal problem asks about the joint distribution of
the partial traces ("marginals") of a uniform random Hermitian operator with
fixed spectrum acting on a space of tensors. We introduce a new approach to
this problem based on studying the mixed moments of the entries of the
marginals. For randomized quantum marginal problems that describe systems of
distinguishable particles, bosons, or fermions, we prove formulae for these
mixed moments, which determine the joint distribution of the marginals
completely. Our main tool is Weingarten calculus, which provides a method for
computing integrals of polynomial functions with respect to Haar measure on the
unitary group. As an application, in the case of two distinguishable particles,
we prove some results on the asymptotic behavior of the marginals as the
dimension of one or both Hilbert spaces goes to infinity.
