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.

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