Moments of random quantum marginals via Weingarten calculus

TL;DR

This paper develops a new method using Weingarten calculus to analyze the joint distribution of quantum marginals for various particle systems, providing explicit formulas and asymptotic results.

Contribution

Introduces a novel approach based on mixed moments and Weingarten calculus to fully characterize quantum marginals for different particle types.

Findings

Derived formulas for mixed moments of marginals

Determined joint distributions for distinguishable particles, bosons, and fermions

Analyzed asymptotic behavior of marginals as system dimension grows

Abstract

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.