The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing

In this paper, we present a method to solve the quantum marginal problem for symmetric $d$-level systems. The is built upon an efficient semi-definite program that determines compatibility conditions of $m$-body reduced density with global $n$-body matrix supported on space. We illustrate applicability in central information problems several exemplary case studies. Namely, (i) fast variational ansatz optimize local Hamiltonians over states, (ii) symmetric, few-body Bell operators states and (iii) set sufficient determine which cannot be self-tested from observables. As by-product our findings, also provide generic, analytical correspondence between arbitrary superpositions $n$-qubit Dicke translationally-invariant diagonal product bond dimension $n$.