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

Understand

In this paper, we present a method to solve the quantum marginal problem for symmetric $d$-level systems.

• The method is built upon an efficient semi-definite program that determines the compatibility conditions of an $m$-body reduced density with a global $n$-body density matrix supported on the symmetric space.
• We illustrate the applicability of the method in central quantum information problems with several exemplary case studies.
• Namely, (i) a fast variational ansatz to optimize local Hamiltonians over symmetric states, (ii) a method to optimize symmetric, few-body Bell operators over symmetric states and (iii) a set of sufficient conditions to determine which symmetric states cannot be self-tested from few-body observables.