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

Abstract

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. As a by-product of our findings, we also provide a generic, analytical correspondence between arbitrary superpositions of $n$-qubit Dicke states and translationally-invariant diagonal matrix product states of bond dimension $n$.

27 pages (21 + appendices), 12 figures

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Cited by in corpus (15)

• Semidefinite programming relaxations for quantum correlations
• Entanglement in highly symmetric multipartite quantum states
• Machine Learning Universal Bosonic Functionals
• Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group
• Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach
• Entanglement transitivity problems
• Certificates of quantum many-body properties assisted by machine learning
• Introduction to quantum entanglement in many-body systems
• Entanglement classification and \emph{non-k}-separability certification via Greenberger-Horne-Zeilinger-class fidelity
• Dicke states as matrix product states
• Symmetric quantum states: a review of recent progress
• Inherent quantum resources in stationary spin chains
• Deriving three-outcome permutationally invariant Bell inequalities
• Refuting spectral compatibility of quantum marginals
• Solving one-body ensemble N-representability problems with spin