Refuting spectral compatibility of quantum marginals

TL;DR

This paper introduces a symmetry-reduced semidefinite programming hierarchy that effectively detects incompatibility in quantum marginals spectral problems, providing complete, dimension-free refutations across various quantum and mathematical contexts.

Contribution

It develops a novel, complete hierarchy for spectral quantum marginal problems that is symmetry-reduced and applicable to multiple related problems.

Findings

Hierarchy detects all incompatible spectra sets

Refutations are dimension-free and certifiable

Applicable to multiple quantum and mathematical problems

Abstract

The spectral variant of the quantum marginal problem asks: Given prescribed spectra for a set of overlapping quantum marginals, does there exist a compatible joint state? The main idea of this work is a symmetry-reduced semidefinite programming hierarchy that detects when no such joint state exists. The hierarchy is complete, in the sense that it detects every incompatible set of spectra. The refutations it provides are dimension-free, certifying incompatibility in all local dimensions. The hierarchy also applies to the sums of Hermitian matrices problem, the compatibility of local unitary invariants, for certifying vanishing Kronecker coefficients, and to optimize over equivariant state polynomials.