Refuting spectral compatibility of quantum marginals

Understand

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.