Refuting spectral compatibility of quantum marginals

Felix Huber; Nikolai Wyderka

doi:10.22331/q-2025-11-20-1918

Refuting spectral compatibility of quantum marginals

¹Division of Quantum Computing, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland
²Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf,Universitätsstraße 1, D-40225 Düsseldorf, Germany

Published:	2025-11-20, volume 9, page 1918
Editor:	Aleksandrs Belovs
Eprint:	arXiv:2211.06349v4
Doi:	https://doi.org/10.22331/q-2025-11-20-1918
Citation:	Quantum 9, 1918 (2025).

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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.

Featured image: Regions of spectral incompatibility: what set of spectra of the reduced density matrices $\varrho_{AB}$, $\varrho_{AC}$, and $\varrho_{BC}$ are compatible with a joint quantum state $\varrho_{ABC}$? We answer this question by providing an efficient semidefinite-programming hierarchy.

► BibTeX data

@article{Huber2025refutingspectral,
  doi = {10.22331/q-2025-11-20-1918},
  url = {https://doi.org/10.22331/q-2025-11-20-1918},
  title = {Refuting spectral compatibility of quantum marginals},
  author = {Huber, Felix and Wyderka, Nikolai},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1918},
  month = nov,
  year = {2025}
}

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