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Entanglement in the Dicke subspace

Aabhas Gulati, Ion Nechita, and Clément Pellegrini

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AbstractWe provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. We establish that PPT entanglement exists for all multipartite systems with local dimension $d\geq3$ and $n\geq3$ parties, disproving a recent conjecture. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.► BibTeX data@article{Gulati2026entanglementindicke, doi = {10.22331/q-2026-07-01-2150}, url = {https://doi.org/10.22331/q-2026-07-01-2150}, title = {Entanglement in the {D}icke subspace}, author = {Gulati, Aabhas and Nechita, Ion and Pellegrini, Cl{\'{e}}ment}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2150}, month = jul, year = {2026} }► References [1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels''.

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The Electronic Journal of Linear Algebra 35, 156–180 (2019). https://doi.org/10.13001/1081-3810.3842Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-01 10:44:19: Could not fetch cited-by data for 10.22331/q-2026-07-01-2150 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-07-01 10:44:19: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. We establish that PPT entanglement exists for all multipartite systems with local dimension $d\geq3$ and $n\geq3$ parties, disproving a recent conjecture. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.► BibTeX data@article{Gulati2026entanglementindicke, doi = {10.22331/q-2026-07-01-2150}, url = {https://doi.org/10.22331/q-2026-07-01-2150}, title = {Entanglement in the {D}icke subspace}, author = {Gulati, Aabhas and Nechita, Ion and Pellegrini, Cl{\'{e}}ment}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2150}, month = jul, year = {2026} }► References [1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels''.

In Oved Shisha, editor, Inequalities. Pages 205–224. Academic Press, New York (1967). [40] Raphael M. Robinson. ``Some definite polynomials which are not sums of squares of real polynomials''.

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