Entanglement in the Dicke subspace

TL;DR

We address entanglement in mixtures of Dicke states within the diagonally symmetric subspace, introducing a tensor-based parametrisation that encodes diagonal entries as a symmetric tensor and a complete dictionary linking entanglement properties to convex cones of tensors. The framework maps separability to completely positive tensors, PPT to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum-of-squares tensors, enabling SDP-based separability and entanglement testing. We prove the existence of PPT entangled states for all multipartite systems with $n\ge3$ and $d\ge3$, show that PPT across the most balanced bipartition implies PPT across all bipartitions, and connect bosonic extendibility to dual hierarchies of non-negative polynomials, yielding practical SDP relaxations. The results bridge multipartite entanglement, semi-algebraic geometry, and polynomial optimization, and provide constructive witnesses and extendibility certificates within a unified framework.

Abstract

In this paper, we 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. Using this framework, we construct explicit PPT entangled states in three or more qutrits. In this class of states, we establish that PPT entanglement exists for all multipartite systems with three qutrits or more, disproving a recent conjecture in [J. Math. Phys. 66, 022203 (2025)]. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies PPT with respect to any other bipartition. 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.

Figures (4)

• Figure 1: The tensor spider $T$ with $6$ legs.
• Figure 2: A tensor contraction: the contracted tensor has now $4$ legs $i_3, i_4, i_5, i_6$. The other two legs are equal $i_1 = i_2$ and summed over.
• Figure 5: The different notions of positivity for Hermitian matrices in the DS subspace (bottom row) and the corresponding class of real symmetric tensors (top row). References for the proof of the correspondence are given, along with the parametrization ($Q$ and/or $W$) realizing the correspondance.
• Figure 6: The different sets explored in this work, with their inclusion and duality structures. On the left-hand side, we have sets of symmetric tensors from $\vee^n \mathbb R^d$, and on the right-hand side, entanglement properties of the mixtures of Dicke states $\mathsf{DS}_d^{(n)}$. Every set in the diagram is in convex duality with the set that is symmetric with respect to the middle horizontal line.

Theorems & Definitions (102)

• Definition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Theorem 2.8