Entanglement in the Dicke subspace
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Katherine Johnson
1R3 1st floor
In this talk, I will present some of the recent results in the entanglement properties of mixtures of Dicke states (https://arxiv.org/abs/2602.15800). These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization in which 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 establish a bridge between multipartite entanglement theory, semialgebraic geometry, and the theory of completely positive and copositive tensors. Within this framework, separability corresponds to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sums-of-squares tensors.
Using this dictionary, we construct explicit PPT entangled states for systems of three or more qutrits. In particular, we show that PPT entanglement exists for all multipartite systems with at least three qutrits, thereby disproving a recent conjecture in J. Math. Phys. 66, 022203 (2025). We further prove that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies PPT with respect to any other bipartition.
Finally, we connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as those introduced by Reznick and Pólya. This connection leads to semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.
This work has been done in collaboration with Ion Nechita from CNRS, LPT Toulouse and Clément Pellegrini from IMT Toulouse.