Dicke states are multipartite quantum states first introduced in [1] to describe the collective excitations of $N$ two-level atoms coupled to a single-mode electromagnetic field. They span a subspace of only $N+1$ dimensions, which contains all fully symmetric states under particle permutations [2]. Such symmetric states arise in many physical systems such as spinor Bose-Einstein condensates [3], photonic platforms [4], or mechanical resonators [5]. With exceptional entanglement and robustness properties, they are harnessed in current quantum-enhanced applications.

Notably, symmetric states are classified as either fully separable or genuinely multipartite entangled, with no intermediate class of partially entangled states [6,7]. However, determining whether a symmetric state belongs to one class or the other is an arduous task, known as the separability problem, which is generally NP-hard [8]. This fact motivated the search for tractable sufficient criteria for entanglement. Arguably, the most renowned of these is the positive partial transposition (PPT), whose violation signals quantum entanglement [9]. The highly structured nature of symmetric states provides advantages in the separability problem [10] by enabling the charting of hard instances corresponding to PPT entangled states (PPTES) in the state-space landscape. PPTES are detected by non-decomposable entanglement witnesses, which are generally challenging to find.

In particular, for two- and three-qubit symmetric states diagonal in the Dicke basis – so-called diagonal symmetric states (DSS), the PPT criterion is necessary and sufficient for separability as PPTES do not exist for this scenario [11]. The same behaviour holds for DSS of two qudits of dimension $d\leq 4$ [12]. It was proven by mapping the separability and PPT properties of bipartite DSS to, respectively, the positivity and copositivity of an associated Hankel matrix. Moreover, such copositive matrices naturally lead to extremal non-decomposable witnesses, which detect entanglement not only in DSS but also in the presence of Dicke coherences [13] . This tight correspondence between separability and semialgebraic geometry is extended in [14] for DSS of arbitrary local dimension $d$ and number of particles $N$. As a result, a complete mathematical theory of separability, PPT, and entanglement in DSS is presented.

The authors put forward a parametrization of DSS in terms of symmetric tensors and use it to construct a dictionary between the separability of DSS and the membership problem in known convex cones of symmetric tensors. In particular, they establish a correspondence between separability in DSS and complete positivity of tensors. Moreover, this formalism maps the PPT property of DSS to semidefinite positivity of tensor flattenings (moment tensors) and leads to the observation that PPT with respect to the most balanced bipartition implies PPT with respect to all remaining bipartitions, as conjectured in [15]. They also discuss the dual problem of entanglement witnesses. Those that are non-decomposable are, by the previous correspondence, mapped to the dual cone of positive moment tensors that are not completely positive, i.e., to positive polynomials that do not admit a sum-of-squares decomposition. Such cones of polynomials are well-studied from a semialgebraic perspective [16]. Thus, the correspondence established in the paper can be used to translate results in this field to the separability problem of DSS. In particular, it allows the construction of PPT-entangled DSS for $N\geq 3$ qutrits, disproving the conjecture that no such states exist for $d\leq 4$ and arbitrary $N$ [15]. Finally, some results on (symmetric) extension hierarchies adapted to DSS are formalized.

This work complements other efforts in characterizing the separability and entanglement of symmetric states. For instance, in [17], the authors demonstrate that PPTES do not exist in a restricted class of symmetric states for any system sizes $N$ and local dimensions $d$. These correspond to states that are diagonal in a different basis than DSS, which is composed of superpositions of Dicke states with a fixed number of total excitations. On the other hand, similar results are reported in [15,18], where specific examples of PPTES for DSS of $N=3$ with $d = 3,4$ are presented. Moreover, the authors of [14] demonstrate that all two-body reductions of pure entangled Dicke states are not PPT, in agreement with [19].

In summary, [14] establishes a successful mapping between the separability of DSS and membership in certain cones of positive tensors, enabling the authors to transfer known properties of such tensors to the separability problem of DSS and address several open questions regarding PPT entanglement in these states. While this study represents a clear advance in the field, it would be beneficial to explore similar strategies for the separability problem of DSS with respect to other bases, and possibly to establish basis-independent conditions that connect this problem to related questions in absolute separability [20] and completely entangled subspaces [21].

Acknowledgments

GM thanks Jordi Romero-Pallejà for checking the text. GM acknowledges financial support by the European Union under ERC Advanced Grant TAtypic, Project No. 101142236.

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