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arXiv:2512.12704 · Quantum Studies Mathematics and Foundations

Entanglement, Coherence, and Recursive Linking in Dicke states : A Topological Perspective

Sougata Bhattacharyya, Sovik Roy

quant-phSubmitted Dec 14, 2025• Announced Dec 16, 2025• 0 citations• Revised Jan 26, 2026 (v2)
v1 · Dec 14, 2025v2 · Jan 26, 2026
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Abstract

This work investigates the topological structure of multipartite entanglement in symmetric Dicke states $|D_n^{(k)}\rangle$. By viewing qubits as topological loops, we establish a direct correspondence between the recursive measurement dynamics of Dicke states and the stability of $n$-Hopf links. We utilize the Schmidt rank to quantify bipartite entanglement resilience and introduce the $l_1$-norm of quantum coherence as a measure of link fluidity. We demonstrate that unlike fragile states such as $ \left| GHZ \right \rangle$ (analogous to Borromean rings), Dicke states exhibit a robust, self-similar topology where local measurements preserve the global linking structure through non-vanishing residual coherence.

Journal reference
Quantum Studies: Mathematics and Foundation, 13 (16), 2026
DOI
10.1007/s40509-026-00392-z

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This work investigates the topological structure of multipartite entanglement in symmetric Dicke states ∣Dn(k)⟩|D_n^{(k)}\rangle∣Dn(k)​⟩. By viewing qubits as topological loops, we establish a direct correspondence between the recursive measurement dynamics of Dicke states and the stability of nnn-Hopf links. We utilize the Schmidt rank to quantify bipartite entanglement resilience and introduce the l1l_1l1​-norm of quantum coherence as a measure of link fluidity. We demonstrate that unlike fragile states such as ∣GHZ⟩ \left| GHZ \right \rangle∣GHZ⟩ (analogous to Borromean rings), Dicke states exhibit a robust, self-similar topology where local measurements preserve the global linking structure through non-vanishing residual coherence.
We propose a technique to investigate multipartite entanglement in the symmetric subspace. Our approach is to map an N -qubit symmetric state onto a bipartite symmetric state of higher local dimension. We show that this mapping preserves separability and allows to characterize the entanglement of the original multipartite state. In particular, we establish a connection between the border rank and the Schmidt rank, and derive lower bounds on entanglement measures. Finally, we reveal the existence of entangled symmetric subspaces, where all bipartite states are entangled.
Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states ($|{R}_{2n+1}\ensuremath{\rangle}$) to read ${2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the nnn-tangle ${\ensuremath{\tau}}_{n}$.
By exploiting the permutation symmetry of Dick states, we derive closed analytical expressions of Schmidt decompositions for {\it all} possible bipartitions of a system described by this kind of state. This allows us to exhaustively compute the entropy of entanglement of the bipartitions and, thus, compare the their entanglement extent. We also address the multipartite character of Dicke states by calculating the purity of balanced bipartitions to determine the potential of multipartite entanglement (the average purity). In particular, we found that the entanglement of WWW states remains constant as the number of qubits is increased. As a final application we define a family of multipartite entanglement witnesses and compute their resistance against random and systematic imperfections. It is shown that in some circumstances, for a fixed white noise fraction, the entanglement becomes detectable only if one {\it increases} the amount of systematic imperfection in the state.
I present methods for detecting entanglement around symmetric Dicke states. In particular, I consider N-qubit symmetric Dicke states with N/2 excitations. I show that for large N these states have the smallest overlap possible with states without genuine multipartite entanglement. Thus these states are particularly well suited for the experimental examination of multipartite entanglement. I present fidelity-based entanglement witness operators for detecting multipartite entanglement around these states. I then consider entanglement criteria, somewhat similar to the spin squeezing criterion, based on the moments or variances of the collective spin operators. Surprisingly, these criteria are based on an upper bound for variances for separable states. I present criteria detecting entanglement in general and criteria detecting only genuine multipartite entanglement. The collective operator measured for this criteria is an important physical quantity: Its expectation value essentially gives the intensity of the radiation when a coherent atomic cloud emits light.
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, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. 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.
This work proposes a structural and topological framework for the description of multipartite quantum entanglement within the Theory of Structural Articulation (TSA). Entanglement is interpreted not as an abstract algebraic property of quantum states in Hilbert space, but as a manifestation of the connectivity of a single structural defect embedded in an underlying medium. The fundamental carrier of quantum structure is modeled as a hypergraph of difference connectivity, whose irreducible hyperedges encode genuine multipartite correlations. Observable entanglement is described as a projection of this hypergraph onto a connectivity graph of pairwise correlations. A topological invariant—the first Betti number of the connectivity graph—is introduced and shown to characterize redundancy of correlation pathways and robustness of multipartite quantum resources against local losses. This provides a unified topological classification of GHZ-type states, W states, and cluster states, and explains qualitative differences in their stability. Beyond static classification, the work formulates a thermodynamic mechanism for the emergence of structural complexity in open quantum systems. Dissipative relaxation of structural conflict drives spontaneous closure of connectivity loops, leading to growth of topological redundancy and formation of highly connected phases. Multipartite entanglement is thereby interpreted as an equilibrium or quasistationary phase of a dense system of structural defects, rather than as the result of fine-tuned unitary control. The results offer a structural interpretation of multipartite entanglement, linking topology, robustness, and self-organization, and suggest new perspectives on the physical origin of scalable quantum resources for quantum information processing and quantum memory.