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

TL;DR

This work develops a topological framework for multipartite entanglement in symmetric Dicke states $|D_n^{(k)}\rangle$, mapping qubits to topological loops and identifying an $n$-Hopf link as the governing connectivity. It jointly uses the Schmidt rank and the $l_1$-norm of coherence to diagnose entanglement persistence and link fluidity under single-qubit projective measurements, showing a robust, self-similar topology for $0<k<n$ with $R=2$ and $C_{l_1}=inom{n}{k}-1$. The analysis demonstrates that after measurement, residual coherences remain positive: $C_{l_1}^{(0)}=\binom{n-1}{k}-1$ and $C_{l_1}^{(1)}=\binom{n-1}{k-1}-1$, enabling recursive preservation and a self-similar hierarchy of $|D_{n}^{(k)}\rangle$ under truncation, with maximal linking density at $k\approx n/2$. The paper thus bridges knot theory and quantum information, offering a topological lens on robust entanglement and suggesting pathways to design resilient quantum networks, while highlighting avenues for future mathematical mapping to knot invariants and extensions to noisy or generalized measurements.

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.