Genuinely multipartite entangled qudits: one rule to link them all

arXiv:2610.33160v1 Announce Type: new Abstract: Links of rings offer an intuitive picture of multipartite entanglement. Cutting one ring can free some of the others, just as losing a particle, i.e. tracing it out, can disentangle some of the remaining ones. This analogy classifies multipartite entanglement according to which groups of particles stay entangled when others are lost, a question at the heart of robust quantum networks. However, pure states realizing a given link class were known only in special cases. Here, we close this gap. Recasting link polynomials as hypergraphs, we give an explicit formula for pure qudit states and prove that, for every link class, their entanglement under any pattern of particle loss matches the link exactly.