Genuinely multipartite entangled qudits: one rule to link them all

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. The same construction also controls genuine multipartite entanglement, determining exactly which of the groups that stay entangled are genuinely multipartite entangled, and all of them can be made so. As a consequence, we prove the conjecture that $m$-resistant pure qudit states, which stay entangled after the loss of any $m$ particles but become fully separable after the loss of $m+1$, exist for any number of particles and any $m$.

The paper asks whether every multipartite entanglement pattern specified by a link—recording which subsets remain entangled after tracing out other particles—has a pure-state representative. Here, a subset is “linked” when its reduced state is entangled across every bipartition, which need not imply genuine multipartite entanglement (GME) for mixed states. It constructs pure qudit representatives for every connected link class. Link-polynomial monomials become hyperedges, and the state is an equal superposition of GHZ states on those hyperedges, with orthogonal local labels identifying each term. Every proper reduced state is block diagonal in these labels: a hyperedge retains GHZ coherence exactly when all its vertices survive. This establishes that linked subsets are exactly the connected induced subhypergraphs, with separability between disconnected components. Local projection and partial-transpose arguments establish the entanglement claims. The construction also characterizes GME: a proper subset has a GME reduced state exactly when it is a hyperedge; the full pure state is GME. Adding all connected subsets as hyperedges therefore makes every linked subset GME. As a consequence, for every $N$ and $0\leq m\leq N-2$, pure states exist whose survivors are GME after any $m$ losses and fully separable after $m+1$, with local dimension $\binom{N}{m}+\binom{N-1}{m}$. The general construction requires $d(v)=|E|+|E_v|\leq2|E|$, potentially exponential in $N$, although encoding needs at most $N+1$ qubits per party. Minimal dimensions, laboratory-friendly preparation, and whether qubits suffice for all link classes or all resistance parameters remain open.