Entanglement Distribution over Quantum Multiplex Hypergraph Networks

Using percolation theory, we identify parameter regions in a quantum network that guarantee a non-zero end-to-end entanglement rate for a given network topology. These parameters govern the probability that a site is present (site intactness) and the probability that a bond exists between those sites (bond intactness). Based on the topology, percolation theory determines the critical parameter regions where global connectivity emerges, meaning there always exists a path connecting the end nodes. We propose an analytical study of fidelity-aware bipartite and multipartite entanglement distribution in a quantum multiplex hypergraph network, where an m-hyperedge corresponds to an m-partite elementary entanglement, and where the parameter region extends beyond the traditional site–bond plane to a three-dimensional space (p^(m) , q^(m) , σ^(m) ). Here, p^(m) is the success probability of an m-partite hyperedge being intact, q^(m) is the success probability of performing an m-partite projective measurement at a site, and σ^(m) is the parameter of a cumulative probability distribution describing the fidelity of cardinality  hyperedge. We assume the fidelity cumulative probability (the probability of having a fidelity above a value) follows a logistic (smooth step) function characterized by σ^(m). Our goal is to determine, based on network topology, the network’s physical and parameter’s logical regions where a non-zero end-to-end entanglement above a target fidelity can be guaranteed.