Multipartite entanglement, a higher-order interaction unique to quantum information, offers various advantages over bipartite entanglement in quantum network (QN) applications. Establishing multipartite entanglement across remote parties in QN requires entanglement routing, which irreversibly transforms the QN topology at the cost of existing entanglement links. Here, we address the question of whether a QN can be topologically transformed into another via entanglement routing. Our key result is an exact mapping from multipartite entanglement routing to Nash-Williams's graph immersion problem, extended to hypergraphs. This generalized hypergraph immersion problem introduces a partial order between QN topologies, permitting certain topological transformations while precluding others, offering discerning insights into the design and manipulation of higher-order network topologies in QNs.

翻译：多体纠缠作为量子信息特有的高阶相互作用，在量子网络应用中相较于二体纠缠具有多种优势。在量子网络中建立远程多方之间的多体纠缠需要纠缠路由技术，该技术会以消耗现有纠缠链路为代价不可逆地改变网络拓扑结构。本文旨在探究量子网络能否通过纠缠路由实现拓扑结构间的相互转化。我们的核心成果是建立了多体纠缠路由与纳什-威廉姆斯图浸入问题的精确映射关系，并将其推广至超图范畴。这种广义超图浸入问题在量子网络拓扑间引入了偏序关系，允许特定拓扑变换而排除其他可能，从而为量子网络中高阶网络拓扑的设计与调控提供了精微的洞见。