Multipartite Entanglement Routing as a Hypergraph Immersion Problem

TL;DR

This work addresses the question of whether a QN can be topologically transformed into another via entanglement routing, and presents an exact mapping from multipartite entanglement routing to Nash-Williams's graph immersion problem, extended to hypergraphs.

Abstract

Multipartite entanglement, linking multiple nodes simultaneously, is a higher-order correlation that offers advantages over pairwise connections in quantum networks (QNs). Creating reliable, large-scale multipartite entanglement requires entanglement routing, a process that combines local, short-distance connections into a long-distance connection, which can be considered as a transformation of network topology. 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.

Figures (13)

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Theorems & Definitions (10)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem II.1
• proof
• Theorem II.2
• proof