Symmetric and asymmetric tripartite states under the lens of entanglement splitting and topological linking

Abstract

This work establishes a direct operational connection between the entanglement structures of specific three-qubit states (i.e., multipartite entanglement) and their corresponding topological links. We investigate the symmetric ∣WW¯⟩ state and the asymmetric ∣Star⟩ state through local projective measurements on individual qubits. The post-measurement states are analyzed via their Schmidt rank to characterize residual bipartite entanglement. For the symmetric ∣WW¯⟩ state, measurement of any qubit consistently results in a non-maximally entangled post-measurement state (Schmidt rank 2), analogous to the behavior of a 3-Hopf link structure, where cutting any ring leaves the remaining two nontrivially linked. On the other hand, the ∣Star⟩ state exhibits a context-dependent fragility. Its behavior predominantly mirrors that of a 3-link chain where severing the central qubit decouples the system, while cutting an outer qubit often preserves a residual link. Crucially, for specific measurement outcomes, the ∣Star⟩ state also exhibits the defining property of the Borromean rings, where the loss of one qubit completely disentangles the remaining two. This analysis provides a concrete interpretation of topological linking structures as a resource for characterizing distributed entanglement and its resilience under local measurement operations, revealing that a single quantum state can contextually embody multiple distinct topological analogues. © The Author(s), under exclusive license to Chapman University 2026.