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Symmetric and asymmetric tripartite states under the lens of entanglement splitting and topological linking
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DOI：10.1007/s40509-026-00390-1.png)
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Abstract
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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 divided by WW & strns;>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid W\overline{W} \rangle $$\end{document} state and the asymmetric divided by Star >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid \text {Star} \rangle $$\end{document} 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 divided by WW & strns;>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid W\overline{W} \rangle $$\end{document} 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 divided by Star >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid \text {Star} \rangle $$\end{document} 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 divided by Star >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid \text {Star} \rangle $$\end{document} 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.
Keywords:
Multipartite entanglement
Topological links
Projective measurement
Schmidt rank
3-Hopf link
3-link chain
Borromean rings
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