Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Studies Mathematics and Foundations

Loops, projective invariants, and the realization of the Borromean topological link in quantum mechanics

Elias Zafiris

Published Jul 11, 2016• 7 citations
Publisher
DOI
10.1007/s40509-016-0081-y

Related papers

Borromean Link in Quantum Theory Loops, Projective Invariants and the Realization of the Borromean Topological Link in Quantum Mechanics

Albrecht von Müller, Elias Zafiris

On thinking•Jan 1, 2018•1 citation

Topological cascade of quantum Borromean rings

Hao Guan, Simone Zuccher, Xin Liu

The evolution and the topological cascade of quantum vortices forming Borromean rings are studied for the first time. The initial configuration of the system is given by three elliptical planar loops linked together, and the evolution is governed by the numerical implementation of the Gross–Pitaevskii equation. It is found that the topological cascade is not unique, but it depends crucially on the initial geometric configuration. Quantum vortices undergo a series of spontaneous reconnections, resulting in various degenerative pathways characterized by different topology and structural complexity triggered by the different inclination of one of the initial ellipses. Typical decaying routes are given by the successive creation of a Whitehead link, a connected sum of two Hopf links, a trefoil knot, a Hopf link, and the final formation of unknotted, unlinked loops. By structural complexity analysis, we show that the generic trend of the vortex decay goes through a series of topological simplifications, resulting in the formation of small-scale planar loops (rings). During the later stage of evolution, the inverse cascade and topological cycles involving the interaction of unknotted loops become more common, remaining sub-dominant to the overall topological simplification process. These results pave the way to investigate the fundamental relations between structural complexity and energy contents.

Physics of Fluids•Feb 1, 2025•3 citations

Geometric and Algebraic Topological Methods in Quantum Mechanics

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

WORLD SCIENTIFIC eBooks•Jan 1, 2005•31 citations

Compatible braidings with Hopf links, multiloop, and Borromean rings in (3+1) -dimensional spacetime

Zhi-Feng Zhang, Peng Ye

Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach towards a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic \particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multi-loop braidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loop braidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601 (2018)]. A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are $\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}$, and $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}$. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.

arXiv:2012.13761•Physical Review Research•May 20, 2021•19 citations

Entanglement, Coherence, and Recursive Linking in Dicke states : A Topological Perspective

Sougata Bhattacharyya, Sovik Roy

This work investigates the topological structure of multipartite entanglement in symmetric Dicke states $|D_n^{(k)}\rangle$. By viewing qubits as topological loops, we establish a direct correspondence between the recursive measurement dynamics of Dicke states and the stability of $n$-Hopf links. We utilize the Schmidt rank to quantify bipartite entanglement resilience and introduce the $l_1$-norm of quantum coherence as a measure of link fluidity. We demonstrate that unlike fragile states such as $ \left| GHZ \right \rangle$ (analogous to Borromean rings), Dicke states exhibit a robust, self-similar topology where local measurements preserve the global linking structure through non-vanishing residual coherence.

quant-pharXiv:2512.12704•Quantum Studies Mathematics and Foundations•Dec 16, 2025•0 citations

Topological Considerations in Quantum Theory

Sami M. Al-Jaber

•Jan 1, 1991•0 citations
The evolution and the topological cascade of quantum vortices forming Borromean rings are studied for the first time. The initial configuration of the system is given by three elliptical planar loops linked together, and the evolution is governed by the numerical implementation of the Gross–Pitaevskii equation. It is found that the topological cascade is not unique, but it depends crucially on the initial geometric configuration. Quantum vortices undergo a series of spontaneous reconnections, resulting in various degenerative pathways characterized by different topology and structural complexity triggered by the different inclination of one of the initial ellipses. Typical decaying routes are given by the successive creation of a Whitehead link, a connected sum of two Hopf links, a trefoil knot, a Hopf link, and the final formation of unknotted, unlinked loops. By structural complexity analysis, we show that the generic trend of the vortex decay goes through a series of topological simplifications, resulting in the formation of small-scale planar loops (rings). During the later stage of evolution, the inverse cascade and topological cycles involving the interaction of unknotted loops become more common, remaining sub-dominant to the overall topological simplification process. These results pave the way to investigate the fundamental relations between structural complexity and energy contents.
Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach towards a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic \particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multi-loop braidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loop braidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601 (2018)]. A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are ZN1,ZN1×ZN2,ZN1×ZN2×ZN3\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}ZN1​​,ZN1​​×ZN2​​,ZN1​​×ZN2​​×ZN3​​, and ZN1×ZN2×ZN3×ZN4\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}ZN1​​×ZN2​​×ZN3​​×ZN4​​. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.
This work investigates the topological structure of multipartite entanglement in symmetric Dicke states ∣Dn(k)⟩|D_n^{(k)}\rangle∣Dn(k)​⟩. By viewing qubits as topological loops, we establish a direct correspondence between the recursive measurement dynamics of Dicke states and the stability of nnn-Hopf links. We utilize the Schmidt rank to quantify bipartite entanglement resilience and introduce the l1l_1l1​-norm of quantum coherence as a measure of link fluidity. We demonstrate that unlike fragile states such as $ \left| GHZ \right \rangle$ (analogous to Borromean rings), Dicke states exhibit a robust, self-similar topology where local measurements preserve the global linking structure through non-vanishing residual coherence.