Multi-boundary entanglement in Chern-Simons theory and link invariants
Abstract We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-04-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | http://link.springer.com/article/10.1007/JHEP04(2017)061 |
| _version_ | 1819170772328906752 |
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| author | Vijay Balasubramanian Jackson R. Fliss Robert G. Leigh Onkar Parrikar |
| author_facet | Vijay Balasubramanian Jackson R. Fliss Robert G. Leigh Onkar Parrikar |
| author_sort | Vijay Balasubramanian |
| collection | DOAJ |
| description | Abstract We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1) k ) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state). |
| first_indexed | 2024-12-22T19:40:42Z |
| format | Article |
| id | doaj.art-0ea04ca27b08416ca66c0ffda0d7f401 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-22T19:40:42Z |
| publishDate | 2017-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-0ea04ca27b08416ca66c0ffda0d7f4012022-12-21T18:14:51ZengSpringerOpenJournal of High Energy Physics1029-84792017-04-012017413410.1007/JHEP04(2017)061Multi-boundary entanglement in Chern-Simons theory and link invariantsVijay Balasubramanian0Jackson R. Fliss1Robert G. Leigh2Onkar Parrikar3David Rittenhouse Laboratory, University of PennsylvaniaDepartment of Physics, University of IllinoisDepartment of Physics, University of IllinoisDavid Rittenhouse Laboratory, University of PennsylvaniaAbstract We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1) k ) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).http://link.springer.com/article/10.1007/JHEP04(2017)061Chern-Simons TheoriesTopological Field TheoriesWilson’t Hooft and Polyakov loops |
| spellingShingle | Vijay Balasubramanian Jackson R. Fliss Robert G. Leigh Onkar Parrikar Multi-boundary entanglement in Chern-Simons theory and link invariants Journal of High Energy Physics Chern-Simons Theories Topological Field Theories Wilson ’t Hooft and Polyakov loops |
| title | Multi-boundary entanglement in Chern-Simons theory and link invariants |
| title_full | Multi-boundary entanglement in Chern-Simons theory and link invariants |
| title_fullStr | Multi-boundary entanglement in Chern-Simons theory and link invariants |
| title_full_unstemmed | Multi-boundary entanglement in Chern-Simons theory and link invariants |
| title_short | Multi-boundary entanglement in Chern-Simons theory and link invariants |
| title_sort | multi boundary entanglement in chern simons theory and link invariants |
| topic | Chern-Simons Theories Topological Field Theories Wilson ’t Hooft and Polyakov loops |
| url | http://link.springer.com/article/10.1007/JHEP04(2017)061 |
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