Geometric entanglement in topologically ordered states
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, col...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2014-01-01
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| Series: | New Journal of Physics |
| Online Access: | https://doi.org/10.1088/1367-2630/16/1/013015 |
| _version_ | 1797751435181424640 |
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| author | Román Orús Tzu-Chieh Wei Oliver Buerschaper Maarten Van den Nest |
| author_facet | Román Orús Tzu-Chieh Wei Oliver Buerschaper Maarten Van den Nest |
| author_sort | Román Orús |
| collection | DOAJ |
| description | Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models. |
| first_indexed | 2024-03-12T16:48:29Z |
| format | Article |
| id | doaj.art-510547e96cc74d40845cba6e9b0ee9af |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:48:29Z |
| publishDate | 2014-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-510547e96cc74d40845cba6e9b0ee9af2023-08-08T11:26:58ZengIOP PublishingNew Journal of Physics1367-26302014-01-0116101301510.1088/1367-2630/16/1/013015Geometric entanglement in topologically ordered statesRomán Orús0Tzu-Chieh Wei1Oliver Buerschaper2Maarten Van den Nest3Institute of Physics, Johannes Gutenberg University , D-55099 Mainz, Germany; Max-Planck-Institut für Quantenoptik , Hans-Kopfermann-Straße 1, D-85748 Garching, Germany; School of Mathematics and Physics, The University of Queensland , QLD 4072, AustraliaC N Yang Institute for Theoretical Physics, State University of New York at Stony Brook , NY 11794-3840, USAPerimeter Institute for Theoretical Physics , 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, CanadaMax-Planck-Institut für Quantenoptik , Hans-Kopfermann-Straße 1, D-85748 Garching, GermanyHere we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.https://doi.org/10.1088/1367-2630/16/1/013015 |
| spellingShingle | Román Orús Tzu-Chieh Wei Oliver Buerschaper Maarten Van den Nest Geometric entanglement in topologically ordered states New Journal of Physics |
| title | Geometric entanglement in topologically ordered states |
| title_full | Geometric entanglement in topologically ordered states |
| title_fullStr | Geometric entanglement in topologically ordered states |
| title_full_unstemmed | Geometric entanglement in topologically ordered states |
| title_short | Geometric entanglement in topologically ordered states |
| title_sort | geometric entanglement in topologically ordered states |
| url | https://doi.org/10.1088/1367-2630/16/1/013015 |
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