Abstract
Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are “strings” connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
A. Einstein, B. Podolsky and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935) 777 [INSPIRE].
A. Yu. Morozov, String theory: what is it?, Sov. Phys. Usp. 35 (1992) 671 [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-boundary entanglement in Chern-Simons theory and link invariants, JHEP 04 (2017) 061 [arXiv:1611.05460] [INSPIRE].
V. Balasubramanian et al., Entanglement entropy and the colored Jones polynomial, JHEP 05 (2018) 038 [arXiv:1801.01131] [INSPIRE].
G. Salton, B. Swingle and M. Walter, Entanglement from topology in Chern-Simons theory, Phys. Rev. D 95 (2017) 105007 [arXiv:1611.01516] [INSPIRE].
S. Chun and N. Bao, Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs, arXiv:1707.03525 [INSPIRE].
S. Dwivedi et al., Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups, JHEP 02 (2018) 163 [arXiv:1711.06474] [INSPIRE].
A.Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
M.H. Freedman, A. Kitaev and Z. Wang, Simulation of topological field theories by quantum computers, Commun. Math. Phys. 227 (2002) 587 [quant-ph/0001071] [INSPIRE].
D. Melnikov, A. Mironov, S. Mironov, A. Morozov and A. Morozov, Towards topological quantum computer, Nucl. Phys. B 926 (2018) 491 [arXiv:1703.00431] [INSPIRE].
L.H. Kauffman, Knot logic and topological quantum computing with Majorana fermions, arXiv:1301.6214 [INSPIRE].
L.H. Kauffman and E. Mehrotra, Topological aspects of quantum entanglement, arXiv:1611.08047.
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1989) 175 [INSPIRE].
M.F. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambriddge U.K. (1990).
M. Dedushenko, Gluing I: integrals and symmetries, arXiv:1807.04274 [INSPIRE].
M. Dedushenko, Gluing II: boundary localization and gluing formulas, arXiv:1807.04278 [INSPIRE].
N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
G. Camilo, D. Melnikov, F. Novaes and A. Prudenziati, Circuit complexity of knot states in Chern-Simons theory, arXiv:1903.10609 [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, Tangle blocks in the theory of link invariants, JHEP 09 (2018) 128 [arXiv:1804.07278] [INSPIRE].
P.K. Aravind, Borromean entanglement of the GHZ state, in Potentiality, entanglement and passion-at-a-distance, R.S. Cohen et al. eds., Kluwer, U.S.A. (1997).
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
H. Gharibyan and R.F. Penna, Are entangled particles connected by wormholes? Evidence for the ER=EPR conjecture from entropy inequalities, Phys. Rev. D 89 (2014) 066001 [arXiv:1308.0289] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement II: it from BC-bit, arXiv:1809.01197 [INSPIRE].
V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
M. Khovanov and L.H. Robert, Foam evaluation and Kronheimer-Mrowka theories, arXiv:1808.09662.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1809.04574
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Melnikov, D., Mironov, A., Mironov, S. et al. From topological to quantum entanglement. J. High Energ. Phys. 2019, 116 (2019). https://doi.org/10.1007/JHEP05(2019)116
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)116