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From topological to quantum entanglement. (English) Zbl 1416.81170
Summary: 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.

MSC:
81T45 Topological field theories in quantum mechanics
81P40 Quantum coherence, entanglement, quantum correlations
58J28 Eta-invariants, Chern-Simons invariants
83E30 String and superstring theories in gravitational theory
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References:
[1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys.81 (2009) 865 [quant-ph/0702225] [INSPIRE].
[2] Einstein, A.; Podolsky, B.; Rosen, N., Can quantum mechanical description of physical reality be considered complete?, Phys. Rev., 47, 777, (1935) · Zbl 0012.04201
[3] Morozov, AY, String theory: what is it?, Sov. Phys. Usp., 35, 671, (1992)
[4] Kitaev, A.; Preskill, J., Topological entanglement entropy, Phys. Rev. Lett., 96, 110404, (2006)
[5] 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].
[6] Dong, S.; Fradkin, E.; Leigh, RG; Nowling, S., Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids, JHEP, 05, 016, (2008)
[7] Balasubramanian, V.; Fliss, JR; Leigh, RG; Parrikar, O., Multi-boundary entanglement in Chern-Simons theory and link invariants, JHEP, 04, 061, (2017) · Zbl 1378.81061
[8] Balasubramanian, V.; etal., Entanglement entropy and the colored Jones polynomial, JHEP, 05, 038, (2018) · Zbl 1391.81177
[9] Salton, G.; Swingle, B.; Walter, M., Entanglement from topology in Chern-Simons theory, Phys. Rev., D 95, 105007, (2017)
[10] S. Chun and N. Bao, Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs, arXiv:1707.03525 [INSPIRE].
[11] Dwivedi, S.; etal., Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups, JHEP, 02, 163, (2018) · Zbl 1387.58029
[12] A.Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys.303 (2003) 2 [quant-ph/9707021] [INSPIRE].
[13] 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].
[14] Melnikov, D.; Mironov, A.; Mironov, S.; Morozov, A.; Morozov, A., Towards topological quantum computer, Nucl. Phys., B 926, 491, (2018) · Zbl 1380.81088
[15] L.H. Kauffman, Knot logic and topological quantum computing with Majorana fermions, arXiv:1301.6214 [INSPIRE].
[16] L.H. Kauffman and E. Mehrotra, Topological aspects of quantum entanglement, arXiv:1611.08047.
[17] Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121, 351, (1989) · Zbl 0667.57005
[18] Atiyah, M., Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math., 68, 175, (1989) · Zbl 0692.53053
[19] M.F. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambriddge U.K. (1990).
[20] M. Dedushenko, Gluing I: integrals and symmetries, arXiv:1807.04274 [INSPIRE]. · Zbl 1388.83787
[21] M. Dedushenko, Gluing II: boundary localization and gluing formulas, arXiv:1807.04278 [INSPIRE].
[22] Lashkari, N., Relative entropies in conformal field theory, Phys. Rev. Lett., 113, (2014)
[23] G. Camilo, D. Melnikov, F. Novaes and A. Prudenziati, Circuit complexity of knot states in Chern-Simons theory, arXiv:1903.10609 [INSPIRE].
[24] Mironov, A.; Morozov, A.; Morozov, A., Tangle blocks in the theory of link invariants, JHEP, 09, 128, (2018) · Zbl 1398.81228
[25] 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).
[26] Maldacena, J.; Susskind, L., Cool horizons for entangled black holes, Fortsch. Phys., 61, 781, (2013) · Zbl 1338.83057
[27] Gharibyan, H.; Penna, RF, Are entangled particles connected by wormholes? Evidence for the ER=EPR conjecture from entropy inequalities, Phys. Rev., D 89, (2014)
[28] M. Van Raamsdonk, Building up spacetime with quantum entanglement II: it from BC-bit, arXiv:1809.01197 [INSPIRE].
[29] Balasubramanian, V.; Hayden, P.; Maloney, A.; Marolf, D.; Ross, SF, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav., 31, 185015, (2014) · Zbl 1300.81067
[30] M. Khovanov and L.H. Robert, Foam evaluation and Kronheimer-Mrowka theories, arXiv:1808.09662.
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