Advertisement

Entanglement on multiple S2 boundaries in Chern-Simons theory

  • Siddharth DwivediEmail author
  • Vivek Kumar Singh
  • P. Ramadevi
  • Yang Zhou
  • Saswati Dhara
Open Access
Regular Article - Theoretical Physics

Abstract

Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint S2 boundaries in SU(N) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU(N) representations, creating four punctures on every S2. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU(N) representations on these four-punctured S2 boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU(N) representations.

Keywords

Chern-Simons Theories Conformal Field Theory Topological Field Theories Wilson ’t Hooft and Polyakov loops 

Notes

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

References

  1. [1]
    R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys.81 (2009) 865 [quant-ph/0702225] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proc. Roy. Soc. LondonA 459 (2003) 2011 [quant-ph/0201143].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    D. Bouwmeester et al., Experimental quantum teleportation, Nature390 (1997) 575 [arXiv:1901.11004].ADSCrossRefGoogle Scholar
  4. [4]
    W. Dür, G. Vidal and J.I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev.A 62 (2000) 062314 [quant-ph/0005115].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Gurvits, Classical deterministic complexity of Edmonds’ problem and quantum entanglement, in Proceedings of the thirty-fifth annual ACM symposium on theory of computing, ACM, (2003), pg. 10.Google Scholar
  6. [6]
    A. Peres, Separability criterion for density matrices, Phys. Rev. Lett.77 (1996) 1413 [quant-ph/9604005] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys.121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    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].ADSCrossRefGoogle Scholar
  9. [9]
    S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids, JHEP05 (2008) 016 [arXiv:0802.3231] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett.96 (2006) 110404 [hep-th/0510092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    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].ADSCrossRefGoogle Scholar
  12. [12]
    D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav.32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-boundary entanglement in Chern-Simons theory and link invariants, JHEP04 (2017) 061 [arXiv:1611.05460] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    G. Salton, B. Swingle and M. Walter, Entanglement from topology in Chern-Simons theory, Phys. Rev.D 95 (2017) 105007 [arXiv:1611.01516] [INSPIRE].ADSMathSciNetGoogle Scholar
  15. [15]
    S. Dwivedi, V.K. Singh, S. Dhara, P. Ramadevi, Y. Zhou and L.K. Joshi, Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups, JHEP02 (2018) 163 [arXiv:1711.06474] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    V. Balasubramanian, M. DeCross, J. Fliss, A. Kar, R.G. Leigh and O. Parrikar, Entanglement entropy and the colored Jones polynomial, JHEP05 (2018) 038 [arXiv:1801.01131] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Chun and N. Bao, Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs, arXiv:1707.03525 [INSPIRE].
  18. [18]
    L.-Y. Hung, Y.-S. Wu and Y. Zhou, Linking entanglement and discrete anomaly, JHEP05 (2018) 008 [arXiv:1801.04538] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    H.J. Schnitzer, Clifford group and stabilizer states from Chern-Simons theory, arXiv:1903.06789 [INSPIRE].
  20. [20]
    G. Camilo, D. Melnikov, F. Novaes and A. Prudenziati, Circuit complexity of knot states in Chern-Simons theory, JHEP07 (2019) 163 [arXiv:1903.10609] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    D. Melnikov, A. Mironov, S. Mironov, A. Morozov and A. Morozov, From topological to quantum entanglement, JHEP05 (2019) 116 [arXiv:1809.04574] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theor.22 (2013) 1350078 [arXiv:1302.5144] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  23. [23]
    J. Gu and H. Jockers, A note on colored HOMFLY polynomials for hyperbolic knots from WZW models, Commun. Math. Phys.338 (2015) 393 [arXiv:1407.5643] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    P.H. Butler, Point group symmetry applications: methods and tables, Springer Science & Business Media, (2012).Google Scholar
  25. [25]
    S. Nawata, P. Ramadevi and V.K. Singh, Colored HOMFLY-PT polynomials that distinguish mutant knots, J. Knot Theor. Ramifications26 (2017) 1750096 [arXiv:1504.00364] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    Zodinmawia and P. Ramadevi, SU(N) quantum Racah coefficients & non-torus links, Nucl. Phys.B 870 (2013) 205 [arXiv:1107.3918] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory and 2D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].
  28. [28]
    S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality in WZW models and Chern-Simons theory, Phys. Lett.B 246 (1990) 417 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    E.J. Mlawer, S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys.B 352 (1991) 863 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    S.G. Naculich and H.J. Schnitzer, Duality between SU(N) kand SU(K) nWZW models, Nucl. Phys.B 347 (1990) 687 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    V.E. Hubeny, R. Pius and M. Rangamani, Topological string entanglement, arXiv:1905.09890 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Theoretical Physics, College of Physical Science and TechnologySichuan UniversityChengduChina
  2. 2.Faculty of PhysicsUniversity of WarsawWarsawPoland
  3. 3.Department of PhysicsIndian Institute of Technology BombayMumbaiIndia
  4. 4.Department of Physics and Center for Field Theory and Particle PhysicsFudan UniversityShanghaiChina

Personalised recommendations