Abstract
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories.
We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ghosh, R.M. Soni and S.P. Trivedi, On the entanglement entropy for gauge theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy for a Maxwell field: numerical calculation on a two dimensional lattice, Phys. Rev. D 90 (2014) 105013 [arXiv:1406.2991] [INSPIRE].
D. Radičević, Notes on entanglement in Abelian gauge theories, arXiv:1404.1391 [INSPIRE].
W. Donnelly, Entanglement entropy and non-Abelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
L.-Y. Hung and Y. Wan, Revisiting entanglement entropy of lattice gauge theories, JHEP 04 (2015) 122 [arXiv:1501.04389] [INSPIRE].
S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, On the definition of entanglement entropy in lattice gauge theories, JHEP 06 (2015) 187 [arXiv:1502.04267] [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [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 wavefunction, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
A. Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
D. Radičević, Entanglement in weakly coupled lattice gauge theories, arXiv:1509.08478 [INSPIRE].
J. Preskill, Lecture notes on quantum computation, http://www.theory.caltech.edu/people/preskill/ph229/.
C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53 (1996) 2046 [quant-ph/9511030] [INSPIRE].
M.A. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2000).
M.M. Wilde, Quantum information theory, Cambridge University Press, Cambridge U.K. (2013) [arXiv:1106.1445].
N. Schuch, F. Verstraete and J.I. Cirac, Nonlocal resources in the presence of supperselection rules, Phys. Rev. Lett. 92 (2004) 087904 [quant-ph/0310124].
N. Schuch, F. Verstraete and J.I. Cirac, Quantum entanglement theory in the presence of supperselection rules, Phys. Rev. A 70 (2004) 042310 [quant-ph/0404079].
K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V.B. Scholz and F. Verstraete, The entanglement of distillation for gauge theories, arXiv:1511.04369 [INSPIRE].
V. Lahtinen, Topological quantum computation. An analysis of an anyon model based on quantum double symmetries, M.Sc. thesis, U. Helsinki, Helsinki Finland (2006).
J.-W. Chen, S.-H. Dai and J.-Y. Pang, Strong coupling expansion of the entanglement entropy of Yang-Mills gauge theories, arXiv:1503.01766 [INSPIRE].
M. Pretko and T. Senthil, Entanglement entropy of U (1) quantum spin liquids, arXiv:1510.03863 [INSPIRE].
J.B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
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: 1510.07455
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Soni, R.M., Trivedi, S.P. Aspects of entanglement entropy for gauge theories. J. High Energ. Phys. 2016, 136 (2016). https://doi.org/10.1007/JHEP01(2016)136
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)136