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Aspects of Entanglement Entropy for Gauge Theories

Ronak M Soni, Sandip P. Trivedi

TL;DR

This paper analyzes a lattice gauge theory definition of entanglement entropy based on embedding gauge-invariant states into an extended Hilbert space. It shows that in Non-Abelian theories the entropy contains an extra classical-like term arising from boundary representation fusion, and that only the sector-conditional quantum part is extractable via LOCC distillation/dilution, while a residual classical piece remains inaccessible. The authors compare this to the electric centre CHR construction, finding key differences in the Non-Abelian case, and demonstrate that the extended definition correctly yields the topological entanglement entropy for non-Abelian toric codes, with the total quantum dimension ${\cal D}=|G|$. They support these findings with a toy model illustrating sector extraction limits and with strong-coupling perturbation theory showing the entanglement is governed by boundary (area-law) terms at leading order. Collectively, the work argues for the extended Hilbert space as a robust framework for gauge-theory entanglement, while clarifying operationally inaccessible classical contributions and their physical significance.

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.

Aspects of Entanglement Entropy for Gauge Theories

TL;DR

This paper analyzes a lattice gauge theory definition of entanglement entropy based on embedding gauge-invariant states into an extended Hilbert space. It shows that in Non-Abelian theories the entropy contains an extra classical-like term arising from boundary representation fusion, and that only the sector-conditional quantum part is extractable via LOCC distillation/dilution, while a residual classical piece remains inaccessible. The authors compare this to the electric centre CHR construction, finding key differences in the Non-Abelian case, and demonstrate that the extended definition correctly yields the topological entanglement entropy for non-Abelian toric codes, with the total quantum dimension . They support these findings with a toy model illustrating sector extraction limits and with strong-coupling perturbation theory showing the entanglement is governed by boundary (area-law) terms at leading order. Collectively, the work argues for the extended Hilbert space as a robust framework for gauge-theory entanglement, while clarifying operationally inaccessible classical contributions and their physical significance.

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.

Paper Structure

This paper contains 19 sections, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Definition
  3. General Discussion
  4. Non-Abelian Theories
  5. The Electric Centre Definition
  6. General Discussion
  7. Non-Abelian Case
  8. Entanglement Entropy
  9. Entanglement Distillation and Dilution
  10. A Toy Model
  11. The Distillation Protocol
  12. The Dilution Protocol
  13. Concluding Comments on the Toy Model
  14. Comments on Gauge Theories
  15. Degeneracy of the Non-Abelian Toric Code
  16. ...and 4 more sections

Figures (2)

  • Figure 1: An example of a region in a lattice. The solid lines are the inside links, and the dotted lines are outside links. $V_{1}$ is an example of what we call a boundary vertex, one which has both inside and outside links ending on it.
  • Figure 2: The topological entanglement entropy can be calculated form three regions of this sort, by the formula, eq. (\ref{['top']}).