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Remote detectability from entanglement bootstrap I: Kirby's torus trick

Bowen Shi, Jin-Long Huang, John McGreevy

Abstract

Remote detectability is often taken as a physical assumption in the study of topologically ordered systems, and it is a central axiom of mathematical frameworks of topological quantum field theories. We show under the entanglement bootstrap approach that remote detectability is a necessary property; that is, we derive it as a theorem. Starting from a single wave function on a topologically-trivial region satisfying the entanglement bootstrap axioms, we can construct states on closed manifolds. The crucial technique is to immerse the punctured manifold into the topologically trivial region and then heal the puncture. This is analogous to Kirby's torus trick. We then analyze a special class of such manifolds, which we call pairing manifolds. For each pairing manifold, which pairs two classes of excitations, we identify an analog of the topological $S$-matrix. This pairing matrix is unitary, which implies remote detectability between two classes of excitations. These matrices are in general not associated with the mapping class group of the manifold. As a by-product, we can count excitation types (e.g., graph excitations in 3+1d). The pairing phenomenon occurs in many physical contexts, including systems in different dimensions, with or without gapped boundaries. We provide a variety of examples to illustrate its scope.

Remote detectability from entanglement bootstrap I: Kirby's torus trick

Abstract

Remote detectability is often taken as a physical assumption in the study of topologically ordered systems, and it is a central axiom of mathematical frameworks of topological quantum field theories. We show under the entanglement bootstrap approach that remote detectability is a necessary property; that is, we derive it as a theorem. Starting from a single wave function on a topologically-trivial region satisfying the entanglement bootstrap axioms, we can construct states on closed manifolds. The crucial technique is to immerse the punctured manifold into the topologically trivial region and then heal the puncture. This is analogous to Kirby's torus trick. We then analyze a special class of such manifolds, which we call pairing manifolds. For each pairing manifold, which pairs two classes of excitations, we identify an analog of the topological -matrix. This pairing matrix is unitary, which implies remote detectability between two classes of excitations. These matrices are in general not associated with the mapping class group of the manifold. As a by-product, we can count excitation types (e.g., graph excitations in 3+1d). The pairing phenomenon occurs in many physical contexts, including systems in different dimensions, with or without gapped boundaries. We provide a variety of examples to illustrate its scope.
Paper Structure (49 sections, 22 theorems, 102 equations, 53 figures, 4 tables)

This paper contains 49 sections, 22 theorems, 102 equations, 53 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Reader's guide
  3. Content of the sections
  4. Why Kirby's torus trick?
  5. Central pillars of entanglement bootstrap
  6. Axioms and existing tools
  7. Immersed regions
  8. Constrained fusion space
  9. Regions dual to excitations
  10. 2d bulk and 3d bulk regions
  11. 2d bulk regions
  12. 3d bulk regions
  13. ${}^\star$Systems with gapped boundaries in 2d and 3d
  14. ${}^\star$Axioms for gapped boundaries in 2d and 3d
  15. ${}^\star$2d: regions adjacent to a gapped boundary
  16. ...and 34 more sections

Key Result

Lemma 2.2

The various constrained sets defined in Definition def:sub-information-convex-set satisfy:

Figures (53)

  • Figure 1: Illustrated is an analog to the growth of a plant from a seed.
  • Figure 2: A summary of the content of the main text. The focus is on the concepts developed and the relations between the sections.
  • Figure 5: Illustration of the three kinds of immersed regions.
  • Figure 6: Immersed versions of embedded regions. For every example shown here, there exists a smooth deformation that connects the two configurations. The deformation is done on $\mathbf{S}^2$. Note that the deformation is more flexible on $\mathbf{S}^2$ compared with $\mathbf{B}^2$.
  • Figure 7: Genus-$g$ handlebodies, where $g=0,1,2,\cdots$.
  • ...and 48 more figures

Theorems & Definitions (104)

  • Definition 2.1: Constrained information convex sets and constrained fusion spaces
  • Remark
  • Lemma 2.2
  • proof
  • Theorem 2.3: Associativity theorem, new form
  • Example 3.1: 3d $S_3$ quantum double
  • Proposition 3.1: Graph sectors for 3d quantum double
  • Remark
  • Definition 4.1: Completion in ${\cal N}$
  • Proposition 4.2
  • ...and 94 more