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DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

Corey Jones

TL;DR

A bimodule version of the DHR tensor category is introduced and it is shown it is an invariant of quasi-local algebras under isomorphisms under isomorphisms with bounded spread.

Abstract

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice $L\subseteq \mathbb{R}^{n}$ satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras $A$ of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric quantum cellular automata (QCA) to $\textbf{Aut}_{br}(\textbf{DHR}(A))$, containing symmetric finite depth circuits in the kernel. For a spin chain with fusion categorical symmetry $\mathcal{D}$, we show the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center $\mathcal{Z}(\mathcal{D})$ . We use this to show that for the double spin flip action $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}$, the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of $S_{3}$, hence is non-abelian, in contrast to the case with no symmetry.

DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

TL;DR

A bimodule version of the DHR tensor category is introduced and it is shown it is an invariant of quasi-local algebras under isomorphisms under isomorphisms with bounded spread.

Abstract

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric quantum cellular automata (QCA) to , containing symmetric finite depth circuits in the kernel. For a spin chain with fusion categorical symmetry , we show the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center . We use this to show that for the double spin flip action , the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of , hence is non-abelian, in contrast to the case with no symmetry.
Paper Structure (16 sections, 28 theorems, 119 equations)

This paper contains 16 sections, 28 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Discrete nets of C*-algebras
  3. Bounded spread isomorphisms and QCA
  4. Physical interpretation of QCA
  5. Discrete DHR Theory
  6. Bimodules of a C*-algebra
  7. DHR functor
  8. Constructing the braiding
  9. 1D spin systems with categorical symmetries
  10. Quantum symmetries: definitions and results
  11. Actions of fusion categories on C*-algebras
  12. Module categories and Q-systems
  13. Realization
  14. DHR categories for fusion spin chains
  15. Examples of QCA
  16. ...and 1 more sections

Key Result

Theorem 1

Let $L$ be a countable metric space with bounded geometry. There is a canonical functor $\textbf{DHR}: \textbf{Net}_{L}\rightarrow \textbf{C*-Tens}$, containing finite depth quantum circuit in the kernel. In particular

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 69 more