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arXiv:1106.4171 · Letters in Mathematical Physics

Haag Duality and the Distal Split Property for Cones in the Toric Code

Pieter Naaijkens

Published Jun 17, 2012• 25 citations
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Abstract

We prove that Haag duality holds for cones in the toric code model. That is, for a cone Λ, the algebra $${\mathcal{R}_{\Lambda}}$$ of observables localized in Λ and the algebra $${\mathcal{R}_{\Lambda^c}}$$ of observables localized in the complement Λ c generate each other’s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if $${\Lambda_1 \subset \Lambda_2}$$ are two cones whose boundaries are well separated, there is a Type I factor $${\mathcal{N}}$$ such that $${\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}$$ . We demonstrate this by explicitly constructing $${\mathcal{N}}$$ .

DOI
10.1007/s11005-012-0572-7

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We prove that Haag duality holds for cones in the toric code model. That is, for a cone Λ, the algebra RΛ{\mathcal{R}_{\Lambda}}RΛ​ of observables localized in Λ and the algebra RΛc{\mathcal{R}_{\Lambda^c}}RΛc​ of observables localized in the complement Λ c generate each other’s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if Λ1⊂Λ2{\Lambda_1 \subset \Lambda_2}Λ1​⊂Λ2​ are two cones whose boundaries are well separated, there is a Type I factor N{\mathcal{N}}N such that RΛ1⊂N⊂RΛ2{\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}RΛ1​​⊂N⊂RΛ2​​ . We demonstrate this by explicitly constructing N{\mathcal{N}}N .
In 1997, Kitaev introduced the toric code as an error-tolerant model for quantum computations, which has become a prominent model in recent years. This thesis is dedicated to providing a detailed exploration of Pieter Naaijkens’ work on the toric code in the thermodynamic limit. The primary focus of this thesis is providing an exposition on local Hamiltonians for the toric code and their ground states, showing that Haag duality holds for cones in the corresponding ground state representation. That is, given a cone Λ, the von Neumann algebras R_Λ and R_Λ_c of observables localized in Λ and Λ_c respectively are each other’s commutant. We also show that given Haag duality, we also have the distal split property, which states that if Λ_1 ⊂ Λ_2 are two cones with sufficiently separated boundaries, then there is a Type I factor M such that R_Λ_1 ⊂ M ⊂ R_Λ_2.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in Rn\mathbb{R}^nRn. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some Z/2\mathbb{Z}/2Z/2-reflection symmetry.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region AAA if and only if the mutual information of the inner region and the exterior, conditioned on the part of AAA inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.
The uniqueness of purifications of quantum states on a system A up to local unitary transformations on a purifying system B is central to quantum information theory. We show that, if the two systems are modeled by commuting von Neumann algebras M_{A} and M_{B} on a Hilbert space H, then uniqueness of purifications is equivalent to Haag duality M_{A}=M_{B}^{'}. In particular, the uniqueness of purifications can fail in systems with infinitely many degrees of freedom-even when M_{A} and M_{B} are commuting factors that jointly generate B(H) and hence allow for local tomography of all density matrices on H. We present a simple argument showing that the uniqueness of purifications, and, hence, Haag duality, fail in ground state sectors of topologically ordered models at renormalization group fixed points on infinite two-dimensional lattices, partitioned into the union of two spatially separated cones and its complement.