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Haag duality and the distal split property for cones in the toric code

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arxiv 1106.4171 v2 pith:N7WJ7OYZ submitted 2011-06-21 math-ph math.MPmath.OAquant-ph

Haag duality and the distal split property for cones in the toric code

classification math-ph math.MPmath.OAquant-ph
keywords lambdaconessubsetalgebracodedistaldualityhaag
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We prove that Haag duality holds for cones in the toric code model. That is, for a cone Lambda, the algebra R_Lambda of observables localized in Lambda and the algebra R_{Lambda^c} of observables localized in the complement Lambda^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 N such that R_{Lambda_1} \subset N \subset R_{Lambda_2}. We demonstrate this by explicitly constructing N.

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  1. Disjoint additivity and local quantum physics

    hep-th 2025-09 conditional novelty 7.0

    Local quantum systems should obey disjoint additivity plus Haag duality, a combination that survives higher-form symmetries and fails for known nonlocal constructions.