For 2D spin systems with mixed-state approximate Haag duality, finite-depth decoherence maps the anyon category of the final state faithfully into that of the initial state, so topological order cannot grow under local noise.
Relative H aag duality for the free field in fock representation
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Mixed state topological order: operator algebraic approach
For 2D spin systems with mixed-state approximate Haag duality, finite-depth decoherence maps the anyon category of the final state faithfully into that of the initial state, so topological order cannot grow under local noise.