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.
Locality and the structure of particle states
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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.