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We study states with intrinsic topological order subjected to localdecoherence from the perspective of separability, i.e., whether a decoheredmixed state can be expressed as an ensemble of short-range entangled purestates. We focus on toric codes and the X-cube fracton state and provideevidence for the existence of decoherence-induced separability transitions thatprecisely coincide with the error-recovery transitions. A key insight is thatlocal decoherence acting on the 'parent' cluster states of these models resultsin a Gibbs state.

Separability transitions in topological states induced by local decoherence