Separability transitions in topological states induced by local decoherence

We study quantum many-body states subjected to decoherence from the perspective of separability, i.e., whether a decohered mixed state can be expressed as an ensemble of short-range entangled pure states. We provide evidence for the existence of 'separability transitions' in a variety of states, including those with intrinsic and symmetry-protected topological order, as well as conventional symmetry broken order. For states with intrinsic topological order, we focus on toric codes and the X-cube fracton state, and provide evidence that the separability transition induced by local decoherence coincides with the error-recovery transition. A key insight is that local decoherence acting on the 'parent' cluster states of these models results in a Gibbs state. This fact is then also employed to study symmetry-enforced separability transitions in various SPTs, including those with higher-form symmetry. In the context of non-stabilizer states, we show that subjecting a p+ip superconductor to a fermion parity breaking decoherence immediately results in a mixed state that can be expressed as a convex sum of non-chiral states, while a fermion-parity preserving channel results in a phase transition at a non-zero threshold.