Separability Transitions in Topological States Induced by Local Decoherence

We study states with intrinsic topological order subjected to local 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 focus on toric codes and the $X$-cube fracton state and provide evidence for the existence of decoherence-induced separability transitions that precisely coincide with the threshold for the feasibility of active error correction. A key insight is that local decoherence acting on the “parent” cluster states of these models results in a Gibbs state. As an example, for the 2D (3D) toric code subjected to bit-flip errors, we show that the decohered density matrix can be written as a convex sum of short-range entangled states for $p>p_c$, where $p_c$ is related to the paramagnetic-ferromagnetic transition in the 2D (3D) random bond Ising model along the Nishimori line.

Figure 1

(a) Topological orders under local decoherence can undergo a separability transition, where only above a certain critical error rate, the decohered mixed state ${\rho }_{\mathrm{dec}}$ can be written as a convex sum of SRE pure states. The bottom depicts the parent cluster states and their offspring models obtained by appropriate measurements (indicated by an arrow); (b) 2D cluster Hamiltonian and 2D toric code; (c) 3D cluster Hamiltonian and 3D toric code; and (d) “Cluster-$X$” Hamiltonian [39] and the $X$-cube Hamiltonian.