Separability transitions in topological states induced by local decoherence

TL;DR

This paper investigates how local decoherence affects topologically ordered states, revealing phase transitions in their separability that align with error correction thresholds, using models like toric codes and fracton states.

Contribution

It demonstrates that local decoherence induces separability transitions in topological states, linking these to error correction thresholds and phase transitions in related statistical models.

Findings

Decoherence leads to separability transitions in topological states.

Transitions coincide with active error correction thresholds.

Decohered states can be expressed as convex sums of SRE states beyond critical error rates.

Abstract

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 (SRE) 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 SRE states for $p > p_c$, where $p_c$ is related to the paramagnetic-ferromagnetic transition in the 2d (3d) random-field bond Ising model along the Nishimori line.