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.

中文翻译：

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局部退相干引起的拓扑状态的可分离性转变

我们从可分离性的角度研究具有局部退相干的固有拓扑序态，即退相干混合态是否可以表示为短程纠缠纯态的系综。我们专注于复曲面码和$X$-立方分形状态，并为退相干引起的可分离性转变的存在提供证据，该转变与主动纠错可行性的阈值精确一致。一个关键的见解是，作用于这些模型的“父”簇状态的局部退相干会导致吉布斯状态。作为一个例子，对于遭受位翻转错误的 2D (3D) 环面码，我们表明去相干密度矩阵可以写成短程纠缠态的凸和：$p>p_C$， 在哪里$p_C$与沿西森线的 2D（3D）随机键 Ising 模型中的顺磁-铁磁转变有关。