We investigate the influence of local decoherence on a symmetry-protected topological (SPT) phase of the two-dimensional (2D) cluster state. Mapping the 2D cluster state under decoherence to a classical spin model, we show a topological phase transition of a ${\mathbb{Z}}_2^{\left(0\right)}\times {\mathbb{Z}}_2^{\left(1\right)}$ SPT phase into a trivial phase occurring at a finite decoherence strength. To characterize the phase transition, we employ three distinct diagnostic methods, namely, the relative entropy between two decohered SPT states with different topological edge states, the strange correlation function of ${\mathbb{Z}}_2^{\left(1\right)}$ charge, and the multipartite negativity of the mixed state on a disk. All the diagnostics can be obtained as certain thermodynamic quantities in the corresponding classical model, and the results of the three diagnostic tests are consistent with each other. Given that the 2D cluster state possesses universal computational capabilities in the context of measurement-based quantum computation, the topological phase transition found here can also be interpreted as a transition in the computational power.

中文翻译：

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开放量子系统中二维对称性保护的拓扑相和跃迁

我们研究了局部退相干对二维 (2D) 簇状态的对称保护拓扑 (SPT) 相的影响。将退相干下的二维簇状态映射到经典自旋模型，我们展示了拓扑相变${\mathbb{Z}}_2^{（0）}\times {\mathbb{Z}}_2^{（1）}$SPT 相位转变为在有限退相干强度下发生的平凡相位。为了表征相变，我们采用了三种不同的诊断方法，即具有不同拓扑边缘状态的两个退相干 SPT 状态之间的相对熵，奇怪的相关函数${\mathbb{Z}}_2^{（1）}$电荷，以及磁盘上混合态的多部分负性。所有诊断都可以作为相应经典模型中的某些热力学量获得，并且三个诊断测试的结果彼此一致。鉴于二维簇状态在基于测量的量子计算中具有通用计算能力，这里发现的拓扑相变也可以解释为计算能力的转变。