The steady states of dynamical processes can exhibit stable nontrivial phases, which can also serve as fault-tolerant classical or quantum memories. For Markovian quantum (classical) dynamics, these steady states are extremal eigenvectors of the non-Hermitian operators that generate the dynamics, i.e., quantum channels (Markov chains). However, since these operators are non-Hermitian, their spectra are an unreliable guide to dynamical relaxation timescales or to stability against perturbations. We propose an alternative dynamical criterion for a steady state to be in a stable phase, which we name uniformity: Informally, our criterion amounts to requiring that, under sufficiently small local perturbations of the dynamics, the unperturbed and perturbed steady states are related to one another by a finite-time dissipative evolution. We show that this criterion implies many of the properties one would want from any reasonable definition of a phase. We prove that uniformity is satisfied in a canonical classical cellular automaton, and we provide numerical evidence that the gap determines the relaxation rate between nearby steady states in the same phase, a situation we conjecture holds generically whenever uniformity is satisfied. We further conjecture some sufficient conditions for a channel to exhibit uniformity and therefore stability. Published by the American Physical Society 2024

中文翻译：

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定义开放量子系统的稳定相


动力学过程的稳态可以表现出稳定的非平凡相，这些相也可以用作容错经典或量子存储器。对于马尔可夫量子（经典）动力学，这些稳态是产生动力学的非厄米特算子的极值特征向量，即量子通道（马尔可夫链）。然而，由于这些算子是非厄米特算子，因此它们的频谱不能可靠地指导动力学弛豫时间尺度或抗扰动稳定性。我们提出了一个稳态处于稳定阶段的另一种动力学标准，我们将其命名为均匀性：非正式地，我们的标准相当于要求，在动力学的足够小的局部扰动下，未扰动和受扰动的稳态通过有限时间耗散演化彼此相关。我们表明，这个标准意味着人们希望从任何合理的相定义中获得的许多属性。我们证明了在规范的经典元胞自动机中满足均匀性，并且我们提供了数值证据，证明间隙决定了同一相位中附近稳态之间的弛豫率，只要满足均匀性，我们就会假设这种情况通常成立。我们进一步推测了通道表现出均匀性并因此表现出稳定性的一些充分条件。 美国物理学会 2024 年出版