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It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.

It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.

由于[[排列置换矩阵]]过于特殊，我们需要能够衡量一般的马尔科夫概率转移矩阵与排列置换矩阵的靠近程度，以度量其[[近似动力学可逆性]]。在文献<ref name="zhang_reversibility">{{cite journal|author1=Jiang Zhang|author2=Ruyi Tao|author3=Keng Hou Leong|author4=Mingzhe Yang|author5=Bing Yuan|year=2024|title=Dynamical reversibility and a new theory of causal emergence|url=https://arxiv.org/abs/2402.15054|journal=arXiv}}</ref>中，作者们提出了一种用矩阵的类[[Schatten范数]]来度量一个马尔科夫概率转移矩阵的[[近似动力学可逆性]]的方法，定义为：

Due to the specific nature of [[Permutation Matrices]], we need a way to measure the proximity of a general Markov transition matrix to a permutation matrix, in order to assess its [[Approximate Dynamical Reversibility]]. In <ref name="zhang_reversibility">{{cite journal|author1=Jiang Zhang|author2=Ruyi Tao|author3=Keng Hou Leong|author4=Mingzhe Yang|author5=Bing Yuan|year=2024|title=Dynamical reversibility and a new theory of causal emergence|url=https://arxiv.org/abs/2402.15054|journal=arXiv}}</ref>, the authors propose using a matrix norm, the [[Schatten Norm]], to quantify the [[Approximate Dynamical Reversibility]] of a Markov transition matrix, defined as:

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