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Therefore, the macroscopic state of this process can be regarded as the parity of the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the microstate. <math>x_{t + 1}^1</math> is a special microstate that always remains consistent with the macroscopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

Therefore, the macroscopic state of this process can be regarded as the parity of the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the microstate. <math>x_{t + 1}^1</math> is a special microstate that always remains consistent with the macroscopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

====Causal Emergence Theory Based on Singular Value Decomposition====

====Causal emergence theory based on singular value decomposition====

[[Erik Hoel's causal emergence theory]] needs to specify a coarse-graining strategy in advance. Rosas' information decomposition theory does not completely solve this problem. Therefore, [[Zhang Jiang]] et al.<ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> further proposed the [[causal emergence theory based on singular value decomposition]].

[[Erik Hoel's causal emergence theory]] needs to specify a coarse-graining strategy in advance. Rosas' information decomposition theory does not completely solve this problem. Therefore, [[Zhang Jiang]] et al.<ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> further proposed the [[causal emergence theory based on singular value decomposition]].

=====Singular Value Decomposition of Markov Chains=====

=====Singular value decomposition of Markov chains=====

Given the [[Markov transition matrix]] <math>P</math> of a system, we can perform [[singular value decomposition]] on it to obtain two orthogonal and normalized matrices <math>U</math> and <math>V</math>, and a diagonal matrix <math>\Sigma</math>: <math>P = U\Sigma V^T</math>. Here, [math]\Sigma = diag(\sigma_1,\sigma_2,\cdots,\sigma_N)[/math], where [math]\sigma_1\geq\sigma_2\geq\cdots\sigma_N[/math] are the singular values of <math>P</math> and are arranged in descending order. <math>N</math> is the number of states of <math>P</math>.

Given the [[Markov transition matrix]] <math>P</math> of a system, we can perform [[singular value decomposition]] on it to obtain two orthogonal and normalized matrices <math>U</math> and <math>V</math>, and a diagonal matrix <math>\Sigma</math>: <math>P = U\Sigma V^T</math>. Here, [math]\Sigma = diag(\sigma_1,\sigma_2,\cdots,\sigma_N)[/math], where [math]\sigma_1\geq\sigma_2\geq\cdots\sigma_N[/math] are the singular values of <math>P</math> and are arranged in descending order. <math>N</math> is the number of states of <math>P</math>.

=====Approximate Dynamical Reversibility and Effective Information=====

=====Approximate dynamical reversibility and effective Information=====

We can define the sum of the <math>\alpha</math> powers of the singular values (also known as the [math]\alpha[/math]-order [[Schatten norm]] of the matrix) as a measure of the [[approximate dynamical reversibility]] of the Markov chain, that is:

We can define the sum of the <math>\alpha</math> powers of the singular values (also known as the [math]\alpha[/math]-order [[Schatten norm]] of the matrix) as a measure of the [[approximate dynamical reversibility]] of the Markov chain, that is:

Moreover, to a certain extent, [math]\Gamma_{\alpha}[/math] can be used instead of <math>EI</math> to measure the extent of causal effect of Markov chains. Therefore, the so-called causal emergence can also be understood as an '''emergence of dynamical reversibility'''.

Moreover, to a certain extent, [math]\Gamma_{\alpha}[/math] can be used instead of <math>EI</math> to measure the extent of causal effect of Markov chains. Therefore, the so-called causal emergence can also be understood as an '''emergence of dynamical reversibility'''.

=====Quantification of Causal Emergence without Coarse-graining=====

=====Quantification of causal emergence without coarse-graining=====

However, the greatest value of this theory lies in the fact that emergence can be directly quantified without a coarse-graining strategy. If the rank of <math>P</math> is <math>r</math>, that is, starting from the <math>r + 1</math>th singular value, all singular values are 0, then we say that the dynamics <math>P</math> has '''clear causal emergence''', and the numerical value of causal emergence is:

However, the greatest value of this theory lies in the fact that emergence can be directly quantified without a coarse-graining strategy. If the rank of <math>P</math> is <math>r</math>, that is, starting from the <math>r + 1</math>th singular value, all singular values are 0, then we say that the dynamics <math>P</math> has '''clear causal emergence''', and the numerical value of causal emergence is:

In summary, the advantage of this method for quantifying causal emergence is that it can quantify causal emergence more objectively without relying on a specific coarse-graining strategy. The disadvantage of this method is that to calculate [math]\Gamma_{\alpha}[/math], it is necessary to perform [[SVD decomposition]] on <math>P</math> in advance. As a result, the computational complexity is [math]O(N^3)[/math], which is higher than the computational complexity of <math>EI</math>. Moreover, [math]\Gamma_{\alpha}[/math] cannot be explicitly decomposed into two components: determinism and degeneracy.

In summary, the advantage of this method for quantifying causal emergence is that it can quantify causal emergence more objectively without relying on a specific coarse-graining strategy. The disadvantage of this method is that to calculate [math]\Gamma_{\alpha}[/math], it is necessary to perform [[SVD decomposition]] on <math>P</math> in advance. As a result, the computational complexity is [math]O(N^3)[/math], which is higher than the computational complexity of <math>EI</math>. Moreover, [math]\Gamma_{\alpha}[/math] cannot be explicitly decomposed into two components: determinism and degeneracy.

=====Specific Example=====

=====Specific example=====

The author gives four specific examples of Markov chains. The state transition matrix of this Markov chain is shown in the figure. We can compare the <math>EI</math> and [[approximate dynamical reversibility]] (the <math>\Gamma</math> in the figure, that is, <math>\Gamma_{\alpha = 1}</math>) of this Markov chain. Comparing figures a and b, we find that for different state transition matrices, when <math>EI</math> decreases, <math>\Gamma</math> also decreases simultaneously. Further, figures c and d are comparisons of the effects before and after coarse-graining. In figure d, the state transition matrix of figure c is coarse-grained by merging the first three states into a macroscopic state. Since the macroscopic state transition matrix in figure d is a [[deterministic system]], the normalized <math>EI</math>, <math>eff\equiv EI/\log N</math> and the normalized [math]\Gamma[/math]: <math>\gamma\equiv \Gamma/N</math> all reach the maximum value of 1.

The author gives four specific examples of Markov chains. The state transition matrix of this Markov chain is shown in the figure. We can compare the <math>EI</math> and [[approximate dynamical reversibility]] (the <math>\Gamma</math> in the figure, that is, <math>\Gamma_{\alpha = 1}</math>) of this Markov chain. Comparing figures a and b, we find that for different state transition matrices, when <math>EI</math> decreases, <math>\Gamma</math> also decreases simultaneously. Further, figures c and d are comparisons of the effects before and after coarse-graining. In figure d, the state transition matrix of figure c is coarse-grained by merging the first three states into a macroscopic state. Since the macroscopic state transition matrix in figure d is a [[deterministic system]], the normalized <math>EI</math>, <math>eff\equiv EI/\log N</math> and the normalized [math]\Gamma[/math]: <math>\gamma\equiv \Gamma/N</math> all reach the maximum value of 1.

====Dynamic independence====

====Dynamic independence====