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Rosas et al. (2020)<ref name=":5" /> proposed a method using [[information decomposition]] theory to define and quantify causal emergence from an [[information theory|information theory]] perspective. [[information decomposition]] is a new method to analyze the complex interrelationships of various variables in [[complex systems]]. Information decomposition represents each piece of partial information as an [[information atom]]. Partial information is projected onto a information atom using an [[information lattice diagram]]. Both [[synergistic information]] and [[redundant information]] can be represented by the corresponding information atoms. This method is based on the [[partial information decomposition]](PID) theory proposed by Williams and Beer <ref name=":16">Williams P L, Beer R D. Nonnegative decomposition of multivariate information[J]. arXiv preprint arXiv:10042515, 2010.</ref>. The paper uses [[partial information decomposition]] (PID) to break down the mutual information between microstates and macrostates. However, the PID framework can only decompose the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the [[integrated information decomposition]] method <math>\Phi ID</math> <ref name=":18">P. A. Mediano, F. Rosas, R. L. Carhart-Harris, A. K. Seth, A. B. Barrett, Beyond integrated information: A taxonomy of information dynamics phenomena, arXiv preprint arXiv:1909.02297 (2019).</ref> and redefine causal emergence by synergistic information. And this information can further be decomposed into [[causal decoupling]] and [[downward causation]] .

Rosas et al. (2020)<ref name=":5" /> proposed a method using [[information decomposition]] theory to define and quantify causal emergence from an [[information theory|information theory]] perspective. [[information decomposition]] is a new method to analyze the complex interrelationships of various variables in [[complex systems]]. Information decomposition represents each piece of partial information as an [[information atom]]. Partial information is projected onto a information atom using an [[information lattice diagram]]. Both [[synergistic information]] and [[redundant information]] can be represented by the corresponding information atoms. This method is based on the [[partial information decomposition]](PID) theory proposed by Williams and Beer <ref name=":16">Williams P L, Beer R D. Nonnegative decomposition of multivariate information[J]. arXiv preprint arXiv:10042515, 2010.</ref>. The paper uses [[partial information decomposition]] (PID) to break down the mutual information between microstates and macrostates. However, the PID framework can only decompose the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the [[integrated information decomposition]] method <math>\Phi ID</math> <ref name=":18">P. A. Mediano, F. Rosas, R. L. Carhart-Harris, A. K. Seth, A. B. Barrett, Beyond integrated information: A taxonomy of information dynamics phenomena, arXiv preprint arXiv:1909.02297 (2019).</ref> and redefine causal emergence by synergistic information. And this information can further be decomposed into [[causal decoupling]] and [[downward causation]] .

=== Recent work ===

=== Recent works ===

Barnett et al. <ref name=":6">Barnett L, Seth AK. Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems. Physical Review E. 2023 Jul;108(1):014304.</ref> proposed the concept of [[dynamical decoupling]], defining emergences based on the decoupling of macroscopic and microscopic dynamics using [[transfer entropy]]. That is, emergence is characterized by the independence of macroscopic and microscopic variables, with no causal relationship, which can be seen as a causal emergence phenomenon.  

Barnett et al. <ref name=":6">Barnett L, Seth AK. Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems. Physical Review E. 2023 Jul;108(1):014304.</ref> proposed the concept of [[dynamical decoupling]], defining emergence based on the decoupling of macroscopic and microscopic dynamics using [[transfer entropy]]. That is, emergence is characterized by the independence of macroscopic and microscopic variables, with no causal relationship, which can be seen as a causal emergence phenomenon.  

Zhang et al. (2024) <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> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea equates causal emergence with dynamical reversibility. The [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value. This index is highly correlated with EI and can also be used to characterize the causal effect strength of dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values, without requiring a coarse-graining scheme.

Zhang et al. (2024) <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> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea is that the strength of cause effect is equivalent to the concept of [[approximate dynamical reversibility]], and therefore, causal emergence is emergence of reversible dynamics. The [[approximate dynamical reversibility]] measure of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value, and [math]\alpha[/math] is a parameter to balance the relative weight of the capability of reserving information in forward and backward dynamics. This index is highly correlated with EI and can also be used to characterize the causal effect strength of dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values of the Markov dynamics, without requiring an operation of coarse-graining.

== Quantification of causal emergence ==

== Quantification of causal emergence ==