  The theory of causal emergence based on effective information(EI) suggests
that complex systems may exhibit a phenomenon called causal emergence(CE),
where the macro-dynamics demonstrate a stronger causal effect than the
micro-dynamics. However, a challenge in this theory is the dependence on the
method used to coarse-grain the system. In this paper, we introduce a fresh
concept of approximate dynamical reversibility and establish a novel framework
for CE based on this and singular value decomposition. Our research not only
reveals a strong correlation between the proximity of a Markov chain being
dynamically reversible and EI in assessing causal effects across various
scenarios but also demonstrates that EI is constrained by the logarithm of the
approximate dynamical reversibility. Leveraging this concept, we present an
innovative approach to quantifying CE that is agnostic to specific
coarse-graining techniques and effectively captures the inherent
characteristics of Markov dynamics. Through empirical evaluations on examples
of boolean networks, cellular automata, and complex networks, we validate our
refined CE definition.
