Information Exchange Fluctuation Theorem Under Coarse-Graining

The fluctuation theorem for information exchange, originally established by Sagawa and Ueda, provides a fundamental framework for understanding the role of correlations in coupled classical stochastic systems. Building upon this foundation, Jinwoo demonstrated that the pointwise mutual information between correlated subsystems captures entropy production as a state function during coupling processes. In this study, we investigate the robustness of this information-theoretic fluctuation theorem under coarse-graining in coupled classical fluctuating systems. We rigorously prove that the fluctuation theorem remains invariant under arbitrary coarse-graining transformations and derive hierarchical relationships between information measures across different scales, thereby establishing its fundamental character as independent of the level of system description. Our results demonstrate that the relationship between information exchange and entropy production is preserved across different scales of observation, providing deeper insights into the thermodynamic foundations of information processing in classical stochastic systems.