Encoded information of mixed correlations: the views from one dimension higher

After reviewing the JT gravity, we discuss the four saddles in the mixed correlation measures of black holes Hawking radiation in the setup of geometric evaporation of \cite{Verheijden:2021yrb}. By looking from $1d$ higher point of view and partial dimensional reduction, we examine the phase structures and the universalities for these four saddles. We also discuss the behavior of quantum error correction codes for each of these four phases, reaching to consistent results. Then, instead of dimension reduction between Einstein gravity and JT, we try to explore the connections between partition functions and saddles of $3d$ Chern-Simons and $2d$ BF theories, $2d$ Liouville and $2d$ Wess-Zumino-Witten models, and also the dimensionally reduced $1d$ Schwarzian and $1d$ particles on group. We specifically sketch on the connections between these theories in the setup of mixed correlations and island formulation.