Universal Characterization of 2D Quantum States via Local Information

The recently introduced Information Lattice provides a framework for classifying quantum states based on their scale-resolved correlations, defining intrinsic length scales through an operational measure of total correlations at each scale. However, the original framework is limited to one dimension, and its key properties, such as locality and conservation of local information, do not straightforwardly extend to higher dimensions. In this work, we show that despite these challenges it is possible to construct new structures that characterize phases in higher-dimensional systems. We illustrate this approach through three examples: the disordered Anderson model, a gapped system with a critical edge, and the Toric code model, where topological features are interpreted via local information.