Higher-Dimensional Information Lattice: Quantum State Characterization through Inclusion-Exclusion Local Information

Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.