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

TL;DR

This paper extends the information lattice framework to higher-dimensional quantum systems, enabling detailed analysis of entanglement, topological order, and critical phenomena through an inclusion-exclusion based local information measure.

Contribution

It introduces a higher-dimensional information lattice using inclusion-exclusion principles, allowing for position- and scale-resolved quantum state characterization in complex geometries.

Findings

Extracted information-based localization lengths

Identified direction-dependent critical exponents

Detected signatures of topological order and non-Abelian fusion channels

Abstract

We generalize the information lattice, originally defined for one-dimensional open-boundary chains, to characterize quantum many-body states in higher-dimensional geometries. In one dimension, the information lattice provides a position- and scale-resolved decomposition of von Neumann information. Its generalization is nontrivial because overlapping subsystems can form loops, allowing multiple regions to encode the same information. This prevents information from being assigned uniquely to any one of them. We address this by introducing a higher-dimensional information lattice in which local information is defined through an inclusion-exclusion principle. The inclusion-exclusion local information is assigned to the lattice vertices, each labeled by subsystem position and scale. We implement this construction explicitly in two dimensions and apply it to a range of many-body ground states with distinct entanglement structures. Within this position- and scale-resolved framework, we extract information-based localization lengths, direction-dependent critical exponents, characteristic edge mode information, long-range information patterns due to topological order, and signatures of non-Abelian fusion channels. Our work establishes a general information-theoretic framework for isolating the universal scale-resolved features of quantum many-body states in higher-dimensional geometries.