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

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.

💡 Research Summary

The authors present a comprehensive extension of the information lattice—a framework that decomposes quantum information into position‑ and scale‑resolved contributions—from one‑dimensional (1D) open chains to higher‑dimensional lattices. In 1D, each contiguous block of sites Cℓⁿ (centered at position n with length ℓ+1) is associated with a vertex on a triangular lattice, and the local information iℓⁿ is defined by an inclusion‑exclusion subtraction of von‑Neumann information of its sub‑blocks. This quantity equals a quantum conditional mutual information and is guaranteed to be non‑negative by strong subadditivity.

In higher dimensions the main obstacle is “overlap redundancy”: different overlapping regions can encode the same bits of information, and their intersections are generally disconnected. To overcome this, the authors restrict the set of subsystems to convex shapes and require closure under intersections, ensuring that any intersection is itself a lattice vertex. With this structure the local information is defined by a Möbius inversion formula (Eq. 6):

iℓⁿ = I(ρCℓⁿ) – Σ₁ I(ρAᵢ) + Σ₂ I(ρAᵢ∩Aⱼ) – Σ₃ I(ρAᵢ∩Aⱼ∩Aₖ) + …

where the Aᵢ are the immediate “children” of Cℓⁿ obtained by removing one layer of sites along any coordinate direction. This inclusion‑exclusion local information (shortened to local information) uniquely attributes bits to a given position and scale, even in the presence of loops of overlapping regions.

The paper implements the construction explicitly for two‑dimensional square lattices using rectangular subsystems labeled by position (nₓ,nᵧ) and scale (ℓₓ,ℓᵧ). The resulting lattice lives in a four‑dimensional index space (two spatial, two scale dimensions). The authors then apply the framework to several representative many‑body ground states, demonstrating its diagnostic power:

1. Localized phase – In a disordered insulating model the local information decays exponentially with distance, allowing a precise extraction of an information‑based localization length ξ₁.

2. Critical phase with a Fermi surface – For a 2D free‑fermion metal the scale‑resolved information exhibits power‑law decay with direction‑dependent exponents αₓ and αᵧ, reflecting anisotropic Fermi‑surface geometry.

3. Chiral edge mode – In a Chern insulator the lattice reveals a pronounced peak of local information confined to vertices along the physical edge, providing a quantitative measure of edge‑state contribution that is absent in bulk entanglement entropy.

4. Topologically ordered state – Using the Kitaev toric‑code model, the authors show that bulk topological order manifests as a uniform, scale‑independent background of local information, while the presence of non‑Abelian anyon defects creates localized patterns that encode the fusion channels. These patterns are invisible to conventional entropy‑based topological entanglement entropy but are captured by the inclusion‑exclusion lattice.

Beyond static diagnostics, the authors discuss the dynamical aspect: the local information obeys a continuity‑like equation, with “information currents” flowing between neighboring vertices under unitary evolution. This hydrodynamic picture parallels recent work on information quasiparticles and suggests new algorithms for simulating non‑equilibrium dynamics by truncating long‑range information currents.

The paper also acknowledges computational challenges: the number of inclusion‑exclusion terms grows combinatorially with subsystem size, and extending the method to non‑convex or irregular geometries remains an open problem. The authors propose future directions such as machine‑learning‑based approximations of higher‑order intersections, multi‑scale compression schemes, and connections to higher‑dimensional topological invariants.

In summary, the work establishes a robust, geometry‑agnostic framework for assigning quantum information to specific positions and scales in higher‑dimensional many‑body systems. By resolving overlap redundancies through inclusion‑exclusion, it enables quantitative extraction of localization lengths, direction‑dependent critical exponents, edge‑mode signatures, long‑range topological patterns, and non‑Abelian fusion information—features that are difficult or impossible to discern from traditional entanglement measures alone. This advances both our conceptual understanding of quantum correlations in complex geometries and provides practical tools for analyzing and simulating high‑dimensional quantum matter.