Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography

TL;DR

This work addresses how to characterize anisotropic correlations in a coordinate-invariant way by mapping short-distance mutual information to a local Riemannian metric. The authors show that a smooth metric emerges only when the MI follows an anisotropic inverse-quadratic law with $X_{loc}=2$, and they decompose the metric as $g_{ij} = \mathcal{V}^{2/D}\gamma_{ij}$ to separate volume from shape/shear. They introduce MI Tomography to reconstruct the local metric from directional MI measurements and validate it on an anisotropic 2D quantum harmonic lattice, where the recovered shape tensor $\gamma_{ij}$ quantitatively reflects the microscopic anisotropy $k_y/k_x$. This framework provides a coordinate-invariant tensor geometry of correlations and a practical protocol for probing anisotropic criticality and nematic-like order in quantum and statistical systems, with open-source code and data to enable broader use.

Abstract

Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)$. We prove that this map yields a smooth Riemannian structure $g_{ij}$ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent $X_{\text{loc}}=2$). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness $g_{ij} \propto δ_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}γ_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $γ_{ij}$). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor $γ_{ij}$ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining $X_{\text{loc}}=2$ branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims.

Figures (1)

• Figure 1: Mutual information tomography on the ground state of an anisotropic 2D quantum harmonic lattice (free scalar). (a) Reconstructed unimodular shape anisotropy, quantified by the eigenvalue ratio $\lambda_x/\lambda_y$ of the shape tensor $\gamma_{ij}$, versus the microscopic coupling ratio $k_y/k_x$ for an $L=2048$ lattice with $m=10^{-3}$, $\ell=1$, and sampling radius $\rho=50$. Open symbols denote all validated runs; filled symbols highlight the "core" subset with $|X_{\mathrm{loc}}-2|\le 0.05$ and DT2 $\le 1\%$. The dashed line is the parameter-free prediction $g_{ij}\propto h_{ij}$ with $h_{ij}\propto\mathrm{diag}(1/k_x,1/k_y)$. (b) Diagnostics for the same runs: normalized DT2 polarization residual (squares) and eigenframe misalignment relative to the lattice axes (triangles), plotted as functions of $k_y/k_x$. Horizontal dashed lines indicate the tolerance thresholds DT2 $=1\%$ and misalignment $=0.5^\circ$ used to define the core subset.