Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography

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 $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+\epsilon)$. 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 $g_{ij} \propto \delta_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}\gamma_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $\gamma_{ij}$). 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 $\gamma_{ij}$ 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.