Scaling of a Mutual-Information Distance in One-dimensional Quantum Spin Chains

We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance $d_E = K_0/\sqrt{I}$, where $I$ is the mutual information. We define a geometric conversion factor, $G \equiv \partial_r d_E$, which quantifies the local scale. We show that $G$ relates directly to $I$ via $G \propto I^{\kappa}$. For systems with power-law correlations $I(r) \sim r^{-X}$, the metric scaling exponent is $\kappa = 1/X - 1/2$. A key consequence is that the geometric scale $G$ is uniform (position-independent) if and only if $\kappa = 0$, which occurs precisely at $X = 2$. This identifies $X = 2$ as the unique condition for a uniform and metric informational distance. We validate this relation using DMRG simulations of the 1D XXZ chain and exact results for the XX model. We demonstrate two falsifiable diagnostics: (i) $G(r)$ is flat in the bulk at criticality ($X \approx 2$) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of $\log G$ versus $\log I$ at the XX benchmark ($X = 2$) yields $\kappa \simeq 0$. This approach provides a coordinate-independent method for identifying scaling regimes that helps to reduce ambiguity from non-universal amplitudes and from the fitting choices in standard power-law analyses, and defines a simple post-processing pipeline that can be applied directly to numerical or experimental mutual-information data.