Emergent Distance and Metricity of Mutual Information in 1D Quantum Chains

Abstract

We develop and formalize a phase diagnostic based on the information-distance \(d_E = K_0/\sqrt{I}\) (mutual information \(I\)) for 1D quantum chains. Calibrating with the Euclidean benchmark \(I(r)\propto r^{-2}\mapsto d_E(r)\propto r\) makes the triangle-inequality test parameter-free and scale-invariant. Under site-averaged, monotone scaling conditions on the 1D line we establish a criterion linking the decay of \(I(r)\) to metric behavior of \(d_E(r)\): power laws \(I(r)\sim r^{-X}\) with \(0<X\le 2\) yield subadditivity (metric scaling), while exponential clustering leads to superadditivity. As an analytic check complementing our earlier numerical study, we verify these predictions in the 1D transverse-field Ising chain using an exact Jordan-Wigner/Bogoliubov-de Gennes solution: at criticality \(I(r)\) follows a power law close to the \(X=2\) benchmark and the equal-legs triangle defect \(\Delta(r,r)=d_E(2r)-2d_E(r)\) is asymptotically non-positive; in gapped regimes \(I(r)\) decays exponentially and \(\Delta(r,r)\gg 0\). The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.