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

• URL: http://arxiv.org/abs/2507.09749v2
• Date: Tue, 04 Nov 2025 01:16:03 GMT
• Title: Emergent Distance and Metricity of Mutual Information in 1D Quantum Chains
• Authors: Beau Leighton-Trudel,
• Abstract summary: Calibrating with the Euclidean benchmark (I(r)propto r-2mapsto d_E(r)propto r) makes the triangle-inequality test parameter-free and scale-invariant.<n>We establish a criterion linking the decay of (I(r)) to metric behavior of (d_E(r)): power laws (I(r)) with (0Xle 2) yield subadditivity (metric scaling), while exponential clustering leads to superaddit
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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