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

TL;DR

This work addresses how to diagnose metricity of correlation geometry in 1D quantum chains using a calibrated information-distance d_E = K_0 / \sqrt{I}. By uniquely calibrating the map with the Euclidean benchmark I(r) ~ r^{-2} → d_E(r) ~ r, the authors derive a sharp link between the decay of mutual information and metricity: d_E is subadditive (a metric) if and only if 0 < X ≤ 2 in I(r) ~ r^{-X}, while exponential clustering violates metricity. Exact solutions of the 1D TFIM show that at criticality I(r) decays as a power law with X near 2 and the triangle defect Δ(r,r) is asymptotically non-positive, whereas in gapped phases I(r) decays exponentially and Δ(r,r) is positive, confirming the predicted dichotomy. The proposed, parameter-free diagnostic relies only on site-averaged two-site MI and provides a practical, falsifiable tool for phase classification in 1D quantum systems, supplemented by public code and data for replication.

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 \(Δ(r,r)=d_E(2r)-2d_E(r)\) is asymptotically non-positive; in gapped regimes \(I(r)\) decays exponentially and \(Δ(r,r)\gg 0\). The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.

Figures (2)

• Figure 1: Metric dichotomy in the 1D TFIM ($L=2048$). (a) Emergent distance $d_E=I^{-1/2}$ versus chord distance $\rho$. At criticality ($h/J=1$) the data follow $d_E\propto\rho^{p}$ with $p=X/2=0.994(1)$ (95% CI); the dotted line shows the Euclidean reference $p=1$. In the gapped phase ($h/J=2$), $d_E$ grows exponentially (non‑metric). (b) Triangle defect $\Delta(r,r)=d_E(2r)-2\,d_E(r)$ (symlog scale): the gapped phase is superadditive ($\Delta>0$), while the critical data are asymptotically subadditive ($\Delta\le0$); the small positive $\Delta$ at short distances reflects pre‑asymptotic corrections.
• Figure 2: Scaling and finite-size checks.(a) Mutual information $I(r)$ vs chord distance $\rho(r)$ for $L{=}2048$. At criticality ($h{=}J$), $I\!\propto\!\rho^{-X}$ with $X=1.988(1)$, close to the $X{=}2$ benchmark (dotted). In the gapped regimes ($h/J{=}0.5,\,2$), $I$ decays exponentially and truncates at the numerical floor ($10^{-12}$ nats). (b) Critical FSS: $d_E/L$ vs $r/L$ collapses for $L\in\{512,1024,2048\}$, consistent with finite-size scaling for open chains; inset, deviation from the $L{=}2048$ curve is $\lesssim 0.5\%$ for $r/L\gtrsim 0.02$.