An identifiability criterion for emergent conformal geometry in mutual information of inhomogeneous critical chains

Abstract

Abstract Whether the spatial geometry implied by a quantum state can be recovered from its entanglement structure is a central question in the "space from Hilbert space" program. We study the sharpest tractable version of this question: can equal-time mutual information (MI) detect metric inhomogeneity, as opposed to topology and dimensionality alone? Working with inhomogeneous critical free fermions, whose emergent curved-CFT metric is known in closed form, we derive an exact expression for the Weyl-dressed conformal separation between two points, D/r = e^([ϕ(a)+ϕ(b)]/2) ⟨e^(−ϕ)⟩ with ϕ = ln v, and expand it in cumulants of ϕ. The leading term is δ1 = 1/2 [ϕ(a) + ϕ(b)] − ⟨ϕ⟩, which vanishes identically for linear ϕ and is small for many gently varying monotonic profiles; when it vanishes, the leading response degrades to 1/2 Var(ϕ). This explains a practical obstruction: the standard linear-v and linear-ln v ramp experiments are second-order-only probes of the emergent metric, and we verify numerically that it improves predictive variance over plain lattice distance by only ∆R^2 ≈ 4 × 10^(−3). The criterion is constructive. Profiles that place the inhomogeneity in the interval interior while matching endpoint velocities restore first-order sensitivity. Exact free-fermion computations on chains of N = 240 sites confirm the resulting predictions: MI is suppressed across a velocity valley and enhanced across a velocity bump at identical lattice separation, a sign reversal no lattice-distance model can produce; and 42 profiles drawn from four functional families collapse onto a single line ln[I_profile/I_flat] = −X_resp δ1 with X_resp = 1.64 and R^2 = 0.9969, improving to R^2 = 0.9991 when the second-order cumulant is included. We conclude that equal-time MI does encode the emergent conformal geometry, but that identifiability—not signal strength—is the binding constraint, and we give a design rule for experiments that respect it. Repository Contents & Replication Scripts This repository provides the complete, standalone codebase used to execute the exact free-fermion correlation-matrix computations and regenerate all numerical data tables, out-of-sample family cross-validations, and data collapse figures presented in the text. manuscript.pdf: Complete unblinded research article text. free_fermion_sim.py: Simulates the inhomogeneous hopping (XX) chains across N=240 site geometries and computes the exact fermionic mutual information matrices via Peschel's correlation matrix method. cumulant_analysis.py: Extracts the response coordinate coordinates, maps the profile variants (Gaussian, sech², Lorentzian, raised cosine), and handles the multi-family data collapse regression profiles. Environment Requirements: Python 3.8+ with standard scientific computing installations (numpy, scipy, matplotlib). Double-Blind Review & Anonymity Compliance Author names, institutional affiliations, contact emails, and regional funding details are permanently withheld from these files to guarantee total anonymity. Content processing and textual structuring were partially optimized using conversational generative AI models, with all underlying mathematical derivations, code architecture, and numerical assertions independently validated by the author.