Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)$. We prove that this map yields a smooth Riemannian structure $g_{ij}$ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent $X_{\text{loc}}=2$). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness $g_{ij} \propto δ_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}γ_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $γ_{ij}$). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor $γ_{ij}$ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining $X_{\text{loc}}=2$ branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims.
Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale Cvac: ds2=Cvac/I(x,x+ε). We prove that this map yields a smooth Riemannian structure gij if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent Xloc=2). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness gij∝δij, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split gij=V2/Dγij, which rigorously separates local density fluctuations (volume V) from directional anisotropy (shape/shear γij). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor γij quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining Xloc=2 branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims.
We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance dE=K0/I, where I is the mutual information. We define a geometric conversion factor, G≡∂rdE, which quantifies the local scale. We show that G relates directly to I via G∝Iκ. For systems with power-law correlations I(r)∼r−X, the metric scaling exponent is κ=1/X−1/2. A key consequence is that the geometric scale G is uniform (position-independent) if and only if κ=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≈2) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of logG versus logI at the XX benchmark (X=2) yields κ≃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.
The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\ensuremath{\tau}$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\ensuremath{\tau}$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\ensuremath{\tau}$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.
We extend the geometric tracking infrastructures established in preceding works to formalize the strict coordinate and dynamical limitations that govern parameterized quantum steering protocols. By constructing a differential-geometric framework mapping smooth control parameters directly to the projective Hilbert space, we isolate the architectural conditions under which optimization protocols undergo structural failure. First, we state and prove the Control Immersion Criterion, demonstrating that any trajectory intersecting a localized coordinate caustic interface triggers a sharp rank deficiency in the pulled-back information metric tensor. This metric non-invertibility forces a catastrophic, asymptotic divergence in the local Levi Civita Christoffel connection symbols (Γmij → ∞), completely destabilizing gradient-driven updates. Second, we evaluate the global phase-space volume element under cyclic parameters, proving that the anti-Hermiticity of the connection generator ensures a strictly real-valued Berry connection and subsequent structural closure of the modified symplectic form (dω=0), maintaining topological conservation despite severe localized metric warping. Finally, we derive a sharp kinematic speed limit bounding non-adiabatic escape rates out of targeted operating subspaces as a direct penalty scaled by the metric tensor. Crucially, this bound exposes an operational duality linking statistical estimation constraints to state-leakage rates through the Quantum Fisher Information Metric structure (gQF IM = 4gij ). Dense grid calculations and dynamic fourth-order Runge-Kutta numerical simulations confirm these analytical constraints down to the machine precision floor, validating this framework for structural pre-mapping routines inside advanced, singularity-aware quantum compilation engines.
Characterizing the nonlocal nature of quantum states is a central challenge in the practical application of large-scale quantum computation and simulation. Quantum mutual information (QMI), a fundamental nonlocal measure, plays a key role in quantifying entanglement and has become increasingly important in studying nonequilibrium quantum many-body phenomena, such as many-body localization and thermalization. However, experimental measurement of QMI remains extremely difficult, particularly for nonequilibrium states, which are more complex than ground states. In this Letter, we employ a multilayer perceptron (MLP) to establish a universal mapping between the QMI and local correlations only up to second order for nonequilibrium states generated by quenches in a one-dimensional disordered XXZ model. Our approach provides a practical method for experimentally extracting QMI, readily applicable in platforms such as superconducting qubits. Moreover, this work will establishes a general framework for reconstructing other nonlocal observables, including Fisher information and out-of-time-ordered correlators.
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.