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.
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.
Whether the spatial geometry implied by a quantum state can be recovered from its entanglementstructure is a central question in the “space from Hilbert space” program. Westudy the sharpest tractable version of this question: can equal-time mutual information(MI) detect metric inhomogeneity, as opposed to topology and dimensionality alone? Workingwith inhomogeneous critical free fermions, whose emergent curved-CFT metric is knownin closed form, we derive an exact expression for the Weyl-dressed conformal separationbetween two points, D/r = e[ϕ(a)+ϕ(b)]/2⟨e−ϕ⟩ with ϕ = ln v, and expand it in cumulantsof ϕ. The leading term is δ1 = 12 [ϕ(a) + ϕ(b)] − ⟨ϕ⟩, which vanishes identically for linearϕ and is small for many gently varying monotonic profiles; when it vanishes, the leadingresponse degrades to 12 Var(ϕ). This explains a practical obstruction: the standard linear-vand linear-ln v ramp experiments are second-order-only probes of the emergent metric, andwe verify numerically that it improves predictive variance over plain lattice distance by onlyΔR2 ≈ 4 × 10−3. The criterion is constructive. Profiles that place the inhomogeneity inthe interval interior while matching endpoint velocities restore first-order sensitivity. Exactfree-fermion computations on chains of N = 240 sites confirm the resulting predictions: MIis suppressed across a velocity valley and enhanced across a velocity bump at identical latticeseparation, a sign reversal no lattice-distance model can produce; and 42 profiles drawnfrom four functional families collapse onto a single line ln[Iprofile/Iflat] = −Xresp δ1 withXresp = 1.64 and R2 = 0.9969, improving to R2 = 0.9991 when the second-order cumulantis 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 designrule for experiments that respect it.
A local measurement of a critical many-body state produces exponentially many microscopic outcomes, each defining a different conditioned quantum state. We ask how much information from one such outcome is actually needed to predict the entanglement that remains between two unmeasured regions. We show that, in critical quantum chains, the universal outcome-resolved Rényi entropy is asymptotically determined by a single compact conformal coordinate extracted directly from the measurement record. Compact-boson theory explains this compression: the universal entropy is insensitive to all spatially varying components of the boundary profile, retaining only a single relative compact zero mode. At the free-fermion XX point, exact lattice identities expose the microscopic structure underlying the decoder, while large-scale outcome-by-outcome calculations confirm its scaling prediction across different subsystem geometries and Rényi indices. Numerical calculations in interacting XXZ chains further show that the same microscopic coordinate continues to decode the conditional entropy without interaction-dependent retuning. Thus an exponentially large space of measurement outcomes is reduced, for this universal observable, to a single conformal degree of freedom, revealing an emergent Luttinger-liquid mechanism for outcome-resolved postmeasurement entanglement.
This paper studies mutual-information-induced graph Laplacians as a diagnostic of quantum phase structure across multiple models of quantum matter. Using exact diagonalization for system sizes N up to 20, and independent DMRG cross-validation up to N=14, mutual information matrices are computed from the ground states of the XXZ chain, the transverse-field Ising model, a classical spin-glass Hamiltonian, and free fermions, and geometric invariants are extracted from the normalized Laplacian spectrum. A classical spin-glass control produces identically zero mutual information, demonstrating that the pipeline does not manufacture spurious structure when correlations are absent. At the two critical points studied, the exact large-N data reveals a genuine, boundary-condition-robust qualitative distinction between the two universality classes: the XXZ spectral gap is fit far better by a form with a logarithmic finite-size correction characteristic of the SU(2) point's marginally irrelevant operator than by a bare power law, while the transverse-field Ising model gap follows the expected power law cleanly. Every code path is covered by an automated test suite that checks each numerical claim in the paper against its underlying data file.
Abstract Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.
A companion paper, The Gravitational Baseline Principle, proposes a minimal pre-geometric quantum substrate and identifies two precise technical conditions that established theorems require before continuum geometry can be said to emerge from it: linear, Lorentz-consistent dispersion, and a mutual-information matrix that encodes a genuine metric distance. This paper tests both directly by exact diagonalization on the critical spin-1/2 antiferromagnetic Heisenberg chain, with a gapped or dimerized chain as a control. The dispersion condition is confirmed to high precision, matching the exact des Cloizeaux-Pearson result and decisively excluding a quadratic alternative. The metric-encoding condition is falsified in its originally hoped-for Euclidean form but not featurelessly: the specific functional form points toward a hyperbolic alternative, confirmed directly via the standard mathematical definition of hyperbolicity, the Gromov four-point delta, rather than a curve fit. An attempt to extend this to a 2D substrate is reported honestly as inconclusive at reachable system sizes, along with a genuine methodological finding: geometrically frustrated lattice shapes can have degenerate ground states that make naive mutual-information calculations non-reproducible unless explicitly checked. A final test sweeps several small dimerization strengths, checking whether the crossover away from hyperbolicity tracks the correlation length's own predicted field-theory scaling. The correlation length itself confirms the predicted scaling cleanly, but whether the geometric diagnostic's own breakdown tracks the same scale is only weakly supported and reported as an open result, not a positive one.
The Parity-Entanglement Bound (PEB) fixes the clock-system mutual information of a ℤ₂ constrained bipartite pure state at exactly two bits. We show that this invariant functions as the zero point of a metric geometry built from information. Six results are given. (i) The deviation Π = 1 - 1/2 I is the unique local-unitary (LU) invariant of the minimal state (pure, ℤ₂, bipartite); relaxing each hypothesis adds a definite number of further axes, which we count numerically (mixedness +1, ℤn+(n-2), tripartite +2; a continuous gauge group opens a continuous axis), (ii) The correlation quantity d = -ln κ is non-negative, symmetric, and vanishes in the coincidence limit; in the gapped regime, at separations above a floor r*(m), it obeys the triangle inequality and is a metric. (iii) The validity window is mapped: the floor r*(m) grows with the gap and the window closes entirely once the correlation length falls below the lattice scale (m ≥ 1.7 in 1D, ≥ 2 in 2D, lattice units); in the opposite, gapless limit the inequality survives with growing slack (0 violations atm m = 0.02) and an asymptotic argument establishes it analytically in the logarithmic regime. Non-Gaussianity per se does not damage the inequality: on a φ⁴ chain the observed violations are reproduced quantitatively by a free field of matched effective gap. (iv) Local noise acts on d as an exactly additive per-region dilation, d → d + δA + δB with δ ≥ 0; pure dephasing leaves d exactly invariant, and local noise can only increase the triangle slack, so the metric is noise-robust by construction. (v) The PEB value equals a combinatorial crossing number of stabiliser generators, hence is independent of system size, clock size, and clock position; it is formally analogous to a topological entanglement entropy. (vi) Following the Dirac-equation construction [6], the two-dimensionality dim ℋC = 2 underlies a distance/time split. The construction is verified by exact and Monte-Carlo simulation (geodesic locking 60/60 across five seeds; zero triangle violations above the floor in more than 1.6 × 10⁴ triples spanning gapped, near-critical, interacting, and noisy regimes) and, as a proof of concept, by a state-preparation study on quantum hardware (IBM FEZ). Formal correspondences to gravitational behaviour geodesic bending in a non-uniform gap and a volume-to-area transition in the crossing count are recorded in Sec. no gravitational derivation is claimed. Statements are labelled [P] proved, [K] known-with-citation, [N] numerical, or [0] open.