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arXiv:2603.10054

Quantization of Ricci Curvature in Information Geometry

Carlos C. Rodriguez

cs.ITcs.LGquant-phSubmitted Mar 9, 2026• Announced Mar 10, 2026
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Abstract

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.

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15 pages, 3 tables

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In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples.
We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.
We introduce a quantum algorithm for computing the Ollivier Ricci curvature, a discrete analogue of the Ricci curvature defined via optimal transport on graphs and general metric spaces. This curvature has seen applications ranging from signaling fragility in financial networks to serving as basic quantities in combinatorial quantum gravity. For inputs given as a point cloud with pairwise distances, we show that our algorithm can achieve an exponential speedup over the best-known classical methods for two particular classes of problem. Our work is another step toward quantum algorithms for geometrical problems that are capable of delivering practical value while also informing fundamental theory.
This paper presents the full, step-by-step construction of the Information Geometry Unification Principle. We start with three rigorous axioms defining quantum information as the fundamental ontology. We derive classical Fisher information geometry, quantum Bures geometry, Levi-Civita connection, Riemann curvature, Ricci tensor, and scalar curvature in full detail. We construct the universal information action and perform complete variational calculus with respect to the quantum state ρ\rhoρ and the information metric gμνg_{\mu\nu}gμν​. Every algebraic step is explicitly shown. We recover Schrödinger mechanics, statistical mechanics, and Einstein gravity as limiting cases. This paper is self-contained and requires no external reference to follow the derivations.
We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a ddd-dimensional lattice with periodic boundary conditions and n=Ldn = L^dn=Ld sites, the Fisher manifold has m=d⋅nm = d \cdot nm=d⋅n dimensions (one per bond), and we find ∣R(Jc)∣∼ndR|\mathcal{R}(J_c)| \sim n^{d_R}∣R(Jc​)∣∼ndR​ with dR=(dν+2η)/(dν+η)d_R = (dν+ 2η)/(dν+ η)dR​=(dν+2η)/(dν+η), where ννν and ηηη are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν=1ν= 1ν=1, η=1/4η= 1/4η=1/4), this predicts dR=10/9d_R = 10/9dR​=10/9, confirmed by exact transfer-matrix computations (L=6L = 6L=6--999: dR=1.1115±0.0002d_R = 1.1115 \pm 0.0002dR​=1.1115±0.0002) and multi-seed MCMC through L=24L = 24L=24. For 3D Ising (ν=0.630ν= 0.630ν=0.630, η=0.0363η= 0.0363η=0.0363), the prediction dR=1.019d_R = 1.019dR​=1.019 is consistent with MCMC on L3L^3L3 tori up to L=10L = 10L=10 (power-law fit: dR=1.040d_R = 1.040dR​=1.040). For 2D Potts q=3q = 3q=3 (predicted 33/29≈1.13833/29 \approx 1.13833/29≈1.138), FFT-MCMC through L=40L = 40L=40 shows deffd_\mathrm{eff}deff​ oscillating non-monotonically around ∼1.20\sim 1.20∼1.20, consistent with O(1/(ln⁡L)2)O(1/(\ln L)^2)O(1/(lnL)2) logarithmic corrections. For q=4q = 4q=4 (predicted 22/1922/1922/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R3=−R1/2R_3 = -R_1/2R3​=−R1​/2, R4=−R2/2R_4 = -R_2/2R4​=−R2​/2 holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.
The classical Gaussian model, a parametric family of the Gaussian distribution, is known to be a space of constant negative curvature if one regards the Fisher information on the model as a Riemannian metric. Constant curvature reflects the scale invariance of the classical Gaussian model, which is well known in information geometry. However, it is shown that if the Kubo–Mori–Bogoliubov Fisher information on the quantum Gaussian model is adopted as a Riemannian metric, then this scale invariance on the Gaussian model is broken due to the quantum effect. In the present study, the connection between the geometry of the classical Gaussian model and its quantum counterpart is clarified using the Taylor expansion with respect to the Planck constant. It is further shown that such a method is not applicable to a finite-dimensional system such as the spin system.
Information theoretic geometry near critical points in classical and quantum systems is well understood for exactly solvable systems. Here, we show that renormalization group flow equations can be used to construct the information metric and its associated quantities near criticality for both classical and quantum systems in a universal manner. We study this metric in various cases and establish its scaling properties in several generic examples. Scaling relations on the parameter manifold involving scalar quantities are studied, and scaling exponents are identified. The meaning of the scalar curvature and the invariant geodesic distance in information geometry is established and substantiated from a renormalization group perspective.
Information theoretic geometry near critical points in classical and quantum systems is well understood for exactly solvable systems. Here we show that renormalization group flow equations can be used to construct the information metric and its associated quantities near criticality, for both classical and quantum systems, in an universal manner. We study this metric in various cases and establish its scaling properties in several generic examples. Scaling relations on the parameter manifold involving scalar quantities are studied, and scaling exponents are identified. The meaning of the scalar curvature and the invariant geodesic distance in information geometry is established and substantiated from a renormalization group perspective.