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arXiv:2607.12559

The Infinitesimal Structure of Quantum Information

Luca Barbieri-Viale

quant-phmath-phmath.AGmath.DGmath.QASubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations• Revised Aug 3, 2026 (v2)
v1 · Jul 14, 2026v2 · Aug 3, 2026
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Abstract

This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras $\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})$, wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As $N \to \infty$, this family converges to a Cauchy-complete power series ring $\mathcal{Q}_\infty$, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.

Comments
Placed the persistent scalar trace background 1/N which defines the exact baricentro of the configuration space representing the maximally mixed state across all dimensions N

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We introduce the Quantum Information Space at Infinity (Quinfinity) Q∞\mathcal{Q}_\inftyQ∞​ as the inverse limit of the symbolic Quantum NNN-Spaces QN\mathcal{Q}_NQN​, identified with the complete local real algebra of formal power series under the adic topology. The pure state pro-variety is formalized as the inverse limit of NNN-level pure state varieties, driven by the algebraic colimit of a direct system of real radical ideals. This real architecture internalizes the imaginary unit, collapsing the metric landscapes onto an invariant Riemannian isometry, which identifies the pro-variety directly with the classical Fisher-Rao statistical manifold. Within this framework, the Heisenberg uncertainty principle undergoes a structural regularization at infinity being intrinsically manifested as a localized truncation obstruction governed by adic algebraic derivations. Concurrently, the continuous Liouville-von Neumann equation operates as an internal derivation tangent to the pro-variety, while the open Gorini-Kossakowski-Sudarshan-Lindblad asymptotic master equation is intrinsically obtained via a non-associative symmetric Jordan product. This dissipative flow acts as a contracting radial vector field that dampens higher-order jet configurations, driving the state trajectories down the hierarchical tree toward the absolute zero element as a non-singular universal attractor.
Mathematical Applications of Science Fiction We present a rigorous derivation of the Quantum FisherInformation Metric (QFIM) within the framework of derivedalgebraic geometry and non-commutative spectral theory.By treating the manifold of quantum states as a modulispace of stable sheaves over a non-commutative C∗-algebra,we establish a canonical isomorphism between the tangentbundle of the statistical manifold and the first derived coho-mology group of the entropic structure sheaf. We introducethe Malcor-Bures Spectral Sequence, which converges to theBures metric, and prove that the entropic dynamics gener-ated by the gradient flow of the von Neumann entropy isgeodesically complete with respect to the Wigner-Yanase-Dyson skew information metric. Furthermore, we analyze thestability of these dynamics using motivic L-functions associ-ated with the modular automorphism group, demonstratingthat the information geometry admits a unique hyper-Kählerstructure in the thermodynamic limit.
Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra J{\mathcal J}J, we exploit the generalized distribution determined by the Jordan product on the dual J⋆{\mathcal J}^{\star}J⋆ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of J{\mathcal J}J, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in J∗{\mathcal J}^{*}J∗ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on J{\mathcal J}J, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of J{\mathcal J}J, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.
This fifth volume completes the core mathematical arc of the series by lifting the classical information-geometric structure on the centre Z(M) (Volume II) to a full quantum Riemannian geometry on the state space of the emergent type-III algebra M. We construct a quantum Fisher information metric on the manifold of normal states of M that is compatible with the direct-integral decomposition and reduces to the classical Fisher metric on the centre. The resulting Riemannian structure is shown to be invariant under the modular flow and to encode the fractal dimension of the avalanche tree (Volume IV) as a curvature invariant. Connections are drawn to recent developments in quantum information geometry on non-commutative algebras and to the programme of fractal quantum states. The unified Riemannian picture furnishes a geometric characterisation of the algebraic Heisenberg cut as the locus where the quantum metric degenerates to its classical projection. All constructions remain anchored in the avalanche-photodiode model while providing a coherent geometric language that unifies the algebraic, modular, fractal and information-geometric threads developed across the series. K 1 S Also available on Academia.edu: https://www.academia.edu/attachments/133072858/download_file?s=portfolio
We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over C∗C^*C∗-algebras. Since the state space of a C∗C^*C∗-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor KKK on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor GGG and a smooth two-form ΩΩΩ. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher--Rao metric in the commutative dominated case, the Fubini--Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form ΩΩΩ is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that ΩΩΩ need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of ΩΩΩ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.
We unify two emergent aspects of quantum theory within a single information-geometricframework based on the Araki relative entropy and the associated Bogoliubov–Kubo–Mori(BKM) metric on a von Neumann algebra state manifold.First, we show that in the commutative reduction to a spatial density subalgebra, theBKM metric becomes the classical Fisher information metric. Its trace gives, after a singlenormalization choice, the expectation value of the Bohmian quantum potential, and thecorresponding Bohmian trajectories arise as Jacobi geodesics of a conformally related metric.The guidance equation is then recovered from the polar decomposition of states satisfyingthe emergent Schr¨odinger equation.Second, we show that the modular flow of the reference state yields the Heisenbergand von Neumann equations, and that the physical Schr¨odinger equation follows under astandard unitary state implementation. In the finite regulated K¨ahler sector, geometricprequantization of the BKM state manifold leads to a discrete horizon area spectrum and touncertainty relations, with an induced gravitational coupling derived from the Seeley–DeWittcoefficient [1].No continuum limit is taken, and no identification with the observed Newton constantis assumed. The results exhibit a common information-geometric framework for Bohmianmechanics, quantum dynamics, and emergent gravitational structures.