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arXiv:2608.27498

Quantum Information at Infinity

L. Barbieri-Viale

quant-phmath-phmath.AGmath.QASubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

We introduce the Quantum Information Space at Infinity (Quinfinity) $\mathcal{Q}_\infty$ as the inverse limit of the symbolic Quantum $N$-Spaces $\mathcal{Q}_N$, 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 $N$-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.

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The Infinitesimal Structure of Quantum Information

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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.

quant-pharXiv:2607.12559•Jul 15, 2026•0 citations

Objective Phase-Space Quantum Information-Capacity Mechanics--An Information Ontology and Mathematical Framework for Quantum Mechanics

Fangzhou Jin

Standard quantum interpretations-based on point trajectories, branching wave functions, or instantaneous projection-rely on two continuum idealizations: infinite spatial information density and infinite measurement bandwidth. A central foundational question is whether complex Hilbert-space kinematics, noncommuting observables, and single-outcome collapse follow from real physical fields under finite information capacity and bandwidth. We formulate Objective Phase-Space Quantum Information-Capacity Mechanics (QICM), specifying the state on a configuration manifold by a real probability density and a compact flow potential. Three axioms govern the theory: the first requires finite Fisher information and defines the potential on a flat circle bundle with quantized holonomy and half-period nodal jumps, resolving Wallstrom's objection; the second sets an elementary resolution quantum from the Bloch-sphere Fisher solid angle; and the third pairs Fisher localization energy with a Nyquist-Planck capacity bound in stochastic equations for closed flow and continuous measurement. On the reduced tangent space, the symplectic form, Fisher-Rao metric, and complex structure form a Kahler geometry inducing the Hermitian inner product, self-adjoint operators, and Heisenberg commutators. The axioms yield uncertainty bounds, the minimum resolution time, oscillator and hydrogen spectra, martingale Born-rule collapse, and the Lindblad equation, and extend to relativistic fields, half-integer spin, Pauli pressure, Compton-scale ultraviolet regularization, and ensemble no-signaling. QICM thus unifies unitary evolution and collapse in a single-world field dynamics without superluminal signaling, yielding testable bounds on compression times, readout horizons, and short-pulse thresholds in circuit-QED, optomechanics, and trapped ions.

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The Variational-Spectral-GKSL Closed Dynamical Theory of a Quantum-Information Field

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Creator:Vitaly Bazarov (Vit-Baz) https://orcid.org/0009-0007-9967-5807 This paper presents a universal, mathematically closed dynamical framework (UAU–ESVG–SAAU–Σ*) that bridges variational calculus, quantum-information geometry, and algebraic quantum field theory. We formulate the evolution of a physical-computational universe as a dissipative gradient flow over a hybrid Wasserstein-Hilbert manifold, rigorously linearized via the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator and Koopman spectral theory. A central theoretical breakthrough of this work is the re-formalization of physical time: time is not postulated as an independent external parameter, but derived emergently as a geometric foliation of the dissipative, non-unitary quantum-information flow over the $C^*$-algebra of observables. We prove the existence of a "timeless invariant core" ($\mathcal{A}^*$) and a strictly defined causal arrow driven by entropy minimization. Furthermore, we provide exact analytic and numerical stationary Gaussian solutions for the $d=2$ spatial reduction. These solutions demonstrate a critical phase transition—an "information black hole" limit ($S_{BH}$) where macroscopic scale collapses, the Fisher information metric diverges, and the embedded observer's proper time freezes. Finally, the system's ability to self-modify its own dynamical laws is closed via an $\omega$-categorical meta-dynamical limit. This framework provides a fully computable foundation for advanced thermodynamic, cosmological, and autonomous AGI-level computational systems.

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Quantum Information in Geometric Quantum Mechanics

G. F. Volkert

A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.

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"Heading towards an Algebraic Heisenberg Cut": Sectorisation, Self-Decoherence, and the Algebraic Foundations of the Quantum–Classical Transition – Volume V: Lifting the Information-Geometric Metric to a Quantum Riemannian Structure on Type-III Algebras and Fractal Quantum States V

Syed Muntasir Mamun

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

•Jun 11, 2026•0 citations

A Lie-Jordan Geometric Formulation of Lindblad Dynamics

Leonel Bixano, Victor Alberto Cruz-Barriguete, Guillermo López-Alvarez +3 more

We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors \(C_{μν}{}^λ\) and \(B_{μν}{}^λ\). Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, \( \mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\}, \) whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations \((B+C)\) and \((B-C)\) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.

quant-pharXiv:2607.04052•Jul 7, 2026•0 citations
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.
This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras QN≡R[ε]/(εN2−1)\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})QN​≡R[ε]/(εN2−1), 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→∞N \to \inftyN→∞, this family converges to a Cauchy-complete power series ring Q∞\mathcal{Q}_\inftyQ∞​, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.
Standard quantum interpretations-based on point trajectories, branching wave functions, or instantaneous projection-rely on two continuum idealizations: infinite spatial information density and infinite measurement bandwidth. A central foundational question is whether complex Hilbert-space kinematics, noncommuting observables, and single-outcome collapse follow from real physical fields under finite information capacity and bandwidth. We formulate Objective Phase-Space Quantum Information-Capacity Mechanics (QICM), specifying the state on a configuration manifold by a real probability density and a compact flow potential. Three axioms govern the theory: the first requires finite Fisher information and defines the potential on a flat circle bundle with quantized holonomy and half-period nodal jumps, resolving Wallstrom's objection; the second sets an elementary resolution quantum from the Bloch-sphere Fisher solid angle; and the third pairs Fisher localization energy with a Nyquist-Planck capacity bound in stochastic equations for closed flow and continuous measurement. On the reduced tangent space, the symplectic form, Fisher-Rao metric, and complex structure form a Kahler geometry inducing the Hermitian inner product, self-adjoint operators, and Heisenberg commutators. The axioms yield uncertainty bounds, the minimum resolution time, oscillator and hydrogen spectra, martingale Born-rule collapse, and the Lindblad equation, and extend to relativistic fields, half-integer spin, Pauli pressure, Compton-scale ultraviolet regularization, and ensemble no-signaling. QICM thus unifies unitary evolution and collapse in a single-world field dynamics without superluminal signaling, yielding testable bounds on compression times, readout horizons, and short-pulse thresholds in circuit-QED, optomechanics, and trapped ions.
Creator:Vitaly Bazarov (Vit-Baz) https://orcid.org/0009-0007-9967-5807 This paper presents a universal, mathematically closed dynamical framework (UAU–ESVG–SAAU–Σ*) that bridges variational calculus, quantum-information geometry, and algebraic quantum field theory. We formulate the evolution of a physical-computational universe as a dissipative gradient flow over a hybrid Wasserstein-Hilbert manifold, rigorously linearized via the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator and Koopman spectral theory. A central theoretical breakthrough of this work is the re-formalization of physical time: time is not postulated as an independent external parameter, but derived emergently as a geometric foliation of the dissipative, non-unitary quantum-information flow over the C∗C^*C∗-algebra of observables. We prove the existence of a "timeless invariant core" (A∗\mathcal{A}^*A∗) and a strictly defined causal arrow driven by entropy minimization. Furthermore, we provide exact analytic and numerical stationary Gaussian solutions for the d=2d=2d=2 spatial reduction. These solutions demonstrate a critical phase transition—an "information black hole" limit (SBHS_{BH}SBH​) where macroscopic scale collapses, the Fisher information metric diverges, and the embedded observer's proper time freezes. Finally, the system's ability to self-modify its own dynamical laws is closed via an ω\omegaω-categorical meta-dynamical limit. This framework provides a fully computable foundation for advanced thermodynamic, cosmological, and autonomous AGI-level computational systems.
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.
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 Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors CμνλC_{μν}{}^λCμν​λ and BμνλB_{μν}{}^λBμν​λ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−12{YX,Z}, \mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\}, D(X,Y)Z=XZY−21​{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C)(B+C)(B+C) and (B−C)(B-C)(B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.