Souriau's Axiomatic Geometric Model of Quantum States: Representation Theory, Positive-Type Functions, Information Geometry, and Extensions Beyond Linear Systems

Syed Muntasir Mamun

Published Jan 1, 2026• 0 citations

Abstract

This paper explores Jean-Marie Souriau's axiomatic framework for quantum mechanics, rooted in representation theory and information geometry, as revived by Frédéric Barbaresco. It delineates quantum states as normalized positive-type functions on dynamical groups, satisfying subordination, Hermitian positivity, and quantization inequalities that yield Hilbert spaces, unitary representations, uncertainty relations, and mixed/Gibbs states. The model unifies classical symplectic geometry with quantum axiomatics, exemplified in linear Heisenberg-Weyl systems producing coherent states. Comparative analysis contrasts Souriau's approach with Kostant's geometric quantization, algebraic QM, and information-geometric metrics (e.g., Balian-Petz), highlighting its group-theoretic strengths and diffeological generality. Extensions propose integrations with infinite-dimensional QFT, quantum thermodynamics, entanglement geometry, and applications in signal processing, machine learning, and quantum computing, forging a cohesive symmetry-driven geometric paradigm for quantum phenomena. Mamun, SM (2026)………………………………..1 Souriau's Axiomatic Geometric Model of Quantum States: Representation Theory, Positive-Type Functions, Information Geometry, and Extensions Beyond Linear Systems Also available on Academia.edu: https://www.academia.edu/attachments/131647634/download_file?s=portfolio

DOI: 10.5281/zenodo.22993442

Related papers

Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I)

Syed Muntasir Mamun

Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio

Mar 29, 20260 citations

Quantum Information Geometric Symmetries: Group Invariance, Maximal Invariants, and Numerical Simulation -KSCCN Volume II

Syed Muntasir Mamun

This Volume II extends the group-invariance framework of Nielsen and Okamura (2026) into quantum information geometry. We provide rigorous definitions for quantum states, Umegaki relative entropy and Petz f-divergences, unitary conjugation actions, the Bures metric, and monotone metrics. We establish the quantum analogues of the classical invariance theorems and derive an explicit maximal-invariant reduction for qubit pairs under simultaneous unitary conjugation: the three scalars (r₁, r₂, cos θ) from Bloch vectors. These form the direct quantum counterpart of the classical relative location + relative scale reduction. A comprehensive QuTiP simulation verifies invariance to machine precision (~10⁻¹³) and demonstrates clean collapse of the quantum relative entropy onto these three invariants. The work unifies the symmetry-reduction principle across classical f-divergences, Fisher–Rao distance, quantum relative entropy, and Bures geometry. Also available on Academia.edu: https://www.academia.edu/attachments/133246712/download_file?s=portfolio

Jan 1, 20260 citations

Quantum Synergies in Nonlinear Optimization: A Comprehensive Review and Extension of Gate-Based Genetic Algorithms and Matrix Subroutine Integrations for Quantum-Enhanced Computational Frameworks

Syed Muntasir Mamun

This paper provides a rigorous academic review of the seminal work by Mamun and Grok xAI (2025) on advancing quantum matrix computations through the integration of geometric mean subroutines and Chebyshev-accelerated chain multiplications for nonlinear and iterative challenges in machine learning and quantum information. Following a thorough summary of its salient contributions, we compare and contrast the approach with contemporary research in quantum genetic algorithms (QGAs), variational quantum eigensolvers (VQEs), and tensor network methods. Strengths in polylogarithmic complexity and hybrid fault-tolerance are highlighted, alongside limitations in NISQ-era decoherence. Extending the framework, we propose novel integrations with fractal-recursive ontologies for non-integer dimensional simulations, the Principle of Co-Adaptation for bivariational dynamics in adaptive quantum circuits, and Souriau's axiomatic geometric model for positive-type state representations in high-dimensional entanglement. These extensions unify quadratic tensor structures with emergent fractal geometries, enabling scalable applications in quantum gravity simulations and autopoietic cognitive systems. Implications for post-quantum cryptography and geopolitical quantum strategies are discussed, advocating for interdisciplinary policies to harness transformative potential while mitigating risks. Also available on Academia.edu: https://www.academia.edu/attachments/131744773/download_file?s=portfolio

Jan 1, 20260 citations

Review of Comparative Geometric Analysis of a Qubit and an RLC Circuit under Time-Embedded and Group-Theoretic Information Geometries

Syed Muntasir Mamun

This review examines the paper by Tomonori Yoshino (2023) on the comparative geometric analysis of a classical RLC circuit and a quantum two-level system (qubit) using two frameworks: group-theoretic information geometry and time-embedded information geometry (TEIG). The original work postulates structural analogies between classical dissipation and quantum decoherence, hypothesizing that TEIG better captures time-dependent and non-Markovian dynamics while group-theoretic geometry excels in symmetry representation. Key results highlight TEIG's ability to incorporate temporal embedding and rupture dynamics, suggesting a unified perspective for classical and quantum systems. Algebraic interpretations involve Lie group actions and time-dependent metrics, while geometric views emphasize manifolds like the Bloch sphere and phase spaces. Secondary research reveals similar efforts in circuit quantum electrodynamics (cQED) and quantum information geometry, indicating ongoing progress toward hybrid classical-quantum models. This review synthesizes these elements, evaluates hypotheses, and discusses future directions in unifying geometric descriptions of physical information systems. Mamun, SM (2026)…………………………………………1 Review of Comparative Geometric Analysis of a Qubit and an RLC Circuit under Time-Embedded and Group-Theoretic Information Geometries Also available on Academia.edu: https://www.academia.edu/attachments/131996529/download_file?s=portfolio

Jan 1, 20260 citations

"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, 20260 citations

Algebraic, Topological, and Mereological Foundations of Quantum Information: A Synthetic Framework for Emergent Geometry, Computational Physics, and Cosmology (Vol-VI)

Syed Muntasir Mamun

This report examines the ontological foundations of quantum mechanics, challenging the notion that geometric space and classical subsystems are primordial axioms of nature. By integrating algebraic quantum mechanics, quantum mereology, discrete homology, and topos theory, the study demonstrates that these features emerge dynamically from purely quantum mechanical precursors. The synthesis utilizes the Gelfand-Naimark-Segal (GNS) construction to derive geometric Hilbert spaces from abstract algebraic relations, effectively relegating the Hilbert space to a derived and contingent structure rather than a fundamental one. The research further employs a "basic shapes" paradigm in discrete homology to efficiently compute the higher-order structures of complex networks, bypassing the exponential computational bottlenecks traditionally associated with high-dimensional topological analysis. By identifying a minimal universal set of atomic configurations, such as the triangular prism or the discrete hypercube, the dimensionality of the boundary matrix is collapsed, achieving median speedups of 47.5x over previous algorithms. KSCCN Mamun, SM (2026) ………………………………….1 Algebraic, Topological, and Mereological Foundations of Quantum Information: A Synthetic Framework for Emergent Geometry, Computational Physics, and Cosmology K S C Furthermore, the report establishes a link between subatomic gauge symmetries and cosmological dark energy through the Axion-Modulated Quintessence framework. Also available on Academia.edu: https://www.academia.edu/attachments/133038377/download_file?s=portfolio

Jan 1, 20260 citations