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
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
This research report presents a comprehensive topos-theoretic architecture to achieve the rigorous unification of classical dissipative systems, exemplified by the Resistor-Inductor-Capacitor (RLC) circuit, and open quantum systems, represented by the qubit. Historically, these domains rely on disparate formalisms (ODEs and symplectic geometry versus complex Hilbert spaces and the Lindblad master equation). By deploying a foundational shift to Topos Theory, the work synthesizes Olivia Caramello's "Toposes as Bridges" methodology, the Time-Embedded Information Geometry (TEIG) framework, and the Isham-Döring topos foundation of quantum physics. The central finding is that classical resistance and quantum decoherence are revealed as strictly Morita-equivalent manifestations of a single geometric invariant residing within a unified Classifying Topos of Dissipative Dynamics. The architecture employs relative toposes to elegantly encapsulate infinite, chaotic environmental noise and utilizes geometric logic, supplemented by categorical Picard-Vessiot theory, to translate continuous differential equations into verifiable logical sequents. Also available on Academia.edu: https://www.academia.edu/attachments/132007728/download_file?s=portfolio
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
This paper synthesizes the "Anatomical Reviews" to redefine the quantum-classical transition through an intrinsic operator-algebraic framework. It moves away from standard environment-induced decoherence toward a model where classicality emerges from the sectorisation of infinite-dimensional Hilbert spaces within isolated systems. Utilizing the Avalanche Photodiode (APD) model, the study demonstrates that macroscopic definiteness arises from double-exponential divergence in state overlap, creating a rigorous structural transition. The work formalizes this transition as a geometric degeneracy in the quantum Fisher information metric, linking fractal avalanche geometry to Tomita-Takesaki modular theory and providing a falsifiable experimental foundation for the algebraic Heisenberg cut. KSCCN Mamun, SM (2026) ……………………………….1 Algebraic Sectorisation, Information Geometry, and the Quantum-Classical Transition A Synthesis of the Anatomical Reviews (Vol-VI) Also available on Academia.edu: https://www.academia.edu/attachments/133083183/download_file?s=portfolio
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
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
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