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
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
While Hilbert spaces remain a cornerstone of functional analysis and quantum mechanics since their formalisation in the early 20th century, the period 2023–2026 has witnessed a surge of novel theoretical refinements, experimental validations, and cross-disciplinary integrations. This update builds directly upon the foundational treatment presented in the preceding paper, emphasising connections to quantum information geometry. Key developments include Hilbert space fragmentation in many-body systems (with experimental realisation), operator-theoretic breakthroughs (e.g., transfinite fixed points and Aluthge transforms), extensions of reproducing kernel Hilbert spaces (RKHS) to machine learning and Gaussian processes, computational advances in rigged Hilbert spaces for generalised eigenfunctions, and geometric reformulations in quantum gravity and holographic duality. We explore these from algebraic, geometric, analytic, and applied perspectives, addressing nuances such as infinite-dimensional subtleties, non-Hermitian extensions, and boundary-condition independence. Implications span quantum technologies, AI, and foundational physics, with rigorous links to the Fubini–Study/Bures metrics and quantum Fisher information. Edge cases (e.g., fragmented subspaces, evolving Hilbert spaces) and future directions are highlighted for completeness. Also available on Academia.edu: https://www.academia.edu/attachments/132280253/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
We develop a comprehensive axiomatic framework for quantum-inspired distance metrics on projective Hilbert spaces, providing a unified foundation that organizes and generalizes existing measures in quantum information theory. Starting from five fundamental axioms, projective invariance, unitary covariance, superposition sensitivity, entanglement awareness, and measurement contextuality, we show that any admissible distance depends solely on state overlap and establish the uniqueness of the Fubini-Study metric as the canonical geodesic distance. Our framework further yields a hierarchy of comparison results relating the Fubini-Study metric, Bures distance, Euclidean distance, measurement-based pseudometrics, and entanglement-sensitive distances. Key contributions include an entanglement-geometry complementarity principle, high-dimensional concentration bounds, and operational interpretations connecting distances to state discrimination and quantum metrology. This work places the geometry of quantum state spaces on a rigorous axiomatic footing, bridging abstract metric theory, information geometry, and operational quantum principles.
This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.
The measurement problem remains one of the most profound foundational issues in quantum mechanics. While the theory excels at predicting probabilities of measurement outcomes via the Schrödinger equation and the Born rule, it fails to provide a consistent, unitary description of the measurement process itself. This paper offers a comprehensive teaching-oriented review: it traces the historical development from the 1920s Copenhagen formulation through Schrödinger's cat paradox, examines the core mathematical and conceptual questions, analyses major interpretations (with a detailed exposition of the Many-Worlds Interpretation and an expanded treatment of decoherence theory), discusses Bell's theorem and its experimental tests, surveys contemporary trends as of 2025–2026 (including quantum computing implications), and evaluates the quantum-to-classical transition. The analysis highlights that the problem is not merely philosophical but has direct implications for quantum technologies, macroscopic superpositions, and our understanding of reality. No single interpretation is endorsed; instead, strengths, limitations, and open questions of each are presented to equip readers with a clear conceptual map. Mamun, SM (2026)……………………………………..1 The Quantum Measurement Problem: Historical Evolution, Fundamental Questions, Interpretations, Bell Tests, and Current Trends (Vol - I) Also available on Academia.edu: https://www.academia.edu/attachments/132618367/download_file?s=portfolio
This paper explores the potential of non-Euclidean geometry to provide a novel framework for understanding and modeling quantum information. Traditional approaches often treat quantum information as a purely mathematical construct, limiting its applicability to purely computational contexts. We propose a radical shift, formalizing quantum information as a geometric object within a non-Euclidean space, leveraging the properties of geometry to reveal deeper connections to the physical world. The core mechanism centers around extending the concept of quantum information beyond classical representations, allowing for the visualization and manipulation of entanglement and other quantum phenomena through a geometric lens. This paper will detail the conceptual groundwork, mathematical framework, and potential implications of this approach, emphasizing its capacity to bridge the gap between abstract mathematics and the complexities of quantum mechanics. Chapter 1: Foundations – Geometry and Information The fundamental concept underpinning this research lies in the recognition that geometry, particularly non-Euclidean geometry, possesses inherent properties that resonate with the nature of quantum information. Classical geometry, with its inherent Euclidean assumptions, struggles to adequately capture the counterintuitive aspects of quantum mechanics. The inherent non-Euclidean nature of space, where distances are not absolute but relative, mirrors the fundamental fuzziness and probabilistic nature of quantum states. Furthermore, the distribution of information, a cornerstone of quantum mechanics, can be visualized as a complex geometric pattern, and non-Euclidean geometry offers a powerful tool to analyze and manipulate this pattern. The challenge, therefore, is to move beyond a purely mathematical abstraction to a geometric understanding of information itself. Early attempts to model quantum information often relied on representations like Hilbert spaces, which, while powerful, represent information as a vector space. However, this representation lacks the inherent geometric richness that is crucial for understanding entanglement and other quantum phenomena. We propose a shift towards representing quantum information as points within a non-Euclidean space, where the distance and orientation of these points encode the quantum state. This shift is predicated on the observation that geometric properties, such as curvature and connectivity, are intimately linked to quantum entanglement and the preservation of information. The core idea is to formulate a geometric model where entanglement is represented not as a correlation between particles but as a distortion of the geometric space itself, influenced by the quantum state. This distortion, driven by the inherent properties of the non-Euclidean space, provides a richer and potentially more intuitive understanding of quantum phenomena. The choice of non-Euclidean geometry is not arbitrary; it is intended to be a 'natural' choice, reflecting the underlying structure of quantum information. Chapter 2: Non-Euclidean Geometry – A Conceptual Framework We introduce a specific type of non-Euclidean geometry, termed "Hyperbolic Geometry for Quantum Information," that incorporates several key features. Unlike Euclidean geometry, Hyperbolic Geometry allows for the representation of multiple, potentially conflicting, states simultaneously. This is achieved through the concept of "Hyperbolic Density," which quantifies the probability of finding a quantum state within a given region of the space. The density of a Hyperbolic Density is influenced by the quantum state, creating a dynamic and evolving geometric representation. The mathematical framework of Hyperbolic Geometry is built around the concept of "Geometric Curvature." This curvature is not simply a measure of distance but rather a representation of the quantum state's inherent orientation. Specifically, we define a curvature function that is sensitive to the quantum state. This curvature is not constant; it adapts to the quantum state, creating a continuously evolving geometric representation. This adaptation is crucial for capturing the inherent randomness and non-determinism of quantum mechanics. Furthermore, we introduce the concept of "Geometric Connectivity," which describes the relationships between points within the Hyperbolic Geometry. These connections are not simply spatial relationships but are defined by the quantum state's inherent properties. The strength and type of connectivity represent the degree to which the quantum state is entangled with the geometric structure. This connectivity is dynamically modified by the curvature function, creating a complex and interconnected geometric pattern. The core mathematical tool for manipulating these geometries is the "Geometric Transformation." This transformation allows us to 'morph' geometric forms into new forms, representing the evolution of quantum information. The transformation is based on a specific set of rules which are intrinsically linked to the nature of entanglement. These rules are carefully designed to preserve the fundamental properties of quantum information.