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
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
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
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 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
N C K SC K SC Recent advances at the intersection of quantum information geometry and machine-learning approaches to the renormalisation group suggest a fertile reinterpretation of the Similarity Renormalization Group (SRG). The continuous unitary transformations that drive a Hamiltonian toward band-diagonal form may be viewed as geodesics on a manifold of quantum states (or of density operators) equipped with a monotone metric, or equivalently as successive layers of a deep network that progressively disentangle degrees of freedom. This independent study develops that correspondence. It reviews the information-geometric structures relevant to quantum field theories and many-body Hamiltonians, examines neural-network renormalisation-group (NN-RG) architectures, and maps the SRG flow—originally clarified in the pedagogical setting of Szpigel & Perry (2000)—onto both geometric and learning-theoretic languages. The resulting perspective yields analytic insights into the optimality of particular generators and practical algorithms for approximate Hamiltonian diagonalisation that combine SRG ideas with modern differentiable programming. Error-scaling diagnostics originating in the two-dimensional laboratory are re-expressed as statements about geodesic deviation and about generalisation error in the associated learning problem. Also available on Academia.edu: https://www.academia.edu/attachments/133749950/download_file?s=portfolio
Wai-Hong Tam, Reza Safari and Hiromichi Matsuyama (2026) introduce Geometric Quantum Physics-Informed Neural Networks (GQPINNs) as a symmetry-aware extension of quantum physics-informed neural networks (QPINNs). Their framework embeds finite-group (Klein four-group (K_4)) and compact Lie-group (SO(2) rotational) symmetries directly into the parametrized quantum circuit ansatz using twirling-based construction of equivariant generator sets. This builds on geometric quantum machine learning (GQML) to enforce solution-level invariance/equivariance by construction, rather than through loss penalties or post-hoc constraints. The paper benchmarks GQPINNs on representative linear and nonlinear PDEs (2D Poisson, 2D diffusion, 1D acoustic wave, 1D viscous Burgers), demonstrating superior mean absolute error (MAE) and parameter efficiency compared to baseline QPINNs and symmetry-adapted classical PINNs. Also available on Academia.edu: https://www.academia.edu/attachments/132740816/download_file?s=portfolio
This research redefines the fundamental ontology of the photon by transitioning from the traditional second quantization paradigm to a framework rooted in first quantization, exact pseudomode electrodynamics, and Entropic Dynamics. The paper posits that photons are localized, information-geometric entities whose physical shape and evolutionary dynamics are deterministically sculpted by the complex permittivity of their environment. By synthesizing non-Hermitian dynamics, non-equilibrium thermodynamics, and information-geometric metrics such as the Quantum Fisher Information and Fubini-Study metric, the framework establishes a unified approach to quantum metrology and structured light. This redefinition resolves the limitations of Markovian approximations and provides a rigorous, high-dimensional basis for advancements in quantum information processing, resilient sensing, and integrated photonics. KSCCN Mamun, SM (2026) …………………………….1 The Information Geometry of the Photon First Quantization, Exact Pseudomode Electrodynamics, and Entropic Redefinitions (Volume VIII) Also available on Academia.edu: https://www.academia.edu/attachments/133123706/download_file?s=portfolio