Spectral Geometric Means, Quantum Relative Entropy and Post-Quantum Primitives on Jordan Algebras An Extension of the Koufany-Lim Framework: Eigenvalue Geometry, Quantum Information and Exceptional Structures (Albert Algebra) in the Mostow Product Setting
The spectral geometric mean of Koufany and Lim (2026), already known to encode the eigenvalue geometry of a pair of positive elements in a Euclidean Jordan algebra, sits at a largely unexplored intersection of quantum information theory and post-quantum cryptography. On the cone of positive-definite Hermitian matrices the spectral mean recovers the Fiedler–Pták construction whose spectrum consists of the geometric means of the eigenvalues of the product; the same construction extends verbatim to any simple Euclidean Jordan algebra, including the exceptional Albert algebra \(\mathfrak{h}_3(\mathbb{O})\). This extension examines three interlocking questions: (i) the relation between the spectral geometric mean and quantum relative entropies (Umegaki, Belavkin–Staszewski, and their Jordan-algebraic analogues); (ii) the possible use of Mostow product metrics \(d_\sigma\) as alternative geometries for quantum state spaces and for divergence-based learning; and (iii) the prospect that the exceptional Jordan structure of the Albert algebra, together with its Mostow decompositions, may furnish new hardness assumptions or algebraic primitives for post-quantum cryptography. Concrete open problems and a stratified ontological reading of the exceptional cone are offered. C S Also available on Academia.edu: https://www.academia.edu/attachments/133701404/download_file?s=portfolio
The spectral geometric mean of Koufany and Lim (2026), already known to encode the eigenvalue geometry of a pair of positive elements in a Euclidean Jordan algebra, sits at a largely unexplored intersection of quantum information theory and post-quantum cryptography. On the cone of positive-definite Hermitian matrices the spectral mean recovers the Fiedler–Pták construction whose spectrum consists of the geometric means of the eigenvalues of the product; the same construction extends verbatim to any simple Euclidean Jordan algebra, including the exceptional Albert algebra (\mathfrak{h}3(\mathbb{O})). This extension examines three interlocking questions: (i) the relation between the spectral geometric mean and quantum relative entropies (Umegaki, Belavkin–Staszewski, and their Jordan-algebraic analogues); (ii) the possible use of Mostow product metrics (d\sigma) as alternative geometries for quantum state spaces and for divergence-based learning; and (iii) the prospect that the exceptional Jordan structure of the Albert algebra, together with its Mostow decompositions, may furnish new hardness assumptions or algebraic primitives for post-quantum cryptography. Concrete open problems and a stratified ontological reading of the exceptional cone are offered. C S Also available on Academia.edu: https://www.academia.edu/attachments/133701404/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
For convex optimization problems Bregman divergences appear as regret functions. Such regret functions can be defined on any convex set, but if a sufficiency condition is added the regret function must be proportional to information divergence and the convex set must be spectral. Spectral sets are sets where different orthogonal decompositions of a state into pure states have unique mixing coefficients. Only on such spectral sets it is possible to define well behaved information theoretic quantities like entropy and divergence. It is only possible to perform measurements in a reversible way if the state space is spectral. The most important spectral sets can be represented as positive elements of Jordan algebras with trace 1. This means that Jordan algebras provide a natural framework for studying quantum information.
Quantum computing has emerged as a transformative paradigm for tackling complex matrix operations that underpin applications in machine learning and quantum information. In this paper, we synthesize recent advancements in quantum computing by integrating the quantum subroutines for matrix geometric means from Liu et al. (2025), which enable efficient solutions to nonlinear matrix equations like algebraic Riccati equations, with the Chebyshev-accelerated quantum matrix multiplication framework from Li et al. (2025), offering quadratic speedups for iterative chain operations on complex matrices. Through detailed reviews of each approach, we highlight shared primitives such as block-encoding, quantum singular value transformation, Hermitianization, and polynomial approximations, demonstrating their complementarity in constructing hybrid algorithms for nonlinear iterative problems. These hybrids achieve polylogarithmic complexities in matrix dimensions while optimizing precision and conditioning dependencies, exemplified in frameworks for Riccati iterations and kernel enhancements. Applications are explored in machine learning, including geometric mean metric learning for anomaly detection in classical and quantum data, and in quantum information, such as optimal fidelity and Rényi entropy estimation with accelerated power computations for error correction. Also available on Academia.edu: https://www.academia.edu/attachments/125700844/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
Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra J, we exploit the generalized distribution determined by the Jordan product on the dual J⋆ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of J, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in J∗ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on J, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of J, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.