We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures, and (iv) a new approach/result in inverse problems for quantum measurements.
We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures, and (iv) a new approach/result in inverse problems for quantum measurements.
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 book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation re
We cannot perform the projective measurement of a momentum on a half line since it is not an observable.Nevertheless, we would like to obtain some physical information of the momentum on a half line.We define an optimality for mea- surement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo.Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self- adjoint and explicitly construct a model Hamiltonian for the measured and probe systems.By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced.We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.
The final goal of this paper is to organize the tools needed to study digital Quantum Communications, where classical information is entrusted to quantum states represented by density operators. A density operator is usually defined starting from a set of kets in the Hilbert space and a probability distribution. A fundamental problem in Quantum Communications is the factorization of such operators of the form ρ=γγ*, where γ is a matrix called a density factor (DF). The environments considered are finite dimensional Hilbert space (discrete variables) and infinite dimensional Hilbert space (continuous variables). Using discrete variables, the multiplicity and the variety of DFs are investigated using the tools of matrix analysis, arriving in particular to establish the DF with minimal size. With continuous variables, the target is the closed-form factorization, which is achieved with recent results for the important class of Gaussian states. Finally, an application is carried out in Quantum Communications with noisy Gaussian states.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H) of bounded operators in Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries:
(i) every gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(Λ)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in H, ar,1≤r≤n are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C and [αrs] is a symmetric matrix in Mn(C);
(ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn) by the estimation of its E2(Cn)-parameters using O(n2) measurements with a finite number of outcomes.