This thesis explores important concepts in the area of quantum information geometry and their relationships. We highlight the unique characteristics of these concepts that arise from their quantum mechanical foundations and emphasize the differences from their classical counterparts. However, we also demonstrate that for Gaussian states, classical analogs can be used to obtain the same mathematical results, providing a valuable tool for simplifying calculations.
To establish the groundwork for the subsequent analysis, we introduce some fundamental ideas from quantum field theory. We then explore the structure of parameter space using the fidelity and the Quantum Geometric Tensor (QGT), which is composed of the Quantum Metric Tensor and the Berry curvature. We also introduce the Quantum Covariance Matrix (QCM) and show its relationship to the QGT. We present how the QCM can be used to study entanglement between quantum systems by obtaining the purity, linear entropy, and von Neumann entropy. To illustrate these concepts, we calculate all these quantities for several systems, including the Stern-Gerlach, a two qubits system, two symmetrically coupled harmonic oscillators, and N coupled harmonic oscillators.
In the final section of this thesis, we examine how the aforementioned quantum concepts can be applied in a classical sense, following the approach taken by Hannay with the Berry phase. We examine classical analogs of the QGT and QCM and since for Gaussian states, all the necessary information to calculate purity, linear entropy, and von Neumann entropy is contained within the QCM, we also generate classical analogs for them. These results in turn can be used to derive measures of separability for classical systems.
In this article, we describe various aspects of categorification of the structures appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F-manifolds, and motivic enrichments.
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned.
We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist picture, yet also operational, such that measurements and classical information arise interdependently as primitive concepts. A derived notion of classical observer caters for a mathematical formulation of Bohr's classical/quantum divide. Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally comp…
We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel" obtained in collaboration with N. Dias and J. Prata. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to s…
Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant $\hbar$) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any val…
We explore the quantization of classical models with position-dependent mass (PDM) terms constrained to a bounded interval in the canonical position. This is achieved through the Weyl-Heisenberg covariant integral quantization by properly choosing a regularizing function $Π(q,p)$ on the phase space that smooths the discontinuities present in the classical model. We thus obtain well-defined operators without requiring the construction of self-adjoint extensions. Simultaneously, the quantization mechanism leads naturally to a semi-classical system, that is, a classical-like model with a well-def…
