Entanglement and the quantum-classical transition

The ubiquitous theory of quantum mechanics has sparked many controversial debates about possible interpretations and its place in fundamental physical theory. Perhaps most sustained of all the questions is the quest for a complete explanation of the process behind the reduction from quantum to classical phenomena. This thesis shall examine this question through detailed investigation of specific models. The models include a careful exposition of quantum entanglement through the original EPR thought experiment for continuous variables and its mathematical transcription. The transcription is compared with existing methods from quantum optics for achieving experimentally verifiable Bell-type inequality violations, which are commonly interpreted as violations of locality. In further development of this famous paradox, the mathematical model is extended to tripartite continuous variable states, and detailed measures of their violation of locality are presented. Having carefully examined quantum phenomena by the EPR-paradox and its extension to tripartite cases, the investigation proceeds by considering the effect on quantum systems by an environment. This involves a re-examination of some well-known quantum system environment models: the spin-1/2 Spin-Boson and Kondo models, in which a two-level quantum system interacts with a bath of bosons or fermions respectively. The technicalities of the interactions are exposed in intricate detail, with a careful description of the constructive bosonisation and transformation methods involved. The thorough analysis leads to a new observation about the elliptic, or fully anisotropic, Kondo model. Importantly, the re-examination of the detailed structure of fermion-gas impurity models and their connection to quantum dissipative systems enables a comprehensive extension of the family of models to include in particular a new three-level dissipative system. To underline the importance of this model, it is shown that the model is exactly solvable by admitting a reparametrisation of the scattering matrix in terms of R-matrices which obey the Yang-Baxter equation. The examination of the interaction between the quantum and the classical concludes with an investigation of entanglement criteria in the system environment models discussed. A variational Ansatz for the ground state is used to demonstrate the numerical calculation of entropy expressions for the three-level systems, while the reynman-Hellmann Theorem is used to give inprinciple exact results for the entropy corresponding to the specific three-level model Hamiltonian introduced in this thesis. Throughout we provide several suggestions for further work, procedures for experimental verification and practical application.