YUSUF IBRAHIM2023-09-212023-09-212011-08-26https://teras.ng/api/asset/document/6c7554d3-59f8-4525-87fb-6ebb90466defhttps://teras.ng/catalog-item/f7bd1700-09a2-41e3-a388-b55a65cb7711http://dspace.teras-network.net:4000/handle/123456789/9121A noncommutative algebra involving operators of the form 𝜌Λ 𝛼 (𝑡) ∙ 𝜌Λ 𝛼 (𝑡) is defined. Using the noncommutative 𝐿𝑝 −spaces technique, we give a constructive approach to quantum Markov evolution of infinite system, based on the notion of the thermodynamic limit. The infinite Markov time evolution is constructed as the thermodynamic limit of the corresponding finite volume (dynamics) evolution. The extended Markov time evolution is then studied, with a view of addressing questions of exponential stability and ergodicity . In quantum spin system the existence of non local physical correlation at a phase transition is a manifestation of the entanglement among the constituents parts. We studied asymptotic entanglement within the frame work of open quantum systems for two independent quantum harmonic oscillators interacting with an environment. The question of separability is addressed using the Peres-Simon equationResearch Theses