Exactly solvable models for qubits coupled to spin environments : decoherence and entanglement.
During the last decade the field of quantum information has seen considerable progress both theoretically and experimentally. The building block of this theory is the so-called qubit, the quantum counterpart of the classical bit. It turns out that spin degrees of freedom of quantum particles are among the most promising candidates for quantum information processing and computation. Thus, a deep understanding of the different processes that govern the dynamics of these objects is of fundamental importance. For instance, the decoherence process, caused the coupling of the qubits to their surrounding environment, is the main obstacle to quantum information processing: it leads them to lose their quantum coherence and thus behave classically. In addition, quantum systems exhibit correlations that have no classical counterpart. This phenomenon is called entanglement and is the vital resource for quantum teleportation and quantum computing. Decoherence and entanglement dynamics were extensively studied within the framework of the Markovian approximation using the master equation approach. However, due to the non-commuting character of quantum observables, only few models are known to be exactly solvable. Moreover, many spin systems display a strongly non-Markovian behavior. This thesis is devoted to the development of new techniques for deriving the exact dynamics of spin qubits, coupled through Heisenberg and/or Ising interactions to spin environments that have internal dynamics. The basic idea behind these techniques is to use the underlying symmetries exhibited by the Hamitonian operators of the composite systems. This allows for the study of problems related to decoherence and entanglement.