Entanglement distillation : a discourse on bound entanglement in quantum information theory

In recent years entanglement has been recognised as a useful resource in quantum information and computation. This applies primarily to pure state entanglement which is, due to interaction with the environment, rarely available. Decoherence provides the main motivation for the study of entanglement distillation. A remarkable effect in the context of distillation is the existence of bound entangled states, states from which no pure state entanglement can be distilled. The concept of entanglement distillation also relates to a canonical way of theoretically quantifying mixed state entanglement. This thesis is, apart from a review chapter on distillation, mainly a theoretical study of bound entanglement and the two major open problems in their classification. The first of these is the classification of PPT bound entanglement (separability problem). After having reviewed known tools we study in detail the multipartite permutation criteria, for which we present new results in their classification. We solve an open problem on the existence of certain PPT states. The Schmidt number of a quantum state is a largely unvalued concept, we analyse it in detail and introduce the Schmidt robustness. The notion of Schmidt number is exploited in the study of the second major problem, that of the existence of NPT bound entanglement. We reformulate this existence problem as particular instance of the separability problem, present a uniform treatment of distillability and numerically demonstrate that bound entanglement is primarily found in low dimensional systems.