This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.