The concepts of the Bures metric and fidelity first was investigated by Donald Bures in $????$ , in an effort to establish a notion of distance on the set of normal states of a von Neumann algebra (\\cite{Bures}) . Later , Uhlmann introduced the concept of transition probability in the theory of quantum mechanics between two state of a $ \\ast $-algebra , based on Bures's work , which was later named as quantum fidelity of two states (\\cite{Uhlmann}) . Jozsa (\\cite{Jozsa}) and Alberti (\\cite{Alberti}) applied these concepts in the context quantum mechanics as a measure of closeness of quantum states . Also , several other authors , including Nilsen and Chuang analysed the concept of fidelity to the distance between quantum states (\\cite{Nielsen}) . Bengtsson and {\\.Z}yczkowski (\\cite{Bengtsson}) provided expository works on the Bures metric for density matrices and they showed that fidelity is a kind of transition probability . Furthermore , Hayashi (\\cite{Hayashi}) worked on the Bures metric for density metrices . After that , there has been a lot of research on this topic . It is considerable that the immense body of literature involving fidelity only utilises the fact that fidelity is defined on normal states of von Neumann algebras . When the von Neumann algebra is bounded linear operators on a Hilbert space $ \\mathcal{H} $ , that is $ \\mathcal{B}\\left(\\mathcal{H}\\right) $ , all the normal states can be described by the ``Trace quot; functional . Because of involving the definition of fidelity in the trace functional , one can formulate the notion of fidelity on arbitrary C$^{\\ast} $-algebras that possess faithful tracial states . \\\\ In general , this thesis considers the study of quantum fidelity , a distinguishability measure in the context of quantum mechanics , in operator algebraic point of view . The notion of fidelity provides a quantitative measure of how close one state of a quantum system is to another state . High fidelity occurs when the two states are very close to each other . Obviously , this concept and metric on the quantum states are closely related together , which this metric on the quantum states known as the Bures metric . In this thesis , fidelity and the Bures metric have been studied in the context of \\begin{itemize} \\item Unital C$ ^{\\ast} $-algebras that possess a faithful positive trace functional . \\item Finite von Neumann algebras that possess a faithful normal positive trace functional . \\end{itemize}