In this thesis, we present the mathematical ramifications of the notion of cleanness for positive operator valued measures. This thesis is based on paper by Douglas Farenick, Remus Floricel and Sarah Plosker (2013) . A quantum system is represented by a Hilbert space H. Let X be a set of measurement outcomes for the system, O(X) is a -algebra of measurement events, and B(H) is the space of all bounded linear operators acting on H. A positive operator valued probability measure is represented a measurement of the system. A quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. In this thesis we introduce a new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and then we characterize this property in the finite-dimensional systems. In chapter two, the necessary background in operator theory is presented. such as spectrum of operator, compact operator, trace type="#_x0000_t75" (for some Hilbert spaces H and K.) The main results are presented in chapter four: Theorem 4.1.1, which gives an analytic description of the order relation for quantum probability measures, and Theorem 4.1.4, which determines the structure of approximately clean quantum probability measures. This chapter contains a number of remarks and observations about the main results. In particular, we show that the relations and are distinct in infinite dimensions, and that there are approximately clean quantum measurements that are not clean.