On the operator-sum formalism

Of crucial importance to the development of quantum computing and information has been the construction of a quantum operations formalism that admits a description of quantum noise while simultaneously revealing the behavior of an open quantum system. The operator-sum representation is such a formalism and has provided a succinct description for set of bounded operators that act on a finite dimensional quantum system. In this paper we derive a basis for the set of bounded operators that act on a $d$-dimensional Hilbert space and we illustrate how this basis set may be extended and identified with a set of elements upon which the operator-sum representation rests.