On the operator-sum formalism

TL;DR

This paper explores the operator-sum formalism in quantum computing, deriving a basis for bounded operators on finite-dimensional systems to better understand quantum noise and open system behavior.

Contribution

It introduces a basis for bounded operators on d-dimensional Hilbert spaces, facilitating the operator-sum representation in quantum noise analysis.

Findings

Derived a basis for bounded operators on finite-dimensional Hilbert spaces

Extended the basis set for operator-sum formalism

Clarified the structure of quantum noise operators

Abstract

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.