Topic
Stinespring factorization theorem
About: Stinespring factorization theorem is a research topic. Over the lifetime, 2 publications have been published within this topic.
Papers
1 Jan 2015
TL;DR: In this paper, the general form of σ-additive probability measures on the complete lattice of orthogonal projections on a Hilbert space and its variations are discussed, and a special case of Stinespring factorization theorem for quantum operations on a separable Hilbert space is shown.
Abstract: This chapter offers some results which will help to understand some foundational aspects of quantum mechanics. It relies on some results presented in the last chapter. The first section discusses the general form of σ-additive probability measures on the complete lattice of orthogonal projections on a Hilbert space (Gleason’s theorem) and its variations. In quantum mechanics and in quantum information theory quantum channels or quantum operations are defined mathematically as completely positive maps between density operators which do not increase the trace (see for instance the book “Quantum Computation and Quantum Information” by M.A. Nielsen and I. L. Chuang, Cambridge University Press 2000). Thus in the next section we determine the general form of quantum operations on a separable Hilbert space, i.e., we prove Kraus’ first representation theorem for operations. Usually quantum information theory studies systems of some finite dimension n and then density operators are just positive \(n \times n\) matrices with complex coefficients which have trace 1. In this context the relevant C\(^*\)-algebra is just the space \(M_n(\mathbb C)\) of all \(n \times n\) matrices with complex entries, for some \(n \in \mathbb{N}\). Therefore in the last section we determine the general form of completely positive maps for these algebras (Choi’s results). Of course, this is a special case of Stinespring factorization theorem, but some important aspects are added.
1 Jan 2015
TL;DR: In this article, the structural analysis of positive mappings on an involutive algebra with unit operators is studied and the Stinespring factorization theorem is derived for the general form of a completely positive map between C\(^*)-algebras.
Abstract: In various parts of quantum physics positive mappings play a fundamental role. These mappings are defined on some involutive algebra (often of operators on some Hilbert space). By definition a positive mapping sends positive elements of its domain to positive elements of the target space. Thus first several characterizations of positive elements in an involutive normed algebra are given. Two types of positive mappings are considered: Positive linear functionals which have values in \(\mathbb{C}\) and completely positive mappings which have values in some other involutive algebra. For the structural analysis of positive mappings the concept of a representation of an involutive algebra is needed. This is introduced and in the case of the involutive algebra \(\mathcal{B}(\mathcal{H})\) of bounded linear operators on a Hilbert space \(\mathcal{H}\) the general form of its representations is determined (Naimark’s theorem). The structure of positive linear functionals on an involutive algebra with unit is presented in the Gelfand-Naimark-Segal (GNS)-construction. Positive linear functionals f on an involutive algebra \(\mathcal{A}\) with unit I such that \(f(I)=1\) are called states. On a weakly closed subalgebra \(\mathcal{A}\) of \(\mathcal{B}(\mathcal{H})\) special states are of the form \(f(A)={\rm{Tr}}(AW)\) where W is a density matrix on \(\mathcal{H}\). These states are characterized in terms of an additional continuity condition (normality, complete additivity), and are called normal states. The Stinespring factorization theorem gives the general form of a completely positive map between C\(^*\)-algebras. When this result is combined with Naimark’s theorem of representations of \(\mathcal{B}(\mathcal{H})\) it allows to determine the general form of completely positive mappings on \(\mathcal{B}(\mathcal{H})\) in more detail.
Performance Metrics
| Year | Papers |
|---|---|
| 2015 | 2 |