Radon-Nikodym derivatives of quantum operations

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon-Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon-Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.Comment: 22 pages; final versio

Partial measures

We study sigma-additive set functions defined on a hereditary subclass of a sigma-algebra and taken values in the extended real line. Analogs of the Jordan decomposition theorem and the Radon-Nikodym theorem are obtained.Comment: 4 pages. Submitted to Lobachevskii Journal of Mathematics ( http://ljm.ksu.ru

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.Comment: 35 page

Bochner Integrals and Vector Measures

This project extends known theorems for scalar valued functions to the context of Banach space valued functions. In particular, it contains generalizations of the classical theory of Lebesgue Integrals, complex measures, Radon-Nikodym theorem and Riesz Representation theorem. We explore some properties of functions whose domains are abstract Banach spaces, where the usual derivatives are replaced by Radon-Nikodym derivatives. The first two Chapters are devoted to infinite dimensional measurable functions and the problem of integrating them. Most of the basic properties of Bochner integration are forced on it by the classical Lebesgue integration and the usual definition of measurability. The Radon-Nikodym theorem for Bochner Integral is the subject to Chapter III. The roles of reflexive spaces, separable anti-dual spaces and the Radon-Nikodym property of Banach spaces are also discussed in this Chapter. One of the most interesting aspects of the theory of the Bochner integral centers about the following questions: When does a vector measure F: Ʒ→X arise as a Bochner integral of an L1(S,X) function (i.e. F(E) = ∫E f dm)? And conversely, if f ∈ L1(S,X). Then, is F: Ʒ→X, defined by F(E) = ∫E f dm, a countably additive vector measure, absolutely continuous with respect to the positive measure m? These two questiones are examined by the Radon-Nikodym theorem and the Riesz Representation theorem. It is worth observing, that the relationsip between these theorems are considered to be just a formality of translating a set of basic definitions from one context to another. There are theories of integration similar to the Bochner Integral, that allow us to integrate functions that are only weakly measurable (The Pettis Integral) with respect to a positive measure. Also, the ultimate generality of the Bochner Integral, the Bartle Integral, for integrating vector valued functions with respect to a general vector measure. However, these theories do not occupy a central role in our study and we limit ourselves to only mentioning [1] as an excellent reference

Lebesgue-radon-nikodym decompositions for operator valued completely positice maps

Ankara : The Department of Mathematics and the Granduate School of Engineering and Science of Bilkent University, 2014.Thesis (Master's) -- Bilkent University, 2014.Includes bibliographical references leaves leaf 28.We discuss the notion of Radon-Nikodym derivatives for operator valued completely positive maps on C*-algebras, first introduced by Arveson [1969], and the notion of absolute continuity for completely positive maps, previously introduced by Parthasarathy [1996]. We begin with the definition and basic properties of positive and complete positive maps and we study the Stinespring dilation theorem which plays an essential role in the theory of Radon-Nikodym derivatives for completely positive maps, based on Poulsen [2002]. Then, the Radon-Nikodym derivative definition and basic properties belonging to Arveson is recorded and finally, we study the Lebesgue type decompositions defined by Parthasarathy in the light of the article Gheondea and Kavruk [2009].Danış, BekirM.S