On disjoint range operators in a Hilbert space

AbstractFor a bounded linear operator M in a Hilbert space H, various relations among the ranges R(M),R(M∗), R(M+M∗) and the null spaces N(M),N(M∗) are considered from the point of view of their relations to the known classes of operators, such as EP, co-EP, weak-EP, GP, DR, or SR. Particular attention is paid to the range projectors of the operators M, M∗ and some further characteristics of these projectors are derived as well

A Uniform Mathematical Representation of Logic and Computation.

The current models of computation share varying levels of correspondence with actual implementation schemes. They can be arranged in a hierarchical structure depending upon their level of abstraction. In classical computing, the circuit model shares closest correspondence with physical implementation, followed by finite automata techniques. The highest level in the abstraction hierarchy is that of the theory of computation.Likewise, there exist computing paradigms based upon a different set of defining principles. The classical paradigm involves computing as has been applied traditionally, and is characterized by Boolean circuits that are irreversible in nature. The reversible paradigm requires invertible primitives in order to perform computation. The paradigm of quantum computing applies the theory of quantum mechanics to perform computational tasks.Our analysis concludes that descriptions at lowest level in the abstraction hierarchy should be uniform across the three paradigms, but the same is not true in case of current descriptions. We propose a mathematical representation of logic and computation that successfully explains computing models in all three paradigms, while making a seamless transition to higher levels of the abstraction hierarchy. This representation is based upon the theory of linear spaces and, hence, is referred to as the linear representation. The representation is first developed in the classical context, followed by an extension to the reversible paradigm by exploiting the well-developed theory on invertible mappings. The quantum paradigm is reconciled with this representation through correspondence that unitary operators share with the proposed linear representation. In this manner, the representation is shown to account for all three paradigms. The correspondence shared with finite automata models is also shown to hold implicitly during the development of this representation. Most importantly, the linear representation accounts for the Hamiltonians that define the dynamics of a computational process, thereby resolving the correspondence shared with underlying physical principles.The consistency of the linear representation is checked against a current existing application in VLSI CAD that exploits the linearity of logic functions for symbolic representation of circuits. Some possible applications and applicability of the linear representation to some open problems are also discussed

A gentle guide to the basics of two projections theory

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik\u27s theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems. (C) 2009 Elsevier Inc. All rights reserved

Perturbative quantum gravity with the Immirzi parameter

We study perturbative quantum gravity in the first-order tetrad formalism. The lowest order action corresponds to Einstein-Cartan plus a parity-odd term, and is known in the literature as the Holst action. The coupling constant of the parity-odd term can be identified with the Immirzi parameter of loop quantum gravity. We compute the quantum effective action in the one-loop expansion. As in the metric second-order formulation, we find that in the case of pure gravity the theory is on-shell finite, and the running of Newton's constant and the Immirzi parameter is inessential. In the presence of fermions, the situation changes in two fundamental aspects. First, non-renormalizable logarithmic divergences appear, as usual. Second, the Immirzi parameter becomes a priori observable, and we find that it is renormalized by a four-fermion interaction generated by radiative corrections. We compute its beta function and discuss possible implications. The sign of the beta function depends on whether the Immirzi parameter is larger or smaller than one in absolute value, and the values plus or minus one are UV fixed-points (we work in Euclidean signature). Finally, we find that the Holst action is stable with respect to radiative corrections in the case of minimal coupling, up to higher order non-renormalizable interactions.Comment: v2 minor amendment

Entanglement-Saving Channels

The set of Entanglement Saving (ES) quantum channels is introduced and characterized. These are completely positive, trace preserving transformations which when acting locally on a bipartite quantum system initially prepared into a maximally entangled configuration, preserve its entanglement even when applied an arbitrary number of times. In other words, a quantum channel $\psi$ is said to be ES if its powers $\psi^n$ are not entanglement-breaking for all integers $n$. We also characterize the properties of the Asymptotic Entanglement Saving (AES) maps. These form a proper subset of the ES channels that is constituted by those maps which, not only preserve entanglement for all finite $n$, but which also sustain an explicitly not null level of entanglement in the asymptotic limit~$n\rightarrow \infty$. Structure theorems are provided for ES and for AES maps which yield an almost complete characterization of the former and a full characterization of the latter.Comment: 26 page

Notes on the type classification of von Neumann algebras

These notes provide an explanation of the type classification of von Neumann algebras, which has made many appearances in recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading these notes will provide you with: (i) an argument for why "factors" are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into "effective density matrices;" (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and II factors in terms of their "standard forms;" and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III$_1$ factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources I have read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.Comment: 38 pages plus 16 pages in appendices; introduction includes a reading guide for which the minimal read is about 28 page

On a partial order defined by the weighted Moore Penrose inverse

The weighted Moore-Penrose inverse of a matrix can be used to define a partial order on the set of m x n complex matrices and to introduce the concept of weighted-EP matrices. In this paper we study the weighted star partial order on the set of weighted-EP matrices. In addition, some properties that relate the eigenprojection at zero with the weighted star partial order are obtained. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Hernández, AE.; Lattanzi, MB.; Thome, N. (2013). On a partial order defined by the weighted Moore Penrose inverse. Applied Mathematics and Computation. 219(14):7310-7318. https://doi.org/10.1016/j.amc.2013.02.010S731073182191

Representation Theory of Chern Simons Observables

Recently we suggested a new quantum algebra, the moduli algebra, which was conjectured to be a quantum algebra of observables of the Hamiltonian Chern Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.Comment: 63 pages, late