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Operator spaces

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator ... More

This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator space theory for readers with less mathematical maturity. The lengthy proofs usually belong to very well-known results (such as Ruan's theorem, or the extension/characterization theorems for completely positive or completely bounded maps). Topics covered include notation and conventions, completely positive maps, operator space duality, operator space tensor products, and duality and tensor products. Notes and historical remarks are presented at the end of the chapter.Less

Basic theory of operator algebras

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

This chapter begins with a discussion of operator algebras and unitizations. It then covers some basic constructions, the abstract characterization of operator algebras, universal constructions of ... More

This chapter begins with a discussion of operator algebras and unitizations. It then covers some basic constructions, the abstract characterization of operator algebras, universal constructions of operator algebras, the second dual algebra, multiplier algebras and corners, and dual operator algebras. Notes and historical remarks are presented at the end of the chapter.Less

Basic Tools For Quantum Mechanics

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave ... More

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.Less

Quantum Quandaries: A Category-Theoretic Perspective

John Baez

in The Structural Foundations of Quantum Gravity

Part of the difficulty of combining general relativity and quantum theory is that they use different sorts of mathematics: one is based on objects such as manifolds, the other on objects such as ... More

Part of the difficulty of combining general relativity and quantum theory is that they use different sorts of mathematics: one is based on objects such as manifolds, the other on objects such as Hilbert spaces. As ‘sets equipped with extra structure’, these look like very different things, so combining them in a single theory has always seemed a bit like trying to mix oil and water. However, work on topological quantum field theory has uncovered a deep analogy between the two. Moreover, this analogy operates at the level of categories. This chapter focuses on two categories. One is the category Hilb whose objects are Hilbert spaces and whose morphisms are linear operators between these. This plays an important role in quantum theory. The other is the category nCob whose objects are (n — 1)-dimensional manifolds and whose morphisms are n-dimensional manifolds going between these. This plays an important role in relativistic theories where spacetime is assumed to be n-dimensional: in these theories the objects of nCob represent possible choices of ‘space’, while the morphisms — called ‘cobordisms’ — represent possible choices of ‘spacetime’.Less

AN INTRODUCTION TO THE THEORY OF C*-ALGEBRAS IN CONSTRUCTIVE MATHEMATICS

Hiroki Takamura

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

This chapter introduces an elementary theory of C*-algebras in the context of Bishop-style constructive mathematics. It givens proof of the Gelfand-Naĭmark-Segal (GNS) construction theorem in ... More

This chapter introduces an elementary theory of C*-algebras in the context of Bishop-style constructive mathematics. It givens proof of the Gelfand-Naĭmark-Segal (GNS) construction theorem in Bishop's constructive mathematics. This important theorem in the theory of operator algebras says that for each C*-algebra and every state, there exists a cyclic representation on some Hilbert space. This chapter's contribution is of particular interest in view of the Bridges-Hellman debate on whether constructive mathematics is able to cope with quantum mechanics. Since quantum mechanics is bound up with the theory of operator algebras on Hilbert spaces, a constructive treatment of the latter has been a challenge for constructive mathematics from the very beginning.Less

THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in ... More

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in a Hilbert or Banach space. The outline of the chapter is as follows. Section 10.1 deals with the theory of maximal monotone operators in Hilbert spaces. Section 10.2 introduces time discretizations and the concepts of mild solutions and the accretive operators in Banach spaces. Section 10.3 applies the theory of accretive operators to the filtration equation. The chapter ends with some new ideas of mass transportation and gradient flows and a review of different extensions to more general equations where new concepts of solution are needed.Less

Differentiability of Lipschitz Maps on Hilbert Spaces

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with ... More

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with the theorem stating that every Lipschitz map of a Hilbert space to a two-dimensional space has points of Fréchet differentiability. This is followed by a lemma, which is stated in an arbitrary Hilbert space but whose validity in the general case follows from its three-dimensional version. The chapter then explains the proof of the theorem and of the lemma stated above. In particular, it considers two cases, one corresponding to irregular behavior and the other to regular behavior.Less

Quantizing

Laura Ruetsche

in Interpreting Quantum Theories

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter ... More

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.Less

Stochastic Dynamics In Hilbert Space

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned ... More

This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.Less

Probability Distributions on Hilbert Space

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out ... More

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.Less

The Basic Theory of Quantum Mechanics

Bas C. van Fraassen

in Quantum Mechanics: An Empiricist View

Covered are Hilbert space, vector, and operator representations of pure and mixed states, measurable physical quantities (observables), Gleason's theorem, Lueders’ Rule, unitary operators, and ... More

Covered are Hilbert space, vector, and operator representations of pure and mixed states, measurable physical quantities (observables), Gleason's theorem, Lueders’ Rule, unitary operators, and Schroedinger's Equation, symmetries of the Hamiltonian and the corresponding conservation laws, and superselection rules.Less

Probability in Hilbert Space

Jochen Rau

in Quantum Theory: An Information Processing Approach

This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it ... More

This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.Less

Fr ´Echet Differentiability of Vector-Valued Functions

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has ... More

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.Less

Hilbert Spaces

Adel N. Boules

in Fundamentals of Mathematical Analysis

This chapter is a good introduction to Hilbert spaces and the elements of operator theory. The two leading sections contains staple topics such as the projection theorem, projection operators, the ... More

This chapter is a good introduction to Hilbert spaces and the elements of operator theory. The two leading sections contains staple topics such as the projection theorem, projection operators, the Riesz representation theorem, Bessel’s inequality, and the characterization of separable Hilbert spaces. Sections 7.3 and 7.4 contain a rather detailed study of self-adjoint and compact operators. Among the highlights are the Fredholm theory and the spectral theorems for compact self-adjoint and normal operators, with applications to integral equations. The section exercises contain problems that suggest alternative approaches, thus allowing the instructor to shorten the chapter while preserving good depth. The last section extends the results to compact operators on Banach spaces. The chapter contains more results than is typically found in an introductory course.Less

Asymptotic Fr echet ´Differentiability

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into ... More

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into higher dimensional spaces the method does not lead to a point of Gâteaux differentiability but constructs points of asymptotic Fréchet differentiability. The proof uses perturbations that are not additive, rather than the variational approach, but still provides (asymptotic) Fréchet derivatives in every slice of Gâteaux derivatives. However, it cannot be used to prove existence of points of Fréchet differentiability of Lipschitz mappings of Hilbert spaces to three-dimensional spaces. The results are negative in the sense that an appropriate version of the multidimensional mean value estimate holds.Less

Monoidal 2-Categories

Chris Heunen and Jamie Vicary

in Categories for Quantum Theory: An Introduction

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional ... More

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional graphical calculus, and show how duality gives a language for oriented surfaces, from which Frobenius algebras emerge in a natural way. We describe 2-Hilbert spaces, categorifications of Hilbert spaces and explore the monoidal 2-category 2Hilb that they give rise to. We then show how we can use dualities in 2Hilb to give a concise and purely topological language to reason about teleportation, dense coding and complementarity.Less

Abstract Hilbert Space

John von Neumann

Nicholas A. Wheeler (ed.)

in Mathematical Foundations of Quantum Mechanics: New Edition

This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, ... More

This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν‎ = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ‎(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω‎). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).Less

Quantum Many-Particle Systems and Second Quantization

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle ... More

Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.Less

Basics

Chris Heunen and Jamie Vicary

in Categories for Quantum Theory: An Introduction

This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these ... More

This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these areas, including categories, functors, natural transformations, vector spaces, Hilbert spaces, tensor products, density matrices, measurement and quantum teleportation.Less