Completely positive linear maps on complex matrices
TL;DR: A linear map from M n to M m is completely positive iff it admits an expression Φ(A)=Σ i V ∗ i AV i where Vi are n×m matrices as mentioned in this paper.
About: This article is published in Linear Algebra and its Applications.The article was published on 1975-06-01 and is currently open access. It has received 2534 citations till now. The article focuses on the topics: Completely positive map & Linear map.
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01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.
14,825 citations
TL;DR: In this article, the basic elements of entanglement theory for two or more particles and verification procedures, such as Bell inequalities, entangle witnesses, and spin squeezing inequalities, are discussed.
Abstract: How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given on the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.
1,639 citations
Book•
01 Jan 2007
TL;DR: In this paper, the authors present a synthesis of the considerable body of new research into positive definite matrices, which have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory and geometry.
Abstract: This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.
1,594 citations
01 Jan 2006
TL;DR: In this article, the space of isospectral 0Hermitian matrices is shown to be the space in which the number 6) and 7) occur twice in the figure, and the discussion between eqs.(5.14) and (5.15) is incorrect.
Abstract: a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)
1,089 citations
Book•
26 Apr 2018TL;DR: In this article, the authors present a self-contained book on the theory of quantum information focusing on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject.
Abstract: This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.
999 citations
Cites methods or result from "Completely positive linear maps on ..."
...The Choi representation of maps is so-named for Choi’s 1975 paper [50] characterizing completely positive maps (as represented by the equivalence of statements 1 and 3 in Theorem 2....
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...22 is an amalgamation of results that are generally attributed to Stinespring [197], Kraus [134, 135], and Choi [50]....
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1,179 citations
Book•
14 Dec 2012TL;DR: In this article, the generalities for positive maps are defined, including Jordan algebras, projection maps, Choi matrices, and dual functionals, as well as the norm of positive maps.
Abstract: Introduction- 1 Generalities for positive maps- 2 Jordan algebras and projection maps- 3 Extremal positive maps- 4 Choi matrices and dual functionals- 5 Mapping cones- 6 Dual cones- 7 States and positive maps- 8 Norms of positive maps- Appendix: A1 Topologies on B(H)- A2 Tensor products- A3 An extension theorem- Bibliography- Index
443 citations
364 citations
TL;DR: In this paper, the authors give some concrete distinctions between positive linear maps and completely positive linear map on C *-algebras of operators, which are written in German type.
Abstract: The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C *-algebras of operators. Herein, C *-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C *-algebras, small Greek letters α, β, γ for complex numbers. We denote by the collection of all n × n complex matrices. () = ⊗ is the C *-algebra of n × n matrices over .
255 citations