The Mathematics of Quantum Information Theory: Geometry of Quantum States
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Quantum Entanglement
MASARYK UNIVERSITY FACULTY}w¡¢£¤¥¦§¨ OF I !"#$%&'()+,-./012345<yA|NFORMATICS Quantum Entanglement MASTER’S THESIS Boris Ranto Brno, 2013 Declaration Hereby I declare, that this paper is my original authorial work, which I have worked out by my own. All sources, references and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Boris Ranto Advisor: Colin Wilmott, M.Sc., Ph.D., prof. RNDr. Jozef Gruska, DrSc. ii Acknowledgement I would like to thank Colin Wilmott and Jozef Gruska for all the sup- port and advice with which they provided me throughout the thesis. iii Abstract The core element of all the phenomena presented in this thesis is quantum entanglement. In order to present and outline the phenom- ena related to the quantum entanglement, the mathematical frame- work of quantum mechanics is presented at the beginning of this thesis. Afterwards, the idea of quantum entanglement along with the uniqueness of this resource and brief application of this resource are outlined. The study of entanglement of two, three and n parties with respect to the tasks of entanglement classification, entanglement de- tection and entanglement distillation follows in order to explore the structure of entanglement. iv Keywords Quantum mechanics, multipartite entanglement, entanglement the- ory, classification of entangled states, detection of entanglement, dis- tillation of entanglement v Contents 1 Introduction & Preliminaries ..................1 1.1 History ............................2 1.2 Hilbert spaces ........................5 1.3 Pure quantum states ....................9 1.4 Products & Operators .................... 11 1.5 Measurements ........................ 15 1.6 Postulates of quantum mechanics ............. 18 2 Introduction to Entanglement ................ -
Alice and Bob Meet Banach the Interface of Asymptotic Geometric Analysis and Quantum Information Theory
Mathematical Surveys and Monographs Volume 223 Alice and Bob Meet Banach The Interface of Asymptotic Geometric Analysis and Quantum Information Theory Guillaume Aubrun Stanisđaw J. Szarek American Mathematical Society 10.1090/surv/223 Alice and Bob Meet Banach The Interface of Asymptotic Geometric Analysis and Quantum Information Theory Mathematical Surveys and Monographs Volume 223 Alice and Bob Meet Banach The Interface of Asymptotic Geometric Analysis and Quantum Information Theory Guillaume Aubrun Stanisđaw J. Szarek American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 46Bxx, 52Axx, 81Pxx, 46B07, 46B09, 52C17, 60B20, 81P40. For additional information and updates on this book, visit www.ams.org/bookpages/surv-223 Library of Congress Cataloging-in-Publication Data Names: Aubrun, Guillaume, 1981- author. | Szarek, Stanislaw J., author. Title: Alice and Bob Meet Banach: The interface of asymptotic geometric analysis and quantum information theory / Guillaume Aubrun, Stanislaw J. Szarek. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Mathe- matical surveys and monographs ; volume 223 | Includes bibliographical references and index. Identifiers: LCCN 2017010894 | ISBN 9781470434687 (alk. paper) Subjects: LCSH: Geometric analysis. | Quantum theory. | AMS: Functional analysis – Normed linear spaces and Banach spaces; Banach lattices – Normed linear spaces and Banach spaces; Banach lattices. msc | Convex and discrete geometry – General convexity – General convex- ity. msc | Quantum theory – Axiomatics, foundations, philosophy – Axiomatics, foundations, philosophy. msc | Functional analysis – Normed linear spaces and Banach spaces; Banach lattices – Local theory of Banach spaces. msc | Functional analysis – Normed linear spaces and Banach spaces; Banach lattices – Probabilistic methods in Banach space theory. -
Arxiv:1701.02187V1 [Quant-Ph] 9 Jan 2017 1
Lectures on Quantum Information Chapter 1: The separability versus entanglement problem Sreetama Das1;2, Titas Chanda1;2, Maciej Lewenstein3;4, Anna Sanpera4;5, Aditi Sen(De)1;2, and Ujjwal Sen1;2 1Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India 2Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India 3ICFO-Institut de Ciencies` Fotoniques,` The Barcelona Institute of Science and Technology, 08034 Castelldefels (Barcelona), Spain 4 ICREA, Passeig de Lluis Companys 23, E-08010 Barcelona, Spain 5 Departament de F´ısica, Universitat Autonoma` de Barcelona, 08193 Bellaterra, Spain Contents I. Introduction 2 II. Bipartite pure states: Schmidt decomposition 2 III. Bipartite mixed states: Separable and entangled states 3 IV. Operational entanglement criteria 3 A. Partial Transposition 4 B. Majorization 5 C. Cross-norm or matrix realignment 6 D. Covariance matrix 7 1. Definition and properties 7 2. Covariance matrix criterion for separability 8 V. Non-operational entanglement criteria 10 3. Technical Preface 10 A. Entanglement Witnesses 10 1. Entanglement Witness from the Hahn-Banach theorem 10 2. Examples 12 B. Positive maps 13 1. Introduction and definitions 13 2. Positive maps and entangled states 14 3. Choi-Jamiołkowski Isomorphism 15 C. Range criterion 16 VI. Bell inequalities 17 A. Detection of entanglement by Bell inequality 18 VII. Quantification of entanglement 18 A. Entanglement of formation 19 arXiv:1701.02187v1 [quant-ph] 9 Jan 2017 1. Concurrence 20 2. Entanglement cost 20 B. Distillable entanglement 21 C. Relative entropy of entanglement 21 D. Negativity and logarithmic negativity 22 VIII. Classification of bipartite states with respect to quantum dense coding 22 A. -
A Study of Entanglement Theory
Iowa State University Capstones, Theses and Creative Components Dissertations Summer 2019 A study of Entanglement theory THANG TRAN Follow this and additional works at: https://lib.dr.iastate.edu/creativecomponents Part of the Algebra Commons Recommended Citation TRAN, THANG, "A study of Entanglement theory" (2019). Creative Components. 353. https://lib.dr.iastate.edu/creativecomponents/353 This Creative Component is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Creative Components by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A Study of ENTANGLEMENT Theory Thang Tran May 24, 2019 Abstract This article is a 50-page study of entanglement theory. First, the basic of quantum mechanics and mathematical definition of entanglement is pre- sented. Second, entanglement is described as a physical phenomenon that set Quantum Mechanics distinct from Classical Physics. Third, entangle- ment as a physical resource for quantum communication is discussed with applications, the distillation process, and quantification problem. Fourth, several methods to detect entanglement are addressed. Detecting entangle- ment is shown to have a closed connection with positive maps in operator theory. Acknowledgement I would like to express my gratefulness to my advisor, Dr. Yiu Tung Poon who not only helped me to correct many mistakes of this writing but also gave me a lot of useful advice for my career path. I have never seen anyone as patient as him in teaching me, no matter how hard I have tried to disappoint him.