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CONVEX ANALYSIS and ITS APPLICATION to QUANTUM INFORMATION THEORY by BEN LI Submitted in Partial Fulfillment of the Requirements

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CONVEX ANALYSIS and ITS APPLICATION to QUANTUM INFORMATION THEORY by BEN LI Submitted in Partial Fulfillment of the Requirements CONVEX ANALYSIS AND ITS APPLICATION TO QUANTUM INFORMATION THEORY by BEN LI Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics, Applied Mathematics and Statistics CASE WESTERN RESERVE UNIVERSITY August, 2018 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Ben Li candidate for the Doctor of Philosophy degree. * Committee Chair Prof. Elisabeth M Werner Committee Member Prof. Harsh Mathur Committee Member Prof. Elizabeth S Meckes Committee Member Prof. Stanisław J Szarek Date of Defense March 22, 2018 *We also certify that written approval has been obtained for any proprietary material contained therein. Copyright by Ben Li, 2018 All Rights Reserved DEDICATED TO MY LOVE iii TABLE OF CONTENTS Dedication ......................................... iii List of Tables ....................................... vi Acknowledgments ..................................... vii Abstract .......................................... ix 1 Introduction and Background ............................ 1 1.1 Introduction and Overview of Results ..................... 1 1.1.1 Mathematical aspects of Quantum Information Theory ....... 1 1.1.2 Geometrization of functions ...................... 2 1.2 Preliminaries .................................. 3 1.2.1 Notation from elementary convex analysis and linear algebra .... 3 1.2.2 Postulates of Quantum Mechanics .................. 7 1.2.3 Functional Calculus .......................... 9 1.3 Mathematical background of Quantum Information Theory ......... 10 1.3.1 Background on Convex Analysis ................... 10 1.3.2 Entanglement and Bell Inequalities .................. 12 1.3.3 Nonlocal Game ............................. 16 1.4 Background on Convex Geometry ....................... 19 Part I ........................................... 21 2 Reverse triangle inequality and its application to Quantum Information Theory 22 2.1 Introductions .................................. 22 2.2 Rotfel’d Inequality and Fuchs-Van de Graaf Inequality ........... 22 2.2.1 Rotfel’d Inequality ........................... 22 2.2.2 Fuchs-Van de Graaf Inequality .................... 25 2.3 The Reverse Triangle Inequality ........................ 29 2.4 Application ................................... 32 iv 3 Exact values of quantum violations in low-dimensional Bell correlation inequalities 35 3.1 Introduction ................................... 35 3.2 Notation and Background ........................... 36 3.3 Some elementary observations ......................... 39 3.4 (2; 2) Bipartite system and (3; 3) Bipartite system .............. 41 3.5 (4; 4) Bipartite system ............................. 45 3.5.1 Geometry of (4; 4) classical correlation polytope ........... 45 3.5.2 Quantum violation for the 41 Bell correlation inequality ...... 46 3.5.3 Quantum violation for the 42 Bell correlation inequality ...... 47 Part II ........................................... 49 4 Floating functions .................................. 50 4.1 Introduction .................................. 50 4.2 Floating functions and floating sets ...................... 51 4.2.1 Log-concave functions ......................... 53 4.3 Main Theorem and consequences ....................... 54 4.4 Proof of Theorem 4.3.1 ............................. 58 References ......................................... 83 v LIST OF TABLES 3.1 Classification of facets of LC2;2, which has 8 vertices and 16 facets. The quan- tum value of a given facet is defined as the maximal violation of its determining Bell inequality on the set of quantum correlation matrices. .......... 42 3.2 Classification of facets of LC3;3, which has 32 vertices and 90 facets. ..... 44 3.3 Classification of facets of LC4;4, which has 128 vertices and 27968 facets. .. 46 vi ACKNOWLEDGMENTS First and foremost, I would like to express the deepest gratitude to my advisors, Dr. Stanisław J. Szarek and Dr. Elisabeth M. Werner, for their constant support and encouragement throughout my graduate studies. Their importance to my growth as a researcher, teacher, and person cannot be overestimated. Without their tireless guidance and support along the way, this thesis would never have been initiated or completed. I could not have imagined having better advisors and mentors for my Ph.D. study. Special thanks go to my defense committee members Dr. Harsh Mathur and Dr. Elizabeth S. Meckes for their invaluable assistance and insights. Another special thanks goes out to Dr. Carsten Schütt and Dr. Julia Dobrosotskaya for their guidance and helpful suggestions. I would also like to give special thanks to Dr. Pankaj Joshi, Dr. Karol Horodecki, Dr. Michał Horodecki, Dr. Paweł Horodecki, Dr. Ryszard Horodecki and Dr. Tomasz Szarek for their effort on the paper [59]. Their generous sharing gives rise to Chapter 2 of this thesis. Thanks are also due to the Institute for Mathematics and its Applications in Min- neapolis and the Institut Henri Poincaré in Paris, and to their staff for their hospitality, and to fellow participants for many inspiring interactions. I would like to thank Dr. David Reeb for comments on the preliminary version of the paper [65] which forms Chapter 3. I would like to thank the Department of Mathematics, Applied Mathematics and Statistics at Case Western Reserve University. In particular, I would like to thank my many mathematics professors and fellow researchers from the past four years at Case: Dr. Florian Besau, Dr. John Duncan, Dr. Julian Grote, Dr. Elizabeth Meckes, Dr. Mark Meckes, Dr. Olaf Mordhorst, Dr. Elizabeth Sell, Dr. David Singer, Dr. Erkki Somersalo, Dr. Stanisław Szarek, Dr. Elisabeth Werner, Dr. Wojbor Woyczynski. You are all masters of your craft and it was a privilege to learn from you. vii I am also indebted to Dr. Susan Kimmel, MD, Ms Qiyu Yu, Dr. Yiling Zhang for their kind assistance and warm encouragement. The completion of this thesis would not have been possible without the help of many others along the way. I am thankful for the guidance and support from every one of you. Lastly, I wish to thank my family, my wife and her family for their support throughout the ups and downs of my life in Cleveland. I am particularly very much indebted to my daughter Claire Li whose smile warms my heart forever. I wish she could understand how much I love her. viii Convex Analysis and its Application to Quantum Information Theory Abstract by BEN LI This dissertation deals with topics in convex analysis and its application in quantum information theory. In the introduction, we collect definitions and background material from analysis, convex geometry and quantum information theory that will be used throughout the paper. Part I addresses problems on mathematical aspects of quantum information. In Chapter 2, we prove the reverse triangle inequality for quantum states. It is a fundamental property of quantum states which is also interesting on its own. It turns out that it is crucial in deriving a quantitative upper bound of quantum violation for the (generalized) Bell inequality with 2 × n inputs. In Chapter 3 we calculate the exact values of quantum violations for all extremal Bell correlation inequalities that appear in the setups involving p up to four measurements; they are all not greater than 2, the Tsirelson bound for the 2×2 setup. While various authors investigated these inequalities via numerical methods, our approach is analytic. We also include tables summarizing facial structure of Bell polytopes in low dimensions. The content of these two chapters appears in [59] and [65]. Part II concentrates on convex geometric analysis. Recently, many notions and re- sults in convex geometry have been extended from the class of convex bodies to classes of functions. In Chapter 4, we introduce floating sets for convex, not necessarily bounded subsets in Rn. This allows us to define floating functions for appropriate classes of func- tions and measures. We establish the asymptotic behavior of the integral difference of a ix log-concave function and its floating function which in turn leads to a new affine invariant for functions. The content of these results appears in [66]. x CHAPTER 1 INTRODUCTION AND BACKGROUND 1.1 INTRODUCTION AND OVERVIEW OF RESULTS 1.1.1 Mathematical aspects of Quantum Information Theory The question of non locality in quantum theory was initiated in the seminal 1935 paper [1] by A. Einstein, B. Podolsky and N. Rosen. The authors were questionning the foundation of quantum theory as a model of nature by pointing out the incompatibility of quantum prediction with the classical understanding of the physical reality. Specifically, outcomes of some experiments may be correlated in a way contradicting “common sense” (this “common sense” is referred to as the local hidden variable model), referred to as the “spooky action at a distance”. This incompatibility was elucidated 30 years later by J. S. Bell [16], who discovered a mathematical description of the boundary of the set of “classical” correlations that can be obtained in a measurement. This description is known as Bell correlation inequalities and it suggests ways to experimentally determine which description of the reality is a correct one. In a general setting, that is, when data are obtained by performing a number of k-value measurements on each part of a multipartite quantum system, the quantum locality prob- lem has been extensively studied during the past dozen or so years within the framework of operator
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