Convex Set Approximation Problems in Quantum Information by Eric Fernandes A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Mathematics and Statistics Guelph, Ontario, Canada c Eric Fernandes, May 2020 ABSTRACT CONVEX SET APPROXIMATION PROBLEMS IN QUANTUM INFORMATION Eric Fernandes Advisor: Dr. Rajesh Pereira University of Guelph, 2019 Co-Advisor: Dr. Bei Zeng This thesis investigates methods to approximate convex sets which involve minimizing the Hausdorff metric between a set and certain subsets. We begin by giving a lower bound for the Hausdorff metric between a hypersphere and a circumscribed simplex. We show that this bound is achieved by the regular simplex. Next, we form a lower bound on the Hausdorff distance between the convex hull of the joint numerical range of positive operator valued-measures and the probability simplex. An entanglement witness is a linear functional that separates the convex compact set of separable states from certain entangled states in the Hilbert space. We investigate the applications of our methods by exploring the problem of finding a polytope generated by entanglement witnesses that has minimal distance to the set of separable states. iii Dedication Sister Bridget iv Acknowledgements Thank you, Dr. Rajesh Pereira, for your guidance and insight throughout my master's experience. Your dedication to your field is inspirational and has encouraged me to remain motivated for any challenges to come. I would like to thank Dr. Bei Zeng for the opportunity to work in the field of quantum information and making my transition from physics to math seamless. Thank you to Ali, Comfort, David, and Kat for being welcoming friends in the department. I would like to thank the lads at 93 Moss for reminding me to take breaks from studies, I couldn't ask for a better group of brothers. Also an honourable mention to my friends at 748 Scottsdale for allowing me to frequently down spike you in smash ultimate. v Contents Abstract ii Dedication iii Acknowledgements iv 1 Introduction and Preliminaries 1 1.1 Outline . .1 1.2 Mathematical Foundations . .3 1.2.1 Euclidean Space . .4 1.2.2 Affine Geometry . .4 1.2.3 Convexity . .5 1.2.4 Hilbert Space . .8 1.2.5 Half Spaces . 11 1.2.6 Simplex . 15 1.2.7 Hausdorff Distance . 15 1.2.8 Joint Numerical Range . 17 1.3 Quantum Information . 19 2 An Application of Blaschke's Selection Theorem 23 2.1 Two problems in convex optimization . 23 2.2 The Blaschke Selection Theorem . 24 2.3 Main Results . 24 3 Optimal Simplex Enclosing a Hypersphere 26 3.1 Introduction . 26 3.2 Circumradius and Inradius . 26 3.3 Circumcircle and Incircle of a Triangle . 27 3.4 Generalization of Euler's Triangle Inequality . 29 3.5 Application of Klamkin and Tsintsifas' Result on the Hausdorff distance . 33 3.6 Conclusion . 35 vi 4 Standard Simplex and the Joint Numerical Range of POVMs 36 4.1 Introduction . 36 4.2 A Hausdorff metric inequality . 40 5 Entanglement Witnesses and Further Work 44 5.1 Introduction . 44 5.2 Separability criterion for low dimension . 44 5.3 Entanglement witnesses . 46 5.3.1 Problem with the entanglement witnesses . 47 5.4 Application of Hausdorff distance with the entanglement witness . 48 Bibliography 49 Chapter 1 Introduction and Preliminaries 1.1 Outline Convexity often arises in functional analysis which is used in quantum information theory. Convex analysis is the study of the properties of convex sets and convex functions. It has major applications in optimization. This thesis will explore the approximations of convex sets and their applications to quantum systems. We study two convex sets A, and B, where B is a subset of A. One of these sets is fixed, and the other set varies among a class of sets. We find a member of that class that minimizes the distance between A and B. Since A and B are sets, the most natural way to define the distance between these two sets is the Hausdorff distance. By bounding the Hausdorff distance, we develop optimal set approximations as well as an understanding of the properties of each set. Quantum information is the manipulation, communication, and storage of information using the laws of quantum mechanics. Quantum mechanics is vastly different from classical mechanics because of phenomena such as superposition and entanglement. This means that entangled particles can have an effect on one another, even when separated by great distances. Particles that do not exhibit this phenomenon are called separable. Even though 1 entanglement is difficult to grasp intuitively, there are mathematical descriptions that help develop an idea of how the process behaves. Whether a particle is entangled or separable, it can be described as a state in a Hilbert space. Quantum states can be described as unit vectors in a Hilbert space. Quantum states that are separable form a convex set in Hilbert space. These convex sets are more often irregularly shaped and difficult to approximate. Entanglement witnesses are used to detect entanglement. An entanglement witness is a linear functional that separates entangled states from the set of separable states. This method of detecting entanglement stems from the Hahn-Banach theorem or the separating hyperplane theorem. The Hahn-Banach theorem dates back to almost a century and is a central result in functional analysis. It has now being used in Quantum Information for the last couple of decades. We begin chapter 1 by explaining the mathematical definitions that are applied later on in the thesis. We introduce methods used to analyze convex sets such as the Hausdorff distance. The Hausdorff distance allows us to measure how close two bounded sets are from each other. This aids in analyzing set approximations and understanding the properties of our sets. A complete inner product space is introduced so that sets of vectors can be applied in quantum information and described as states. In chapter 2, we introduce two generalized problems that apply to the remainder of the thesis. We apply two completely different methods in solving special cases of these problems in chapter 3 and chapter 4. Using the Blaschke Selection Theorem; we will show that both of these methods indeed always have a solution. In chapter 3, we study the problem of finding the simplex containing the closed unit ball of dimension n which has minimal Hausdorff distance to the closed unit ball. In order to do this, we introduce the geometric properties and definitions that arise with convex sets and their enclosed convex subsets. When calculating the distance between these objects the inradius and circumradius are introduced as the main parameters in this chapter. A 2 relationship between these radii is introduced by Klamkin and Tsintsifas in [16] that inspires the main result of this chapter which bounds the Hausdorff distance between the hypersphere and the simplex that contains it. In chapter 4, the simplex, or more specifically the probability simplex is fixed while the convex hull of the joint numerical range of positive operator valued-measure (POVM's) is en- closed inside. We make statements about types of POVM's and bound the distance between the set and the convex hull. Chapter 4 is where the main theorem of the thesis is proposed and proved. Finally, chapter 5 is focused on further research. We begin with the problem of separa- bility of pure states. Separability of mixed states is more difficult and requires operational entanglement criterion. Operational entanglement criterion are different ways to identify entanglement other than Schmidt decomposition. There are still limitations to identifying entanglement with this method. This introduces the idea of the Entanglement Witness seper- ating entangled states from separable states. An entanglement witness is a linear functional used to distinguish separable states from entangled states. We conclude the thesis with the application of the Hausdorff distance with using the entanglement witness. 1.2 Mathematical Foundations This chapter introduces the basic mathematical tools necessary used throughout the thesis. We begin with an introduction to the types of spaces and operations used in these spaces. We also introduce the notion of entanglement and other quantum properties. 3 1.2.1 Euclidean Space Definition 1.1. [14] A norm on a vector space V is a rule which, given any v 2 V , specifies a real number jjvjj such that 1. jjvjj > 0 if v 6= 0, and jj0jj = 0; 2. jjavjj = jaj · jjvjj for any v 2 V and any scalar a; 3. jjv + wjj ≤ jjvjj + jjwjj for any v; w 2 V: A normed space is a vector space V with a given norm. Remark 1.2. The norm is used to measure the distance between two vectors. Lemma 1.3. (Cauchy-Schwarz Inequality) For any vectors v; w 2 Rn n 2 n n X X 2 X 2 viwi ≤ jvij jwij : i=1 i=1 i=1 Definition 1.4. [11] A Cauchy sequence of elements of a normed space is a sequence (xn) such that for any > 0 there is a number N such that jjxn − xmjj < for all n; m ≥ N: Definition 1.5. [11] A normed space M is called complete if every Cauchy sequence in M converges in M. Theorem 1.6. [2] (Bolzano-Weierstrass) Each bounded sequence of real (or complex) numbers contains a convergent subsequence. 1.2.2 Affine Geometry We assume the reader has a basic understanding of linear algebra including such topics as; linear dependence and independence, dimension, linear maps and so forth. For the majority of the thesis, we will be working in a space of dimension n. When studying problems that are invariant under translations, it is more natural to work in the setting of affine geometry.