The Structure of the Real Numerical Range and the Surface Area Quantum Entanglement Measure by Matthew Kazakov 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 Matthew Kazakov, December, 2018 ABSTRACT THE STRUCTURE OF THE REAL NUMERICAL RANGE AND THE SURFACE AREA QUANTUM ENTANGLEMENT MEASURE Advisors: Matthew Kazakov Dr. Rajesh Pereira University of Guelph, 2018 Dr. David Kribs An extensive analysis has been done on the numerical range of an operator, how- ever, little research has been done on its real analogue. In this thesis we give a number of results and properties regarding the real numerical range, and real higher rank nu- merical range. We motivate this study by providing the reader with an application of how the real higher rank numerical range may be used in the study of conic sections. Finally, we end the paper with a short introduction into the field of quantum infor- mation theory, eventually building up to introduce a new measure of entanglement for pure symmetric states. iii ACKNOWLEDGEMENTS First and foremost, I would like to thank both of my parents, Pat and Dragi Kazakov. They have been on this journey with me from day one through till the very end. All of those long nights in the office solving problems, deriving new expressions and ideas are dedicated to you both. This would all have been a dream were it not for you both to make it a reality. I am forever grateful for all the love and support you both have given me. Thanks mom and dad! Next, I would like to take this opportunity to thank each of my advisors indi- vidually. To Dr. David Kribs, thank you for all your time and advice on all things operator and quantum information theory related. These fields can, at times, seem to be very complicated and confusing in nature. However, with your knowledge and experience, you were able to simplify many concepts for me so that I was able to better my overall understanding and thinking abilities, not just in these fields, but in mathematics as a whole. Thank you David, it has been a pleasure! To Dr. Rajesh Pereira, I cannot say enough good things! So many times I have been blown away at the immense knowledge you have in this ever-expanding universe that is mathematics. You are one of the most intelligent people I have had the pleasure of knowing and working with. I thank you for all of our insightful conversations. I have walked away with so much more knowledge about so many areas of mathematics and a better abstract problem solver. Thank you Rajesh. Lastly I would like to thank the University of Guelph- Mathematics and Statistics iv department for funding my Masters research. v Contents Abstract ii Acknowledgements iii List of Figures vii 1 Introduction 1 2 Background 3 2.0.1 Some basic definitions and notation . 3 2.0.2 The numerical range W (A) ................... 11 2.0.3 The real numerical range R(A) . 13 2.0.4 The complex higher rank numerical range Λk(A) . 18 2.0.5 The real higher rank numerical range Rk(A) . 20 2.0.6 The joint numerical range (real and complex) . 20 2.0.7 Hyperboloids and ellipsoids . 21 3 Conic sections and the real k- rank numerical range 29 3.0.1 Results on the real numerical range . 29 3.0.2 Hyperspherical cross sections of ellipsoids and hyperboloids . 44 4 Connections to quantum information and the surface area entangle- ment measure 47 4.1 Quantum information preliminaries . 47 4.1.1 An introduction to basic quantum information theory . 49 4.1.2 Quantum error correction and the (complex) k- rank numerical range . 53 4.1.3 A synopsis of entanglement . 57 4.2 Introducing the new measure . 63 4.2.1 Regular polygons and special case polyhedra . 67 4.2.2 The surface area entanglement measure . 74 vi 4.2.3 A possible extension of the measure . 80 4.2.4 Known results for the n point problem . 81 4.2.5 A link to the Thomson problem in physics . 86 5 Conclusions and future work 88 5.0.1 The real higher rank numerical range . 88 5.0.2 The surface area entanglement measure . 89 vii List of Figures 2.1 Image of a one-sheeted hyperbola in three dimensional space. Taken from [35]. 24 2.2 Image of a two sheeted hyperbola in three dimensional space. Taken from [36]. 24 4.1 Depiction of a maze. Item taken from Google images. 48 4.2 Image of the Bloch sphere taken from [1]. 50 4.3 A regular polygon decomposed into n triangles. 68 4.4 The inscribed QSP for j i ........................ 77 4.5 The inscribed QSP for the GHZ state in 3 particles as depicted by the red triangle. 79 4.6 Figure taken form [29] depicting the maximal volume polyhedra for the cases of 28, 29 and 30 vertices. 85 1 Chapter 1 Introduction In this thesis we examine both the higher rank numerical range (what will be re- ferred to as the complex higher rank numerical range throughout this thesis) as well as its real analogue. The terms higher rank numerical range and rank-k numerical range are used interchangeably. In chapter 2 we provide an extensive amount of prerequisite information to out- line the results and conclusions given in the latter sections. This chapter includes the classical numerical range which we denote W (A), the real numerical range, denoted R(A), the complex higher rank numerical range, denoted Λk(A), the real higher rank numerical range, denoted Rk(A), the joint complex higher rank numerical range and the joint real higher rank numerical range. A natural ordering presents itself here based on the timeline in which these objects were constructed. 2 In chapter 3 we give the main results concerning the real higher rank numerical range. One of the main theorems we derive is the real elliptical range theorem. This result and others are presented here. In chapter 4, our focus is primarily on quantum information theory. The begin- ning remarks here are designed to introduce the topic and notation used further on in the section. A brief connection is defined, stating how the complex higher rank numerical range appears in quantum error correction. We then give a detailed intro- duce to entanglement theory which serves as a stepping stone for the entanglement measure we create. Finally, we introduce the surface area entanglement measure and a possible extension. In chapter 5 we summarize the important results and key concepts of the paper. A short discussion regarding future work is also discussed here. 3 Chapter 2 Background We introduce the notation to be used throughout the course of the paper, along with a detailed overview of the different numerical ranges that will be seen. We end off with a discussion of conic sections as they do manifest themselves naturally in this context. 2.0.1 Some basic definitions and notation We provide here a complete set of definitions and relevant facts needed throughout the remainder of the paper. Much of what follows here is a discussion about matrices and some definitions from analysis. Definition 2.1. An eigenvalue of a matrix A is a scalar λ that satisfies the equality Ax = λx for some non-zero vector x. Here x is said to be an eigenvector. th Definition 2.2. The singular value of a matrix A, denoted si (for the i singular 4 value) are the square roots of the eigenvalues of the matrix A∗A. They are usually written in descending order, i.e. s1 ≥ s2 ≥ · · · ≥ sn Throughout this paper, we will typically use λ to denote eigenvalues of operators. We also will use the letters a; b; c; d; α; β to denote complex numbers, though this should be clear based on the context they come up in. It should also be stated that all operators mentioned in this paper are finite dimensional, hence we will use this interchangeably with the term, matrix. In the definitions that follow, we will take A∗ to mean the conjugate transpose of the matrix A. We now introduce some notions and remarks regarding types of matrices, proper- ties and two well known decompositions. Definition 2.3. A matrix H is said to be Hermitian (or self-adjoint) if H = H∗. Lemma 2.1. Hermitian matrices have an entirely real spectrum. Proof. Suppose Hx = λx for some norm 1 vector x. Then we see that λ = x∗Hx = (x∗H∗x)∗ but by the hermiticity of H we get that (x∗H∗x)∗ = (x∗Hx)∗ = λ,¯ and hence we have shown λ = λ¯ which implies λ must have imaginary part equal to zero, thus making it entirely real. 5 Definition 2.4. A matrix U is said to be unitary if it satisfies U ∗U = UU ∗ = I. Definition 2.5. A matrix P is said to be positive definite if it is symmetric and all of its eigenvalues are strictly greater than zero. Equivalently, P is positive definite if for all non-zero x 2 Rn we have x∗P x > 0. A similar definition holds for positive semidefinite operators, allowing there to be zero eigenvalues (and with x∗P x ≥ 0). Lemma 2.2. Let P be a positive semidefinite matrix. Then there exists a matrix B such that P = B∗B. Proof. If P is positive definite, there exists an eigen-decomposition (see statement immediately after proof) where P = U ∗ΛU for some unitary matrix U and diagonal matrix Λ.