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Convex Geometric Connections to Information Theory CONVEX GEOMETRIC CONNECTIONS TO INFORMATION THEORY by Justin Jenkinson Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Department of Mathematics CASE WESTERN RESERVE UNIVERSITY May, 2013 CASE WESTERN RESERVE UNIVERSITY School of Graduate Studies We hereby approve the dissertation of Justin Jenkinson, candi- date for the the degree of Doctor of Philosophy. Signed: Stanislaw Szarek Co-Chair of the Committee Elisabeth Werner Co-Chair of the Committee Elizabeth Meckes Kenneth Kowalski Date: April 4, 2013 *We also certify that written approval has been obtained for any proprietary material contained therein. c Copyright by Justin Jenkinson 2013 TABLE OF CONTENTS List of Figures . .v Acknowledgments . vi Abstract . vii Introduction . .1 CHAPTER PAGE 1 Relative Entropy of Convex Bodies . .6 1.1 Notation . .7 1.2 Background . 10 1.2.1 Affine Invariants . 11 1.2.2 Associated Bodies . 13 1.2.3 Entropy . 15 1.3 Mean Width Bodies . 16 1.4 Relative Entropies of Cone Measures and Affine Surface Areas . 24 1.5 Proof of Theorem 1.4.1 . 30 2 Geometry of Quantum States . 37 2.1 Preliminaries from Convex Geometry . 39 2.2 Summary of Volumetric Estimates for Sets of States . 44 2.3 Geometric Measures of Entanglement . 51 2.4 Ranges of Various Entanglement Measures and the role of the Di- mension . 56 2.5 Levy's Lemma and its Applications to p-Schatten Norms . 60 2.6 Concentration for the Support Functions of PPT and S ...... 64 2.7 Geometric Banach-Mazur Distance between PPT and S ...... 66 2.8 Hausdorff Distance between PPT and S in p-Schatten Metrics . 67 2.9 Maximum Trace Distance from a PPT state to S ......... 70 iii Summary . 74 APPENDIX A Constants in the Spherical Isoperimetric Inequality . 75 Bibliography . 81 iv LIST OF FIGURES FIGURE PAGE 1 The Minkowski sum of a triangle and a circle. .2 2 The polar of the cube is the tetrahedron. .3 3 The polar of an ellipse is another ellipse. .4 1.1 The Gauss map of an ellipse, K.....................8 1.2 The support function of K........................ 10 A.1 The plots of q3 and q4........................... 79 v ACKNOWLEDGMENTS Many thanks to my advisors Stanislaw Szarek and Elisabeth Werner for providing so much inspiration and motivation and for leading me through this endeavor. The collaboration with Karol and Michal Horodecki was essential to many of the results in Chapter 3 and I am grateful for their participation and comments. I would also like to thank my committee members for many useful comments and suggestions. I would also like to thank the Department of Mathematics, the School of Graduate Studies, and Case Western Reserve University in general. This research has been partially supported by grants DMS-0503642, DMS-0652722 and DMS-0801275 from the National Science Foundation. vi ABSTRACT Convex Geometric Connections to Information Theory by JUSTIN JENKINSON Convex geometry is a field of mathematics that has experienced rapid growth in recent years and has proven to be an extremely useful perspective in areas of research. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geom- etry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. vii Finally, the optimal constants for the spherical isoperimetric inequality are pro- vided and generalizations of a concentration inequality are suggested. viii INTRODUCTION Convex geometry is a relatively young field of mathematics, yet it is based on simple notions of volume and distance known even to the ancient Greeks. Modern convex geometry has its roots with Hermann Brunn and Hermann Minkowski around the turn of the 20th century. The body of knowledge that grew from their work became known as the Brunn-Minkowski theory. Rolf Schneider [81] describes the Brunn- Minkowski theory as \...the result of merging two elementary notions for point sets in Euclidean space: vector addition and volume." One of the central concepts is n Minkowski addition. If K; L ⊂ R their Minkowski sum is given by: K + L = fx + y : x 2 K ; y 2 Lg (see figure 1). Minkowski addition of sets is a fundamental notion in convex geometry and leads to many important results such as isoperimetric inequalities, Hadwiger's theorem, and the Brunn-Minkowski inequality. The Brunn-Minkowski inequality n states that for any compact subsets, K and L, of R jK + Lj1=n ≥ jKj1=n + jLj1=n ; or, equivalently, for t 2 [0; 1] jtK + (1 − t)Lj ≥ jKjt jLj1−t : Here and in what follows |·|, when applied to a set, denotes the volume of the set in the appropriate dimension. The content of the Brunn-Minkowski inequality can 1 Figure 1: The Minkowski sum of a triangle and a circle. be summarized nicely (if somewhat informally) by noticing that the statements are equivalent to saying that |·|1=n is concave (or, equivalently, that log(|·|) is concave) with respect to Minkowski addition. The Brunn-Minkowski inequality is in fact re- sponsible for much of the modern theory [23, 81]. It was through this inequality that many important connections to other fields where made. Variants of the Brunn- Minkowski inequality show up in many fields and have been proven to be extremely useful. The classical isoperimetric inequality compares the length of a closed curve in the plane and the area which it encloses. This idea has been generalized and variants exist in many fields of mathematics. The classic survey by Osserman [73] shows how pervasive these ideas are. Isoperimetric inequalities will be discussed in more detail in section 1.2 and A. A geometric perspective is extremely useful for many fields of study. A few that have benefitted in particular are functional analysis, probability, graph theory, linear programming, and information theory. n Another important notion in convex geometry is duality. For a set K ⊂ R , we 2 define the polar of K (or the dual of K) as ◦ n K = fy 2 R : hx; yi ≤ 1 for all x 2 Kg Notice that K◦ is always convex (a set L is convex if x; y 2 L then for all t 2 [0; 1], tx + (1 − t)y 2 L). If K is also convex and contains the origin in its interior, then ◦◦ n n K = K. Additionally, the set K0 of convex bodies in R containing the origin as an interior point is closed under the polar operation, K 7! K◦. The polar operation is also contravariant. That is, if K ⊂ L then L◦ ⊂ K◦ and (rK)◦ = 1=rK◦ for all n K; L ⊂ R and r 2 R. For example, figure 2 demonstrates the polar operation on 3 the cube in R and figure 3 demonstrates that the polar of an ellipse is again another ellipse. An important fact is that If X is a normed space with unit ball K, then the dual space, X∗, has unit ball K◦. This is the reason the term \dual" is sometimes used instead of \polar". This is also illustrated in figure 2: the polar of the l1 ball is the l1 ball and vise versa. This duality is a powerful tool for exploring the structure of convex sets and normed spaces. Figure 2: The polar of the cube is the tetrahedron. 3 Figure 3: The polar of an ellipse is another ellipse. Information theory has its roots with Claude Shannon [86] who wished to quantify information and to discover the best method of propagation. Information is prop- agated by a process of encoding, transmitting, and decoding. Shannon wished to find bounds on the amount of information that could be passed through circuits and to find encoding schemes that were optimal for these circuits. Being able to dis- tinguish information from noise is crucial to encoding, decoding, and cryptography and quickly leads to questions of distinguishability of probability distributions [13]. Therefore, central notions in information theory involve measures of distinguisha- bility of probability distributions. These notions of distinguishability (Kolmogorov distance, Bhattacharyya coefficient, Shannon distinguishability [13, 22]) give us a new interpretation of certain mathematical objects and have quantum analogs as well. Quantum information theory arose from the work of Richard Feynman who, in 1982 in an entertaining paper [21] asked, \Can physics be simulated by a universal 4 computer?" The field shifted with the pioneering work of Bennett and Brassard [7] who considered a quantum cryptography. This led the way to quantum comput- ing and quantum algorithms. That is, using quantum interactions to propagate and manipulate information. In 1995 Peter Shor [88] showed that a quantum computer could, in theory, factorize prime numbers in polynomial time, a feat that is thought to be impossible with classical (non-quantum) computers. Shor's groundbreaking result elevated quantum information theory from a novelty to a priority for cryptographers and security administrations. The secret ingredient that makes quantum information theory powerful and unique from classical information theory is the notion of entan- glement.
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