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Functional Analysis for Quantum Information Functional Analysis for Quantum Information Jason Crann David W. Kribs Vern I. Paulsen ii Contents 1 Metric and Topological Spaces 1 1.1 Topicstobecovered ....................... 1 1.2 MetricSpaces........................... 2 1.2.1 EquivalentMetrics .................... 4 1.2.2 Cauchy Sequences and Completeness . 4 1.2.3 Compact Sets . 5 1.2.4 Uniform Convergence . 6 1.2.5 Baire’s Theorem . 8 1.3 QubitsandtheBlochSphere . 9 1.3.1 Qubit Metrics and Fidelity . 10 1.4 Topological Spaces . 11 1.4.1 Nets and Directed Sets . 12 1.4.2 Unordered vs Ordered Sums . 14 1.4.3 The Key Separation Axiom. 15 2 Normed Spaces and Quantum Information Basics 17 2.1 Topicstobecovered ....................... 17 2.2 Banach Spaces . 18 2.2.1 Bounded and Continuous . 22 2.2.2 Equivalence of Norms . 23 2.3 Consequences of Baire’s Theorem . 24 2.3.1 Principle of Uniform Boundedness . 25 2.3.2 Open Mapping, Closed Graph, and More! . 26 2.4 Hahn-Banach Theory . 28 2.4.1 Krein-Milman Theory . 30 2.4.2 Application in Quantum Information . 30 2.5 Dual Spaces . 31 2.5.1 Banach Generalized Limits . 32 2.5.2 The Double Dual and the Canonical Embedding . 34 iii iv CONTENTS 2.5.3 The Weak Topology . 35 2.5.4 The Weak* Topology . 36 2.5.5 Completion of Normed Linear Spaces . 37 2.6 Banach space leftovers . 37 2.6.1 Quotient Spaces . 38 2.6.2 Conditions for Completeness . 38 2.6.3 Norms and Unit Balls . 39 2.6.4 Polars and the Absolutely Convex Hull . 40 2.7 Hilbert Spaces . 41 2.7.1 Hilbert Space Completions . 41 2.7.2 Qubits and Dirac Notation . 43 2.8 Bases in Hilbert Space . 43 2.8.1 DirectSums........................ 45 2.8.2 Bilinear Maps and Tensor Products . 46 2.8.3 Tensor Products of Hilbert Spaces . 48 2.8.4 Infinite Tensor Products . 50 2.9 Basic Quantum Information Nomenclature . 52 2.9.1 Postulates of Quantum Mechanics . 52 3 Operators on Hilbert Space 53 3.1 Topicstobecovered ....................... 53 3.2 Operators on Hilbert Space . 54 3.2.1 The dual of a Hilbert Space . 55 3.2.2 The Hilbert Space Adjoint . 56 3.2.3 The Banach Space Adjoint . 57 3.3 Some Important Classes of Operators . 57 3.4 Spectral Theory . 60 3.5 Numerical Ranges . 63 3.6 Majorization and Quantum State Convertibility . 64 3.7 Spectral Mapping Theorems and Functional Calculi . 64 3.8 The Riesz Functional Calculus . 65 3.8.1 The Spectrum of a Hermitian . 67 3.9 Normal Operators . 68 3.9.1 Positive Operators . 69 3.9.2 Polar Decomposition . 70 3.10 Density Operators and Mixed States . 70 3.11 Compact Operators . 71 3.11.1 The Schatten Classes . 73 3.11.2 Hilbert-Schmidt and Tensor Products . 75 3.12 Unbounded Operators . 76 CONTENTS v 3.13 Introductory Quantum Algorithms . 82 3.14 Von Neumann Algebras . 82 4 Operator Algebras 87 4.1 Topicstobecovered ....................... 87 4.2 von Neumann algebras . 87 4.3 C⇤-algebras . 92 4.3.1 Representation Theory . 95 4.3.2 States and positivity . 99 4.3.3 Finite-Dimensional C⇤-algebras . 100 4.3.4 Free products and POVMS . 100 4.3.5 Tensor Products . 101 4.4 Operator Algebra Quantum Error Correction . 103 4.5 Non-local Games and Quantum Correlations . 103 4.5.1 Non-local Games . 103 4.5.2 C⇤-algebra perspective . 109 4.5.3 Quantum Correlations and Conne’s Embedding Con- jecture...........................112 4.6 Advanced Theory of von Neumann Algebras . 112 5 Completely Positive Maps and Operator Systems 113 5.1 Topics to be covered . 113 5.2 Quantum Channels . 113 5.2.1 Open quantum systems . 113 5.2.2 Dual Perspective . 115 5.2.3 Completely Positive Maps . 117 5.2.4 Structure of Completely Positive Maps . 121 5.2.5 Multiplicative Domains and Quantum Error Correction129 5.2.6 Continuity of Stinespring Dilations . 130 5.2.7 Quantum Privacy and Complementarity . 130 5.2.8 Operator Systems . 130 5.2.9 Arveson’s extension theorem . 130 vi CONTENTS Chapter 1 Metric and Topological Spaces 1.1 Topics to be covered In this chapter we cover basic topics in analysis and topology that provide a mathematical foundation for the topics from functional analysis presented in subsequent chapters. We also include an introductory discussion on a key notion from quantum information: the quantum bit or “qubit”. Readers with a background that includes a good course in real or complex analysis can move swiftly through this chapter, whereas those coming to the text from di↵erent fields should carefully work through the material. Review of Metric Spaces • Definition of metric and examples Convergence of sequences ✏ δ definition of Continuity − Open, closed and compact sets Sequential characterizations Connected and pathwise connected sets Equivalence and Uniform equivalence of metrics Cauchy sequences and completeness C(R) as a metric space Baire’s theorem 1 2 CHAPTER 1. METRIC AND TOPOLOGICAL SPACES Qubits and the Bloch Sphere • Qubit metrics and fidelity General Topological Spaces and Nets • Open and closed sets, continuity Nets and directed sets Closed sets, compact sets, and continuity in terms of nets 1.2 Metric Spaces Many mathematical investigations require various notions of distance to be quantified. This is accomplished at a most basic level through metrics. Definition 1.1. Given a set X a metric on X is a function, d : X X R ⇥ ! that satisfies for all x, y, z X: 2 1. d(x, y) 0, ≥ 2. d(x, y) = 0 if and only if x = y, 3. d(x, y)=d(y, x), 4. d(x, z) d(x, y)+d(y, z). The pair (X, d) is called a metric space. If in place of 2) we only have that d(x, x) = 0 for all x X,thend is called a pseudometric. 2 Example 1.2. Some standard examples of metrics on real spaces include the following (with complex versions defined similarly): (R,d)whered(x, y)= x y . • | − | n n p 1/p (R ,dp)wheredp(x, y)= xi yi , for any 1 p<+ . • i=1 | − | 1 When p = 2 we call this the Euclidean distance. P n (R ,d )whered (x, y) = max xi yi :1 i n . • 1 1 {| − | } 1 x = y X any non-empty set, define d(x, y)= 6 . This is called the • (0 x = y discrete metric. Given a metric space (X, d), and a subset W X, (W, d) is also a • ✓ metric space. This is called a metric subspace of X. 1.2. METRIC SPACES 3 Definition 1.3. Let (X, d ), (Y,d ) be metric spaces, let f : X Y be a 1 2 ! function, and let x X. We say that f is continuous at x provided 0 2 0 that for every ✏>0, there exists δ>0 such that d1(x, x0) <δimplies d2(f(x),f(x0)) <✏. We call f continuous if it is continuous at every x X. 0 2 Example 1.4. Consider the subset ( 1, +1) R of (R,d). Observe that − ✓ x the function f : R ( 1, +1) given by f(x)= 1+ x is one-to-one, onto and ! − t | | continuous. The inverse function g(t)= 1 t is also continuous. −| | Definition 1.5. Let (X, d) be a metric space. A subset is open provided O that whenever x ,thenthereexistsδ>0 such that the set 2O B(x ; δ):= x d(x ,x) <δ . 0 { | 0 }✓O The set B(x0; δ) is called the ball centered at x0 of radius δ.Asetis called closed if its complement is open. These concepts also characterize continuity. Proposition 1.6. Let (X, d ), (Y,d ) be metric spaces. Let f : X Y . 1 2 ! The following conditions are equivalent: 1. f is continuous, 2. for every open set U Y , the set f 1(U):= x X : f(x) U is ✓ − { 2 2 } open, 3. for every closed set C Y , the set f 1(C) is closed. ✓ − Convergence of sequences also gives good characterizations of continuity. Definition 1.7. Let (X, d) be a metric space. A sequence xn n N con- { } 2 verges to x0 provided that for every ✏>0thereisN such that n>N implies d(x ,x ) <✏.Wewritelim x = x or x x . n 0 n n 0 n ! 0 Proposition 1.8. Let (X, d ), (Y,d ) be metric spaces and let f : X Y . 1 2 ! Then 1. f is continuous at x X if and only if for every sequence x such 0 2 { n} that limn xn = x0 we have that limn f(xn)=f(x0). 2. C X is closed if and only if for every convergent sequence x C ✓ { n}✓ we have that lim x C. n n 2 4 CHAPTER 1. METRIC AND TOPOLOGICAL SPACES 1.2.1 Equivalent Metrics Definition 1.9. Let X be a set and let d1 and d2 be metrics on X.We say that d1 and d2 are equivalent provided that a set is open in the d1 metric if and only if it is open in the d2 metric. We say that d1 and d2 are uniformly equivalent provided that there are constants, A, B > 0such that d (x, y) Ad (x, y) and d (x, y) Bd (x, y) for all x, y X. 1 2 2 1 2 It is easy to see that if two metrics are uniformly equivalent then they are equivalent. Also, two metrics are equivalent if and only if the function id : X X is continuous from (X, d )to(X, d ) and from (X, d )to(X, d ).
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