DOCSLIB.ORG

Mathematical Introduction to Quantum Information Processing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Introduction to Quantum Information Processing Mathematical Introduction to Quantum Information Processing (growing lecture notes, SS2019) Michael M. Wolf June 22, 2019 Contents 1 Mathematical framework 5 1.1 Hilbert spaces . .5 1.2 Bounded Operators . .8 Ideals of operators . 10 Convergence of operators . 11 Functional calculus . 12 1.3 Probabilistic structure of Quantum Theory . 14 Preparation . 15 Measurements . 17 Probabilities . 18 Observables and expectation values . 20 1.4 Convexity . 22 Convex sets and extreme points . 22 Mixtures of states . 23 Majorization . 24 Convex functionals . 26 Entropy . 28 1.5 Composite systems and tensor products . 29 Direct sums . 29 Tensor products . 29 Partial trace . 34 Composite and reduced systems . 35 Entropic quantities . 37 1.6 Quantum channels and operations . 38 Schrödinger & Heisenberg picture . 38 Kraus representation and environment . 42 Choi-matrices . 45 Instruments . 47 Commuting dilations . 48 1.7 Unbounded operators and spectral measures . 51 2 Basic trade-offs 53 2.1 Uncertainty relations . 53 Variance-based preparation uncertainty relations . 54 Joint measurability . 55 2 CONTENTS 3 2.2 Information-disturbance . 56 No information without disturbance . 56 2.3 Time-energy . 58 Mandelstam-Tamm inequalities . 58 Evolution to orthogonal states . 59 These are (incomplete but hopefully growing) lecture notes of a course taught in summer 2019 at the department of mathematics at the Technical University of Munich. 4 CONTENTS Chapter 1 Mathematical framework 1.1 Hilbert spaces This section will briefly summarize relevant concepts and properties of Hilbert spaces. A complex Hilbert space is a vector space over the complex numbers, equipped with an inner product h·; ·i : H × H ! C and an induced norm k k := h ; i1=2 w.r.t. which it is complete.1 Hence, every Hilbert space is in particular a Banach space. We will use the physicists convention that the inner prod- uct is linear in the second and conjugate-linear in its first argument so that h ; c'i = ch ; 'i = hc ; 'i, 8c 2 C. The most import inequality for the inner product is the Cauchy-Schwarz inequality, which immediately follows2 from the identity 1 2 k k2 k'k2 − jh ; 'ij2 = k'k2 − h'; i' ≥ 0: k'k2 This also shows that equality holds iff ' and are linearly dependent. A characteristic property of any norm that is induced by an inner product is that it satisfies the parallelogram law k + 'k2 + k − 'k2 = 2 k k2 + k'k2 : (1.1) In fact, whenever a norm satisfies Eq.(1.1) for all ; ', then we can reconstruct a corresponding inner product via the polarization identity, which in the case of a complex space reads 3 1 X k k 2 h ; 'i = i ' + i : (1.2) 4 k=0 1That is, every Cauchy sequence converges. 2Note that the derivation of Cauchy-Schwarz does not use that h ; i = 0 ) = 0. It only requires that h ; i ≥ 0. 5 6 CHAPTER 1. MATHEMATICAL FRAMEWORK A central concept that is enabled by an inner product is orthogonality: ; ' are called orthogonal if h ; 'i = 0. In that case k + 'k = k − 'k so that the parallelogram law becomes the Pythagoras identity k + 'k2 = k k2 + k'k2. For any subset S ⊆ H the orthogonal complement S? is defined as the subset of H whose elements are orthogonal to every element in S. S? is then necessarily a closed linear subspace. Every closed linear subspace H1 ⊆ H, in turn, gives rise to a unique decomposition of any element 2 H as = 1 + 2, where ? 1 2 H1 and 2 2 H2 = H1 . In this way, the Hilbert space decomposes into an orthogonal direct sum H = H1 ⊕H2. The i’s can equivalently be characterized as those elements in Hi closest to . Uniqueness of the i’s enables the definition of two orthogonal projections Pi : H!Hi; 7! i, which are linear idempotent maps related via P2 = 1 − P1, where 1 denotes the identity map on H. If we think the idea of orthogonal decompositions of a Hilbert space further, we are led to the concept of an orthonormal basis. An orthonormal basis is a set feig ⊆ H whose linear span is dense in H and whose elements satisfy hei; eji = δij. Its cardinality defines the dimension of the Hilbert space. Separability of H means that there is a countable orthonormal basis. In that case, for P every 2 H we have = ihei; iei (converging in norm) and the Parseval 2 P 2 identity k k = i jhei; ij holds. An orthonormal set of vectors can always be extended to an orthonormal basis. Another property that Hilbert spaces share with their Euclidean ancestors is expressed by the Riesz representation theorem: it states that every continuous linear map from H into C is of the form 7! h'; i for some ' 2 H, and vice versa. In other words, there is a conjugate linear bijection between H and its topological dual space H0 (i.e. the space of all continuous linear functionals). The possible identification of H and H0 motivates the so-called Dirac-notation that writes j i for elements of H and h'j for elements of H0. These symbols are then called ket and bra, respectively and the inner product in this nota- tion reads h'j i (forming a “bra(c)ket”). When we would restrict ourselves to Euclidean spaces, kets and bras would be nothing but column vectors and row vectors, respectively. Dirac notation also enables the introduction of a ket-bra j ih'j : H!H that defines a map jφi 7! j ih'jφi. Using ket-bras, a necessary and sufficient condition for a set of orthonormal vectors to form a basis of a separable Hilbert space is given by X jekihekj = 1: (1.3) k To write expressions of this form even more compactly, the elements of a fixed orthonormal basis are often simply specified by their label so that one writes jki instead of jeki. So far, this has been abstract Hilbert space theory. Before we proceed, some concrete examples of Hilbert spaces: Example 1.1. Cn becomes a Hilbert space when equipped with the standard Pn inner product h ; 'i = i=1 i'i. 1.1. HILBERT SPACES 7 N CN P 2 Example 1.2. The sequence space l2( ) := 2 j k j kj < 1 be- comes a Hilbert space when equipped with the standard inner product h ; 'i = P k k'k. The standard orthonormal basis in this case is given by sequences ek; k 2 N such that the l’th element in ek equals δlk. R R C R 2 Example 1.3. The function space L2( ) := f : ! R j (x)j dx < R 2 1 = ∼ where ∼ ' , R j (x) − '(x)j dx = 0 becomes a separable Hilbert R space with h ; 'i = R (x)'(x)dx. Example 1.4. The space Cn×m of complex n × m matrices becomes a Hilbert space with hA; Bi = tr [A∗B]. Two Hilbert spaces H1 and H2 are called isomorphic if there is a bijection U : H1 !H2 that preserves all inner products. U, which is called a Hilbert space isomorphism, is then necessarily linear and it turns out that Hilbert spaces are isomorphic iff they have the same dimension. Hence, all separable Hilbert n spaces are isomorphic to either C or l2(N), in particular, L2(R) ' l2(N). Sometimes one has to deal with inner product spaces that are not complete. In these cases the following theorem comes in handy and allows to ‘upgrade’ every such space to a Hilbert space: Theorem 1.1 (Completion theorem). For every inner product space X there is a Hilbert space H and a linear map V : X!H that preserves all inner products3 so that V (X ) is dense in H and equal to H if X is complete. The space H is then called the completion of X . It is unique in the sense that if (V 0; H0) give rise to another completion, then there is a Hilbert space isomorphism U : H!H0 s.t. V 0 = U ◦ V . As in the more general case of metric spaces, the completion is constructed by considering equivalence classes of Cauchy-sequences in X . Usually, this con- struction is, however, hardly used beyond the proof of this theorem, and it is sound to regard H as a superspace of X that has been constructed from X by adding all the elements that were missing for completeness. We finally state a property that distinguishes Hilbert spaces from almost all other normed spaces and has various applications in the form of a dimension reduction argument: Lemma 1.2 (Johnson-Lindenstrauss). There is a universal constant c 2 R such that for any 2 (0; 1], Hilbert space H, n 2 N, 1; : : : ; n 2 H there is a linear map L : H!Hd that is a multiple of an orthogonal projection onto a d-dimensional subspace Hd with c d ≤ log n; 2 so that for all i; j 2 f1; : : : ; ng : 2 2 2 (1 − ) k i − jk ≤ kL i − L jk ≤ (1 + ) k i − jk : (1.4) 3In other words, V is an isometry; see next section for the definition. 8 CHAPTER 1. MATHEMATICAL FRAMEWORK This is often stated and used for real Hilbert spaces, but equally valid for complex ones. From now on, we will tacitly assume that all Hilbert spaces H; H1; H2; etc. are complex and separable. Exercise 1.1. Show that the closed unit ball of any Hilbert space is strictly convex. Exercise 1.2. Show that any linear map U : H1 !H2 that preserves norms also preserves inner products. Exercise 1.3.
Load more

Recommended publications
  • L2-Theory of Semiclassical Psdos: Hilbert-Schmidt And
    2 LECTURE 13: L -THEORY OF SEMICLASSICAL PSDOS: HILBERT-SCHMIDT AND TRACE CLASS OPERATORS In today's lecture we start with some abstract properties of Hilbert-Schmidt operators and trace class operators. Then we will investigate the following natural t questions: for which class of symbols, the quantized operator Op~(a) is a Hilbert- Schmidt or trace class operator? How to compute its trace? 1. Hilbert-Schmidt and Trace class operators: Abstract theory { Definitions via singular values. As we have mentioned last time, we are aiming at developing the spectral theory for semiclassical pseudodifferential operators. For a 2 S(m), where m is a decaying t 2 n symbol, we showed that Op~(a) is a compact operator acting on L (R ). As a t consequence, either Op~(a) has only finitely many nonzero eigenvalues, or it has infinitely nonzero eigenvalues which can be arranged to a sequence whose norms converges to 0. However, in general the eigenvalues of a compact operator A are non-real. A very simple way to get real eigenvalues is to consider the operator A∗A, which is 1 a compact self-adjoint linear operator acting on L2(Rn). Thus the eigenvalues of A∗A can be list2 in decreasing order as 2 2 2 s1 ≥ s2 ≥ s3 ≥ · · · : The numbers sj (which will always be taken to be the positive one) are called the singular values of A. Definition 1.1. We say a compact operator3 A is a Hilbert-Schmidt operator if !1=2 X 2 (1) kAkHS := sj < +1: j The number kAkHS is called the Hilbert-Schmidt norm (or Schatten 2-norm) of A.
    [Show full text]
  • Quantum Information
    Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding .
    [Show full text]
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • Singular Value Decomposition (SVD)
    San José State University Math 253: Mathematical Methods for Data Visualization Lecture 5: Singular Value Decomposition (SVD) Dr. Guangliang Chen Outline • Matrix SVD Singular Value Decomposition (SVD) Introduction We have seen that symmetric matrices are always (orthogonally) diagonalizable. That is, for any symmetric matrix A ∈ Rn×n, there exist an orthogonal matrix Q = [q1 ... qn] and a diagonal matrix Λ = diag(λ1, . , λn), both real and square, such that A = QΛQT . We have pointed out that λi’s are the eigenvalues of A and qi’s the corresponding eigenvectors (which are orthogonal to each other and have unit norm). Thus, such a factorization is called the eigendecomposition of A, also called the spectral decomposition of A. What about general rectangular matrices? Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/22 Singular Value Decomposition (SVD) Existence of the SVD for general matrices Theorem: For any matrix X ∈ Rn×d, there exist two orthogonal matrices U ∈ Rn×n, V ∈ Rd×d and a nonnegative, “diagonal” matrix Σ ∈ Rn×d (of the same size as X) such that T Xn×d = Un×nΣn×dVd×d. Remark. This is called the Singular Value Decomposition (SVD) of X: • The diagonals of Σ are called the singular values of X (often sorted in decreasing order). • The columns of U are called the left singular vectors of X. • The columns of V are called the right singular vectors of X. Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/22 Singular Value Decomposition (SVD) * * b * b (n>d) b b b * b = * * = b b b * (n<d) * b * * b b Dr.
    [Show full text]
  • Banach J. Math. Anal. 2 (2008), No. 2, 59–67 . OPERATOR-VALUED
    Banach J. Math. Anal. 2 (2008), no. 2, 59–67 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) http://www.math-analysis.org . OPERATOR-VALUED INNER PRODUCT AND OPERATOR INEQUALITIES JUN ICHI FUJII1 This paper is dedicated to Professor Josip E. Peˇcari´c Submitted by M. S. Moslehian Abstract. The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y ∗X as an operator- valued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold. 1. Introduction Inequality plays a basic role in analysis and consequently in Mathematics. As surveyed briefly in [6], operator inequalities on a Hilbert space have been discussed particularly since Furuta inequality was established. But it is not easy in general to give a simple condition that the equality in an operator inequality holds. In this note, we observe basic operator inequalities and discuss the equality conditions. To show this, we consider simple linear algebraic structure in operator spaces: For Hilbert spaces H and K, the symbol B(H, K) denotes all the (bounded linear) operators from H to K and B(H) ≡ B(H, H). Then, consider an operator n n matrix A = (Aij) ∈ B(H ), a vector X = (Xj) ∈ B(H, H ) with operator entries Xj ∈ B(H), an operator-valued inner product n ∗ X ∗ Y X = Yj Xj, j=1 Date: Received: 29 March 2008; Accepted 13 May 2008. 2000 Mathematics Subject Classification. Primary 47A63; Secondary 47A75, 47A80. Key words and phrases. Schwarz inequality, Jensen inequality, Operator inequality.
    [Show full text]
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
    [Show full text]
  • Studies in the Geometry of Quantum Measurements
    Irina Dumitru Studies in the Geometry of Quantum Measurements Studies in the Geometry of Quantum Measurements Studies in Irina Dumitru Irina Dumitru is a PhD student at the department of Physics at Stockholm University. She has carried out research in the field of quantum information, focusing on the geometry of Hilbert spaces. ISBN 978-91-7911-218-9 Department of Physics Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020 Studies in the Geometry of Quantum Measurements Irina Dumitru Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 13.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, and digitally via video conference (Zoom). Public link will be made available at www.fysik.su.se in connection with the nailing of the thesis. Abstract Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally- Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs. This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements.
    [Show full text]
  • A Mini-Introduction to Information Theory
    A Mini-Introduction To Information Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case. arXiv:1805.11965v5 [hep-th] 4 Oct 2019 Contents 1 Introduction 2 2 Classical Information Theory 2 2.1 ShannonEntropy ................................... .... 2 2.2 ConditionalEntropy ................................. .... 4 2.3 RelativeEntropy .................................... ... 6 2.4 Monotonicity of Relative Entropy . ...... 7 3 Quantum Information Theory: Basic Ingredients 10 3.1 DensityMatrices .................................... ... 10 3.2 QuantumEntropy................................... .... 14 3.3 Concavity ......................................... .. 16 3.4 Conditional and Relative Quantum Entropy . ....... 17 3.5 Monotonicity of Relative Entropy . ...... 20 3.6 GeneralizedMeasurements . ...... 22 3.7 QuantumChannels ................................... ... 24 3.8 Thermodynamics And Quantum Channels . ...... 26 4 More On Quantum Information Theory 27 4.1 Quantum Teleportation and Conditional Entropy . ......... 28 4.2 Quantum Relative Entropy And Hypothesis Testing . ......... 32 4.3 Encoding
    [Show full text]
  • Quantum Information Theory
    Lecture Notes Quantum Information Theory Renato Renner with contributions by Matthias Christandl February 6, 2013 Contents 1 Introduction 5 2 Probability Theory 6 2.1 What is probability? . .6 2.2 Definition of probability spaces and random variables . .7 2.2.1 Probability space . .7 2.2.2 Random variables . .7 2.2.3 Notation for events . .8 2.2.4 Conditioning on events . .8 2.3 Probability theory with discrete random variables . .9 2.3.1 Discrete random variables . .9 2.3.2 Marginals and conditional distributions . .9 2.3.3 Special distributions . 10 2.3.4 Independence and Markov chains . 10 2.3.5 Functions of random variables, expectation values, and Jensen's in- equality . 10 2.3.6 Trace distance . 11 2.3.7 I.i.d. distributions and the law of large numbers . 12 2.3.8 Channels . 13 3 Information Theory 15 3.1 Quantifying information . 15 3.1.1 Approaches to define information and entropy . 15 3.1.2 Entropy of events . 16 3.1.3 Entropy of random variables . 17 3.1.4 Conditional entropy . 18 3.1.5 Mutual information . 19 3.1.6 Smooth min- and max- entropies . 20 3.1.7 Shannon entropy as a special case of min- and max-entropy . 20 3.2 An example application: channel coding . 21 3.2.1 Definition of the problem . 21 3.2.2 The general channel coding theorem . 21 3.2.3 Channel coding for i.i.d. channels . 24 3.2.4 The converse . 24 2 4 Quantum States and Operations 26 4.1 Preliminaries .
    [Show full text]
  • Using Functional Analysis and Sobolev Spaces to Solve Poisson’S Equation
    USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON'S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye to- wards defining weak solutions to elliptic PDE. Using Lax-Milgram we prove that weak solutions to Poisson's equation exist under certain conditions. Contents 1. Introduction 1 2. Banach spaces 2 3. Weak topology, weak star topology and reflexivity 6 4. Lower semicontinuity 11 5. Hilbert spaces 13 6. Sobolev spaces 19 References 21 1. Introduction We will discuss the following problem in this paper: let Ω be an open and connected subset in R and f be an L2 function on Ω, is there a solution to Poisson's equation (1) −∆u = f? From elementary partial differential equations class, we know if Ω = R, we can solve Poisson's equation using the fundamental solution to Laplace's equation. However, if we just take Ω to be an open and connected set, the above method is no longer useful. In addition, for arbitrary Ω and f, a C2 solution does not always exist. Therefore, instead of finding a strong solution, i.e., a C2 function which satisfies (1), we integrate (1) against a test function φ (a test function is a Date: September 28, 2016. 1 2 YI WANG smooth function compactly supported in Ω), integrate by parts, and arrive at the equation Z Z 1 (2) rurφ = fφ, 8φ 2 Cc (Ω): Ω Ω So intuitively we want to find a function which satisfies (2) for all test functions and this is the place where Hilbert spaces come into play.
    [Show full text]
  • Inverse and Implicit Function Theorems for Noncommutative
    Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains Mark E. Mancuso Abstract Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity. Keywords: Noncommutive functions, operator noncommutative functions, free anal- ysis, inverse and implicit function theorems, strong operator topology, dilation theory. MSC (2010): Primary 46L52; Secondary 47A56, 47J07. INTRODUCTION Polynomials in d noncommuting indeterminates can naturally be evaluated on d-tuples of square matrices of any size. The resulting function is graded (tuples of n × n ma- trices are mapped to n × n matrices) and preserves direct sums and similarities. Along with polynomials, noncommutative rational functions and power series, the convergence of which has been studied for example in [9], [14], [15], serve as prototypical examples of a more general class of functions called noncommutative functions. The theory of non- commutative functions finds its origin in the 1973 work of J. L. Taylor [17], who studied arXiv:1804.01040v2 [math.FA] 7 Aug 2019 the functional calculus of noncommuting operators. Roughly speaking, noncommutative functions are to polynomials in noncommuting variables as holomorphic functions from complex analysis are to polynomials in commuting variables.
    [Show full text]
  • Notes on the Spectral Theorem
    Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner prod- uct space V has an adjoint. (T ∗ is defined as the unique linear operator on V such that hT (x), yi = hx, T ∗(y)i for every x, y ∈ V – see Theroems 6.8 and 6.9.) When V is infinite dimensional, the adjoint T ∗ may or may not exist. One useful fact (Theorem 6.10) is that if β is an orthonormal basis for a finite dimen- ∗ ∗ sional inner product space V , then [T ]β = [T ]β. That is, the matrix representation of the operator T ∗ is equal to the complex conjugate of the matrix representation for T . For a general vector space V , and a linear operator T , we have already asked the ques- tion “when is there a basis of V consisting only of eigenvectors of T ?” – this is exactly when T is diagonalizable. Now, for an inner product space V , we know how to check whether vec- tors are orthogonal, and we know how to define the norms of vectors, so we can ask “when is there an orthonormal basis of V consisting only of eigenvectors of T ?” Clearly, if there is such a basis, T is diagonalizable – and moreover, eigenvectors with distinct eigenvalues must be orthogonal. Definitions Let V be an inner product space. Let T ∈ L(V ). (a) T is normal if T ∗T = TT ∗ (b) T is self-adjoint if T ∗ = T For the next two definitions, assume V is finite-dimensional: Then, (c) T is unitary if F = C and kT (x)k = kxk for every x ∈ V (d) T is orthogonal if F = R and kT (x)k = kxk for every x ∈ V Notes 1.
    [Show full text]