The mathematical structure of scattering theory is developed within the framework of so-called linear spaces. These linear spaces involve generalisations of the familiar concepts which were introduced in elementary courses in vector algebra and the analysis of functions of a real variable. In these early courses we invariably worked with the value of the function at some point in its domain of definition rather than with the abstract quantity of the function itself. This strategy will no longer be adequate for our purposes. However, in order to be able to work with such abstract elements we shall need generalisations of concepts such as addition, multiplication by a scalar, magnitude and distance. The required generalisations are obtained by extending the familiar notions of vector algebra and geometry in R n . We shall find that this leads quite naturally to the notions of a metric, a generalisation of the familiar Euclidean distance function in R n , a metric space and a normed linear space. Furthermore, it will no longer be sufficient to work in finite dimensional spaces such as R n and C n but instead we must be prepared to work in infinite dimensional spaces. This we shall be able to do very conveniently in the structure of so-called Hilbert spaces.
The mathematical structure of scattering theory is developed within the framework of so-called linear spaces. These linear spaces involve generalisations of the familiar concepts which were introduced in elementary courses in vector algebra and the analysis of functions of a real variable. In these early courses we invariably worked with the value of the function at some point in its domain of definition rather than with the abstract quantity of the function itself. This strategy will no longer be adequate for our purposes. However, in order to be able to work with such abstract elements we shall need generalisations of concepts such as addition, multiplication by a scalar, magnitude and distance. The required generalisations are obtained by extending the familiar notions of vector algebra and geometry in R n . We shall find that this leads quite naturally to the notions of a metric, a generalisation of the familiar Euclidean distance function in R n , a metric space and a normed linear space. Furthermore, it will no longer be sufficient to work in finite dimensional spaces such as R n and C n but instead we must be prepared to work in infinite dimensional spaces. This we shall be able to do very conveniently in the structure of so-called Hilbert spaces.
Quantum theory, in its conventional formulation, is built on the theory of Hilbert spaces and operators. In this chapter we go through this basic material, which is central for the rest of the book. Our treatment is mainly intended as a refresher and a summary of useful results. It is assumed that the reader is already familiar with some of these concepts and elementary results, at least in the case of finite-dimensional inner product spaces. We present proofs for propositions and theorems only if the proof itself is considered to be instructive and illustrative. This gives us the freedom to present the material in a topical order rather than in the strict order of mathematical implication. Good references for this chapter are the functional analysis textbooks by Conway [45], Pedersen [113] and Reed and Simon [121]. These books also contain the proofs that we skip here. Hilbert spaces As an introduction, before a formal definition is given, one may think of a Hilbert space as the closest possible generalization of the inner product spaces c d to infinite dimensions. Actually, there are no finite-dimensional Hilbert spaces (up to isomorphisms) other than c d spaces. The crucial requirement of completeness becomes relevant only in infinite-dimensional spaces. This defining property of Hilbert spaces guarantees that they are well-behaved mathematical objects, and many calculations can be done almost as easily as in c d .
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
Quantum mechanics is based on linear algebra. The general theory uses infinite dimensional vector spaces. Fortunately for us, to describe spin or polarization we need only finite dimensions, which makes things much easier. In fact, we need only a few tools. At the end of this chapter I have given a list. The rest of the chapter explains how to use these tools and what the calculations mean. There are many examples. It is important to work carefully through all of them. The mathematics introduced here is essential to everything that follows. Like much mathematics, it can seem complicated when it is first introduced, but it becomes almost second nature with practice. The actual computations only involve addition and multiplication of numbers, along with an occasional square root and trigonometric function.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.