The set of natural numbers {1, 2, 3, . . .} is denoted by N. The set of integers {. . . ,−2,−1, 0, 1, 2, . . .} is denoted by Z. Q denotes the set of rational numbers. Finally R and C denote the sets of real numbers and complex numbers, respectively. Observe that N ⊂ Z ⊂ Q ⊂ R ⊂ C The vector spaces encountered in physics are mostly real vector spaces and complex vector spaces. Classical mechanics and electrodynamics are formulated mainly in real vector spaces while quantum mechanics (and hence this book) is founded on complex vector spaces. In the rest of this chapter, we briefly summarize vector spaces and matrices (linear maps), taking applications to quantum mechanics into account.
The set of natural numbers {1, 2, 3, . . .} is denoted by N. The set of integers {. . . ,−2,−1, 0, 1, 2, . . .} is denoted by Z. Q denotes the set of rational numbers. Finally R and C denote the sets of real numbers and complex numbers, respectively. Observe that N ⊂ Z ⊂ Q ⊂ R ⊂ C The vector spaces encountered in physics are mostly real vector spaces and complex vector spaces. Classical mechanics and electrodynamics are formulated mainly in real vector spaces while quantum mechanics (and hence this book) is founded on complex vector spaces. In the rest of this chapter, we briefly summarize vector spaces and matrices (linear maps), taking applications to quantum mechanics into account.
In this chapter we fix our terminology and notation, mostly related to (real and complex) linear algebra. We will consider only algebraic properties. Infinite-dimensional vector spaces will not be equipped with any topology. Let us stress that using precise terminology and notation concerning linear algebra is very useful in describing various aspects of quantization and quantum fields. Even though the material of this chapter is elementary, the terminology and notation introduced in this chapter will play an important role throughout our work. In particular we should draw the reader's attention to the notion of the complex conjugate space (Subsect. 1.2.3), and of the holomorphic and antiholomorphic subspaces (Subsect. 1.3.6). Throughout the book K will denote either the field ℝ or ℂ, all vector spaces being either real or complex, unless specified otherwise. Elementary linear algebra The material of this section is well known and elementary. Among other things, we discuss four basic kinds of structures, which will serve as the starting point for quantization: (1) Symplectic spaces – classical phase spaces of neutral bosons, (2) Euclidean spaces – classical phase spaces of neutral fermions, (3) Charged symplectic spaces – classical phase spaces of charged bosons, (4) Unitary spaces – classical phase spaces of charged fermions.
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.
Quantum mechanics is like a pot: it is almost indestructible and extremely rigid, but also very flexible because you can use any ingredients for your soup. Göran Lindblad Quantum mechanics Although our first four chapters have been very mathematical, quantum mechanics has never been very far away. Let us recall how the mathematical structure of quantum mechanics is usually summarized at the end of a first course in the subject. First of all the pure states are given by vectors in a Hilbert space. If that Hilbert space is finite dimensional it is simply the vector space equipped with a scalar product of the particular kind that we called a Hermitian form in Eq. (3.82). Actually a pure state corresponds to an entire equivalence class of vectors; this is usually treated in such a way that the vectors are normalized to have length one and afterwards vectors differing by an overall phase e iø are regarded as physically equivalent. In effect then the space of pure states is the complex projective space; as always in this book n = N – 1. The notation used in quantum mechanics differs from what we have used so far. We have denoted vectors in by Z α , while in quantum mechanics they are usually denoted by a ket vector. We can think of the index α as just a label telling us that Z α is a vector, and then these two notations are in fact exactly equivalent.
Linear vector space is to quantum mechanics what calculus is to classical mechanics. In this chapter the essential ideas of linear vector spaces will be discussed. The reader is already familiar with vector calculus in three-dimensional Euclidean space E 3 (Chapter 1). We therefore present our discussion as a generalization of elementary vector calculus. The presentation will be, however, slightly abstract and more formal than the discussion of vectors in Chapter 1. Any reader who is not already familiar with this sort of discussion should be patient with the first few sections. You will then be amply repaid by finding the rest of this chapter relatively easy reading. Euclidean n - space E n In the study of vector analysis in E 3 , an ordered triple of numbers ( a 1 , a 2 , a 3 ) has two different geometric interpretations. It represents a point in space, with a 1 , a 2 , a 3 being its coordinates; it also represents a vector, with a 1 , a 2 , and a 3 being its components along the three coordinate axes (Fig. 5.1). This idea of using triples of numbers to locate points in three-dimensional space was first introduced in the mid-seventeenth century. By the latter part of the nineteenth century physicists and mathematicians began to use the quadruples of numbers ( a 1 , a 2 , a 3 , a 4 ) as points in four-dimensional space, quintuples ( a 1 , a 2 , a 3 , a 4 , a 5 ) as points in fivedimensional space etc. We now extend this to n -dimensional space E n , where n is a positive integer. Although our geometric visualization doesn't extend beyond three-dimensional space, we can extend many familiar ideas beyond three-dimensional space by working with analytic or numerical properties of points and vectors rather than their geometric properties.
Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.
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.