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.
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.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
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.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian 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.
Philosophy is written in that great book which continually lies open before us (I mean the Universe). But one cannot understand this book until one has learned to understand the language and to know the letters in which it is written. It is written in the language of mathematics, and the letters are triangles, circles and other geometric figures. Without these means it is impossible for mankind to understand a single word; without these means there is only vain stumbling in a dark labyrinth. Galileo Galilei Quantum theory is cast in the language of complex vector spaces. These are mathematical structures that are based on complex numbers. We learned all that we need about such numbers in Chapter 1. Armed with this knowledge, we can now tackle complex vector spaces themselves. Section 2.1 goes through the main example of a (finite-dimensional) complex vector space at tutorial pace. Section 2.2 provides formal definitions, basic properties, and more examples. Each of Section 2.3 through Section 2.7 discusses an advanced topic. Reader Tip . The reader might find some of this chapter to be “just boring math.” If you are eager to leap into the quantum world, we suggest reading the first two or three sections before moving on to Chapter 3. Return to Chapter 2 as a reference when needed (using the index and the table of contents of find specific topics).