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.
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.
The content of this chapter may serve as yet another supplemental topic to meet the needs and interests beyond those of a usual course curriculum. Here we shall present an over-simplified, but hopefully totally transparent, description of some of the fundamental ideas and concepts of quantum mechanics, using a pure linear algebra formalism. Vectors in ℂ n and Dirac bracket Consider the vector space ℂ n , consisting of column vectors, and use { e 1 , …, e n } to denote the standard basis of ℂ n . For u , υ ∈ ℂ n with recall that the Hermitian scalar product is given by so that { e 1 , …, e n } is a unitary basis, satisfying ( e i , e j ) = δ ij , i, j = 1, …, n . In quantum mechanics, it is customary to rewrite the scalar product (9.1.2) in a bracket form, 〈 u | v 〉. Then it was Dirac who suggested to view 〈 u |υ〉 as the scalar pairing of a ‘bra’ vector 〈 u | and a ‘ket’ vector |υ〉, representing the row vector u † and the column vector υ. Thus we may use | e 1 〉, …, | e n 〉 to denote the standard basis vectors of ℂ n and represent the vector u in ℂ n as Therefore the bra-counterpart of | u 〉 is simply given as As a consequence, the orthonormal condition regarding the basis { e 1 , …, e n } becomes and the Hermitian scalar product of the vectors | u 〉 and |υ〉 assumes the form For the vector | u 〉 given in (9.1.3), we find that a i = 〈 e i | u 〉, i = 1, …, n . Now rewriting | u 〉 as and inserting (9.1.7) into (9.1.8), we obtain which suggests that the strange-looking ‘quantity’, should naturally be identified as the identity mapping or matrix, which readily follows from the associativity property of matrix multiplication. Similarly, we have Thus (9.1.10) can be applied to both bra and ket vectors symmetrically and what it expresses is simply the fact that | e 1 〉, …,| e n form an orthonormal basis of ℂ n .
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.
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.
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.
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
According to Dirac's bra-ket notation, in an inner-product space, the inner product ⟨x∣y⟩ of vectors x,y can be viewed as an application of the bra ⟨x∣ to the ket ∣y⟩. Here ⟨x∣ is the linear functional ∣y⟩↦⟨x∣y⟩ and ∣y⟩ is the vector y. But often -- though not always -- there are advantages in seeing ∣y⟩ as the function a↦a⋅y where a ranges over the scalars. For example, the outer product ∣y⟩⟨x∣ becomes simply the composition ∣y⟩∘⟨x∣. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.