Mathematical Preliminary I: Linear Vector Space

Songbo Xie

V. Balakrishnan

G. F. Roach

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.

A. R. Marlow

In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.

Ralph E. Christoffersen

Phillip Kaye, Raymond Laflamme, Michele Mosca

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.