Introduction

To make this thesis self-contained, a brief summary of quantum mechanics is given in this chapter. A reader who is familiar with quantum mechanics can skim or skip this chapter. For a thorough treatment of quantum mechanics, one can refer to [25] or [12]. A good introduction can be found in The Feynman's Lectures on Physics [26]. The notation used in this thesis is mostly due to P. A. M. Dirac [12]. A brief summary of quantum computation is also presented in Section 3.5. For more details, the reader can refer to [27]. For quantum computation in general and its applications, the reader can refer to [28] and the references therein.

To begin with, quantum mechanics is one of the greatest triumphs of modern science. Indeed, it is the most important foundation of modern physics. Perhaps more importantly, quantum mechanics provides an adequate account for atomic events which in turn offer a theoretical foundation for chemistry and molecular biology. This chain goes further and further and, as many believe, will eventually encompass all natural sciences^3.1.

Quantum mechanics is a theory describing the physical world of very small scale. In fact, any theory of atoms -- or any other elementary building blocks of matter -- intrinsically has an absolute concept of what is large and what is small, for otherwise the substance can be further divided into yet smaller parts, ad infinitum, according to the continuity of physical substance and physical laws. As Dirac stated,

[I]n order to give an absolute meaning to size, such as is required for any theory of the ultimate structure of matter, we have to assume that there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance -- a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer (p.3-4 [12].)

This is the fundamental principle of quantum mechanics known as Heisenberg's Uncertainty Principle. Specifically, Heisenberg's Uncertainty Principle states

$$\Delta p \Delta q \ge {1\over2} \hbar$$ (3.1)

where $p$ and $q$ being canonical momentum and coordinate; $\hbar=h /2 \pi$ with $h$ being the Planck Constant ( $h=6.62608\,{10}^{-34}$ Joule Second); $\Delta S=\sqrt {\langle (S-\langle S \rangle)^2 \rangle}$, for $S \in \{p,q\}$; $\langle \cdot \rangle$ is the expectation value. Before delving into the formalism of quantum mechanics, we start with two experiments which, we hope, can disclose the key properties (and indeed ``strangeness'') of quantum mechanics.