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Institute of Physics eBooks

Two-level quantum systems

Shinil Cho

Published Jul 1, 2022• 0 citations
Publisher

Abstract

Chapter 1 summarizes concepts and rules of vectors and matrices used in this book, and fundamental knowledge of quantum mechanics. They are applied throughout this book.
DOI
10.1088/978-0-7503-3963-6ch1

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In this chapter, I will introduce the concepts of quantum information that are necessary to the understanding of quantum cryptography. I first review the main principles of quantum mechanics. I then introduce concepts that somehow translate information theory into the quantum realm and discuss their unique features. Finally, I conclude this chapter with some elements of quantum optics. Fundamental definitions in quantum mechanics In classical mechanics, a system is described with physical quantities that can take certain values at a given moment in time; we say that is has a state. For instance, the state of an elevator comprises its position and speed at a given time. As dictated by common sense, if the elevator is at a given height, it cannot be simultaneously found at another location. With quantum mechanics, things are fundamentally different and elementary particles can behave against common sense. For instance, a quantum system can simultaneously be in different levels of energy. This is not because our knowledge of the state of the system is incomplete: the behavior of the quantum system is consistent with the fact that it is in simultaneous levels of energy. It would go beyond the scope of this book to describe quantum mechanics in detail. Instead, I propose a synthetic introduction suitable for the understanding of quantum cryptography. For a more detailed introduction, please refer to the books listed at the end of this chapter.
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.