A qubit is the classical version of a bit in the sense that it can take one of two values. But the key idea of the quantum world is that it can, in fact, take both values at the same time. Here we explore the physics of the qubit and use it as a vehicle to better understand some of the stranger features of quantum mechanics.

In this chapter, we present an introduction to an important area of contemporary quantum physics: quantum information and quantum entanglement. After a brief introduction regarding why and how linear algebra is so useful in this area, we first consider the concepts of quantum bits and quantum gates in quantum information theory. We next explore some geometric features of quantum bits and quantum gates. Then we study the phenomenon of quantum entanglement. In particular, we shall clarify the notions of untangled and entangled quantum states and establish a necessary and sufficient condition to characterize or divide these two different categories of quantum states. Finally, we present Bell’s theorem which is of central importance for the mathematical foundation of quantum mechanics implicating that quantum mechanics is nonlocal.

This chapter surveys several different mathematical methods for time-dependent change of quantum states using quantum field theory. The Bloch sphere method is introduced, which can be used to show the physics discussed in Chapter 3, that electronic transitions, or “jumps,” are not instantaneous.

This chapter introduces the basic mathematical formalism for working with quantum information. We discover qubits, or quantum bits, how to combine them using the tensor product, and how to measure them by choosing a basis. We discuss unitary operations, which are elementary transformations on qubits. The chapter ends with a convenient representation of qubits as vectors on the 3-dimensional Bloch sphere, and a useful “cheat sheet,” which summarizes useful definitions and identities.

Appendix D: two-level quantum mechanical systems, or qubits. Description in terms of Bloch vector. Poincaré sphere. Expression of purity. Projection noise in an energy measurement. Description of a set of N coherently driven qubits by a collective Bloch vector.