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.

In this chapter we discuss the application of entanglement to quantum optical interferometry and to quantum information processing. Quantum random number generation is discussed. Quantum cryptography is discussed, as is quantum computing. The quantum optical realization of some quantum gates is discussed.

One of the first applications of quantum information to cryptography to be discovered is to the creation of money that cannot be copied. Due to the no-cloning principle, which states that there is no procedure that can copy an arbitrary quantum state, we can hope to create perfectly secure money based on quantum information. In this chapter we study how this can be done by following Wiesner’s idea from the 1970s. To analyze the security of Wiesner’s scheme we develop a formalism for general quantum attacks by studying quantum channels, and encounter some limitations of Wiesner’s scheme.

Appendix C: treatment of composite systems with the help of the tensor product of Hilbert spaces and partial trace operation. No-cloning theorem.