Chapter 2 serves as an introduction to the fundamental principles of quantum mechanics, focusing on closed systems. It begins with the historic Stern–Gerlach experiment, highlighting the discovery of quantum spin. The narrative then shifts to the mathematical framework of quantum mechanics, covering inner product spaces, Hilbert spaces, and linear operators. These concepts are crucial for understanding the behavior and manipulation of quantum states, the core of quantum information theory.

The chapter further explores the encoding of information in quantum states, emphasizing qubits, and discusses quantum measurements, revealing the probabilistic nature of quantum mechanics. Additionally, it addresses hidden variable models, offering insights into the deterministic versus probabilistic interpretations of quantum phenomena.

Unitary evolution and the Schrödinger equation are introduced as mechanisms for the time evolution of quantum states, showcasing the deterministic evolution in the absence of measurements. This section underscores the dynamic aspect of quantum systems, pivotal for advancements in quantum information theory.

We introduce inner product spaces. After proving the Cauchy-Schwarz inequality we show that any inner product induces a norm and that the norm then satisfies the parallelogram identity. We show that the inner product can be recovered from the induced norm (via the polarisation identity). We define Hilbert spaces as complete inner product spaces and show that the spaces l^2 and L^2 are Hilbert spaces.