Chapter 5 delves into divergences and distance measures, which are crucial for comparing quantum states. It begins with classical divergences such as the Kullback–Leibler and Jensen–Shannon, then advances to their quantum counterparts, discussing their optimal characteristics. Influenced by quantum resource theories, these quantum extensions provide foundational insights into the robust tools of resource theories. The chapter concentrates on particular divergences that serve as true metrics, including the trace distance and a variant of the fidelity, and explores the concept of distance between subnormalized states, which is essential in the context of quantum measurements. It emphasizes the purified distance, a useful tool for understanding the entanglement cost of quantum systems, setting the stage for further exploration in later chapters. The chapter offers a mathematically approachable survey of these measures, underscoring their practical importance in quantum information theory.

How do we define knowledge, and, crucially for cryptography, ignorance? In this chapter we lay the basis for future security proofs by formalizing the notion of knowledge of a quantum party, such as the memory of an eavesdropper, about a classical piece of information, such as a secret key. For this we introduce an appropriate measure of conditional entropy, the min-entropy, and introduce important tools to bound it using guessing games.