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.

After a general introduction to multivariate statistical analyses, we focus on describing the task of multivariate classification, distinguishing its non-hierarchical and hierarchical forms. Focusing on hierarchical agglomerative classification methods (cluster analysis), we highlight the important decisions that must be made regarding the measurement of dissimilarity (distance) among objects. Following this, we explain the construction of dendrograms representing this hierarchical classification. We also briefly mention divisive classification methods, focusing on the TWINSPAN method. The methods described in this chapter are accompanied by a carefully-explained guide to the R code needed for their use, in this case employing the cluster package.