  This document is meant as a pedagogical introduction to the modern language
used to talk about quantum theory, especially in the field of quantum
information. It assumes that the reader has taken a first traditional course on
quantum mechanics, and is familiar with the concept of Hilbert space and
elementary linear algebra. As in the popular textbook on quantum information by
Nielsen and Chuang, we introduce the generalised concept of states (density
matrices), observables (POVMs) and transformations (channels), but we also
characterise these structures from an algebraic standpoint, which provides many
useful technical tools, and clarity as to their generality. This approach also
makes it manifest that quantum theory is a direct generalisation of probability
theory, and provides a unifying formalism for both fields. The focus on
finite-dimensional systems allows for a self-contained presentation which
avoids many of the technicalities inherent to the more general $C^*$-algebraic
approach, while being appropriate for the quantum information literature.
