# Algebraic approach to quantum theory: a finite-dimensional guide. (arXiv:1505.03106v4 [quant-ph] UPDATED)

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.