Quantum Harmonic Analysis of the Density Matrix: Basics

In this Review we will study rigorously the notion of mixed states and their density matrices. We mostly give complete proofs. We will also discuss the quantum-mechanical consequences of possible variations of Planck's constant h. This Review has been written having in mind two readerships: mathematical physicists and quantum physicists. The mathematical rigor is maximal, but the language and notation we use throughout should be familiar to physicists.