Abstract In this chapter we discuss statistical mixtures of pure quantum states, resulting in so-called mixed states, and their representation using density operators. The properties of density operators for pure and mixed states are analysed and we examine their time evolution governed by the Liouville-von Neumann equation. As particular examples of interest we consider the density operators for thermal states, focusing on harmonic oscillators and the application of this model to calculate the average photon number for the modes of the electromagnetic field at thermal equilibrium. We then turn to two-level systems, or qubits, which are of particular relevance in quantum information theory and we inspect the geometry of the state space of density operators as a convex subspace of the Hilbert-Schmidt space. Finally, we take a closer look at density operators for bipartite systems which forms the mathematical basis for considering entanglement in later chapters.
In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix
ρ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of
ρ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.