On some properties of the time evolution of quadratic open quantum systems

The paper is aimed at revisiting some mathematical properties that appear in open quantum systems with quadratic Hamiltonians. In particular, a discussion is included of some properties related to the notions of purity, separability, and entanglement, introduced by physicists in quantum information theory. The results are applied, among other things, to coupled harmonic oscillators appearing in the quantum Brownian motion. Moreover, the master equation is revisited, which was obtained for the time evolution of an open system coupled to an environment modeled by any quadratic Hamiltonians.