In this lecture, we start with an introduction to the Gaussian states on the CCR algebra of Weyl operators. Let pj,qj, 1≤j≤n, be the momentum and position observables in L2(Rn). A state ρ in L2(Rn) is said to be Gaussian if for every xj,yj∈R, 1≤j≤n, the observable ∑j=1n(xjpj+yjqj) has a normal distribution on the real line in the state ρ. We explain a characterization for Gaussian states and give some basic properties of Gaussian states.

Next, we discuss Gaussian Markov semigroups, namely, quantum dynamical systems which preserve the Gaussian states. Finally, we consider a bipartite Gaussian system interacting with environments. We show that such a system may possess a unique Gaussian entangled stationary state and that any initial state converges towards this stationary state. We investigate the dependence of entanglement on temperature and interaction strength.