Convergence to pure steady states of linear quantum systems

We study the convergence of density operators of linear quantum systems towards their steady states in trace norm. We give compact and intuitive necessary and sufficient conditions for such linear quantum systems to have pure Gaussian steady states in terms of the Hamiltonian and coupling operators. Furthermore, using the nullifiers concept in Gu et al (2009 Phys. Rev. A 79), we show that set of the nullifiers of these states are spanned by the coupling operators and other operators obtained via the commutation of Hamiltonian and coupling operators. (paper)