On Irreducibility of Gaussian Quantum Markov Semigroups

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a $d$-mode Fock space is represented in a generalized Gorini-Kossakowski-Lindblad-Sudharsan form
$$ x\mapsto G^*x + \sum_{\ell}L_\ell^*\, x L_\ell + x\, G $$
with an operator $G$ quadratic in creation and annihilation operators and Kraus operators $L_1,\dots,L_m$ linear in creation and annihilation operators. Kraus operators, commutators $[G,L_\ell]$ and iterated commutators $[G,[G,L_\ell]],\dots$ up to the order $2d-m$, as linear combinations of creation and annihilation operators determine a vector in $\mathbb{C}^{2d}$. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate $\mathbb{C}^{2d}$, under the technical condition that the domains of $G$ and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with $G$ is fully non-commutative. We discuss open problems and illustrate them by examples.