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 GKLS form 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 ${ℂ}^{2d}$. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate ${ℂ}^{2d}$, 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.

上有界算子代数上的高斯量子马尔可夫半群的生成元$d$-mode Fock 空间以广义 GKLS 形式表示，带有一个运算符$G$创造和湮灭算子和 Kraus 算子的二次方${\mathrm{大号}}_{\mathrm{1个}},\dots \dots ,{\mathrm{大号}}_{米}$线性产生和湮灭算子。Kraus 运营商，换向器$[G,{\mathrm{大号}}_{\ell }]$和迭代换向器$[G,[G,{\mathrm{大号}}_{\ell }]],\dots \dots$直到订单$\mathrm{2个}d-米$，因为创造和湮灭算子的线性组合决定了一个矢量${ℂ}^{\mathrm{2个}d}$. 我们表明，如果此类向量生成，则高斯量子马尔可夫半群是不可约的${ℂ}^{\mathrm{2个}d}$，在以下领域的技术条件下$G$和数字运算符重合。相反，我们表明，如果克劳斯算子及其迭代换向器生成的线性空间具有$G$是完全不可交换的。