Relationship between the G-cond and quantum-diffusion criteria for irreducibility

Determine whether the condition that the complex linear span of the iterated commutators of the generalized GKLS drift operator G with the linear jump operators contains all creation and annihilation operators is equivalent to the condition that no nontrivial Z-invariant subspace of C^{2d} is contained in the kernel of the quantum diffusion matrix C_Z, without imposing additional technical assumptions or the existence of a normal invariant state.

Background

The paper reviews two algebraic criteria for irreducibility of Gaussian quantum Markov semigroups. The first, denoted G-cond, requires the complex span of the jump operators and all their iterated commutators with G to contain every creation and annihilation operator. The second, denoted C_Z-cond, requires that the kernel of the quantum diffusion matrix C_Z contain no nontrivial Z-invariant subspace.

Prior work had established equivalence in one mode and under additional hypotheses in multimode settings, while C_Z-cond had also been characterized under the assumption that the semigroup admits a normal invariant state. The paper subsequently proves the equivalence in the general GQMS setting, so the issue is explicitly identified as an open problem in the literature but is resolved by the paper itself.