Yikang Zhang, with Thomas Barthel, characterized Davies irreducibility in quantum dynamical semigroups. In their finite-dimensional criterion, the multiplicative algebra generated by the Lindblad operators and must be the full operator algebra, with .[ref]
Criteria for Davies Irreducibility of Markovian Quantum Dynamics by Yikang Zhang and Thomas Barthel. Main criterion; corrected drift-operator normalization in the current version
This makes dynamical accessibility an algebraic question. Changing a Hamiltonian term or a dissipator can change irreducibility even when much of the model is left in place.
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