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Journal of Physics A: Mathematical and Theoretical 57, 115301. By Yikang Zhang and Thomas Barthel.

Verified version 4, section 2, Theorem 2.2: scalar commutant implies uniqueness when a faithful stationary state exists. Section 2 explains why that hypothesis matters.

Thomas Barthel and Yikang Zhang established algebraic tests for Davies irreducibility of Markovian open-system quantum evolution. Their criterion connects the Lindblad operators and effective drift operator with the algebra they generate.^[ref]

For a finite-dimensional irreducible system, the stationary density matrix is unique and full rank. The work distinguishes different notions of reducibility, preventing a convenient algebraic test from being applied to the wrong dynamical definition.

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 $K=iH+\frac{1}{2}{\sum }_a{L}_a^{†}{L}_a$ must be the full operator algebra, with $\hslash =1$.^[ref]

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.