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]
Criteria for Davies Irreducibility of Markovian Quantum Dynamics by Yikang Zhang and Thomas Barthel. Abstract and main algebraic irreducibility criterion
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.
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