 Davies irreducibility

A finite-dimensional quantum dynamical semigroup is Davies irreducible when no proper nonzero subspace can support a state for all future times. Equivalently, if a projection $P$ satisfies ${\Phi }_t(P\rho P)=P{\Phi }_t(P\rho P)P$ for every state $\rho$ and every $t\ge 0$, then $P=0$ or $P=I$. This is a property of invariant supports, rather than merely uniqueness of a stationary density matrix.

For a two-level system with only spontaneous decay, $L=\sqrt{\gamma }|0⟩⟨1|$ and zero Quantum Hamiltonian, the ground-state line is invariant. The stationary state is unique but pure, so the dynamics is reducible. Adding a positive excitation rate $L^{\prime }=\sqrt{\kappa }|1⟩⟨0|$ removes that invariant line; the stationary populations are $\gamma /(\gamma +\kappa )$ and $\kappa /(\gamma +\kappa )$ and the dynamics is irreducible. In finite dimensions irreducibility ensures a unique faithful stationary state, which has full rank. Related notions of quantum irreducibility must be distinguished.^[ref]