= 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 $t\ge0$, 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, $L=\sqrt\gamma\,|0\rangle\langle1|$ 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 $L'=\sqrt\kappa\,|1\rangle\langle0|$ 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.<Criteria for Davies Irreducibility of Markovian Quantum Dynamics by Yikang Zhang and Thomas Barthel>[Sections 3 and 6, invariant supports and two-level examples]{ref}
Back to article page
