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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]

G. Lindblad characterized generators of completely positive quantum dynamical semigroups. In finite-dimensional trace-preserving dynamics, the corresponding Lindblad equation can be written, with $\hslash =1$,where $H$ is a Quantum Hamiltonian, $\rho$ a density matrix and $L_a$ the Lindblad operators.^[ref]

Complete positivity preserves positivity even when the system is considered together with an untouched auxiliary quantum system. The semigroup and continuity assumptions are substantive: the displayed time-independent Markovian form does not describe every possible reduced evolution with memory.

A quantum dynamical semigroup is a continuous family $({\Phi }_t{)}_{t\ge 0}$ of linear maps on density matrices satisfying ${\Phi }_0=I$ and ${\Phi }_{t+s}={\Phi }_t\circ {\Phi }_s$, with each map completely positive and trace preserving. The composition law expresses time-homogeneous evolution with no dependence on how the interval is divided.

In finite dimensions, continuity gives ${\Phi }_t=e^{t\mathcal{L}}$, whose generator has Lindblad equation form. For a qubit dephasing example,Off-diagonal decay factors multiply under composition, while populations stay fixed. The map is the mixture $(1+e^{-\gamma t})\rho /2+(1-e^{-\gamma t})Z\rho Z/2$, which verifies complete positivity. A semigroup need not have physical inverse maps at positive times: undoing this decay would generally amplify coherence beyond valid states.^[ref]

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.