  The dynamics of Markovian open quantum systems are described by Lindblad
master equations, generating a quantum dynamical semigroup. An important
concept for such systems is (Davies) irreducibility, i.e., the question whether
there exist non-trivial invariant subspaces. Steady states of irreducible
systems are unique and faithful, i.e., they have full rank. In the 1970s,
Frigerio showed that a system is irreducible if the Lindblad operators span a
self-adjoint set with trivial commutant. We discuss a more general and powerful
algebraic criterion, showing that a system is irreducible if and only if the
multiplicative algebra generated by the Lindblad operators $L_a$ and the
operator $K=iH+\sum_a L^\dagger_aL_a$, involving the Hamiltonian $H$, is the
entire operator space. Examples for two-level systems, show that a change of
Hamiltonian terms as well as the addition or removal of dissipators can render
a reducible system irreducible and vice versa. Examples for many-body systems
show that a large class of spin chains can be rendered irreducible by
dissipators on just one or two sites. Additionally, we discuss the decisive
differences between (Davies) reducibility and Evans reducibility for quantum
channels and dynamical semigroups which has lead to some confusion in the
recent physics literature, especially, in the context of boundary-driven
systems. We give a criterion for quantum reducibility in terms of associated
classical Markov processes and, lastly, discuss the relation of the main result
to the stabilization of pure states and argue that systems with local Lindblad
operators cannot stabilize pure Fermi-sea states.
