  We investigate under which conditions a mixed state on a finite-dimensional
multipartite quantum system may be the unique, globally stable fixed point of
frustration-free semigroup dynamics subject to specified quasi-locality
constraints. Our central result is a linear-algebraic necessary and sufficient
condition for a generic (full-rank) target state to be frustration-free
quasi-locally stabilizable, along with an explicit procedure for constructing
Markovian dynamics that achieve stabilization. If the target state is not
full-rank, we establish sufficiency under an additional condition, which is
naturally motivated by consistency with pure-state stabilization results yet
provably not necessary in general. Several applications are discussed, of
relevance to both dissipative quantum engineering and information processing,
and non-equilibrium quantum statistical mechanics. In particular, we show that
a large class of graph product states (including arbitrary thermal graph
states) as well as Gibbs states of commuting Hamiltonians are frustration-free
stabilizable relative to natural quasi-locality constraints. Likewise, we
provide explicit examples of non-commuting Gibbs states and non-trivially
entangled mixed states that are stabilizable despite the lack of an underlying
commuting structure, albeit scalability to arbitrary system size remains in
this case an open question.
