  We investigate whether a generic multipartite pure state can be the unique
asymptotic steady state of locality-constrained purely dissipative Markovian
dynamics. In the simplest tripartite setting, we show that the problem is
equivalent to characterizing the solution space of a set of linear equations
and establish that the set of pure states obeying the above property has either
measure zero or measure one, solely depending on the subsystems' dimension. A
complete analytical characterization is given when the central subsystem is a
qubit. In the N-partite case, we provide conditions on the subsystems' size and
the nature of the locality constraint, under which random pure states cannot be
quasi-locally stabilized generically. Beside allowing for the possibility to
approximately stabilize entangled pure states that cannot be exact steady
states in settings where stabilizability is generic, our results offer insights
into the extent to which random pure states may arise as unique ground states
of frustration free parent Hamiltonians. We further argue that, to high
probability, pure quantum states sampled from a t-design enjoy the same
stabilizability properties of Haar-random ones as long as suitable dimension
constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a
connection between the tasks of quasi-local state stabilization and unique
state reconstruction from local tomographic information, and provide a
constructive procedure for determining a generic N-partite pure state based
only on knowledge of the support of any two of the reduced density matrices of
about half the parties, improving over existing results.
