  We study sequences (both cyclic and randomized) of idempotent
completely-positive trace-preserving quantum maps, and show how they
asymptotically converge to the intersection of their fixed point sets via
alternating projection methods. We characterize the robustness features of the
protocol against randomization and provide basic bounds on its convergence
speed. The general results are then specialized to stabilizing en- tangled
states in finite-dimensional multipartite quantum systems subject to a resource
constraint, a problem of key interest for quantum information applications. We
conclude by suggesting further developments, including techniques to enlarge
the set of stabilizable states and ensure efficient, finite-time preparation.
