On the Role of Hamiltonians for Dissipative Entanglement Engineering*

Determining whether an entangled state of interest can be asymptotically prepared by realistic open-system dynamics has important applications across quantum engineering. This problem has been recently solved for purely dissipative quasi-local dynamics described by a continuous-time Markovian semigroup. Here, we extend our previous analysis by addressing the role of internal Hamiltonian dynamics as well as of Hamiltonian control resources for achieving the same task. We show how Hamiltonians that are not frustration-free can genuinely extend the class of stabilizable states. In particular, we present stabilizing Hamiltonians, along with necessary and sufficient conditions for their existence, for maximally entangled GHZ-states and translationally invariant W-states, none of which are generally stabilizable by dissipation alone.