  Distilling highly entangled quantum states from weaker ones is a process that
is crucial for efficient and long-distance quantum communication, and has
implications for several other quantum information protocols. We introduce the
notion of distillation under limited resources, and specifically focus on the
energy constraint. The corresponding protocol, which we call the canonical
distillation of entanglement, naturally leads to the set of canonically
distillable states. We show that for non-interacting Hamiltonians, almost no
states are canonically distillable, while the situation can be drastically
different for interacting ones. Several paradigmatic Hamiltonians are
considered for bipartite as well as multipartite canonical distillability. The
results have potential applications for practical quantum communication
devices.
