  Quantum technologies exploiting bipartite entanglement could be made more
efficient by using states having the minimum amount of energy for a given
entanglement degree. Here, we study how to generate these states in the case of
a bipartite system of arbitrary finite dimension either by applying a unitary
transformation to its ground state or through a zero-temperature thermalization
protocol based on turning on and off a suitable interaction term between the
subsystems. In particular, we explicitly identify three possible unitary
operators and five possible interaction terms. On one hand, two of the three
unitary transformations turn out to be easily decomposable in terms of local
elementary operations and a single nonlocal one, making their implementation
easier. On the other hand, since the thermalization procedures can be easily
adapted to generate many different states, we numerically show that, for each
degree of entanglement, generating minimum-energy entangled states costs, in
general, less than generating the vast majority of the other states.
