  We compute the minimal energy cost for extracting entanglement from the
ground state of a bosonic or fermionic quadratic system. Specifically, we find
the minimal energy increase $\Delta E_{\mathrm{min}}$ in the system resulting
from replacing an entangled pair of modes, sharing entanglement entropy $\Delta
S$, by a product state, and we show how to construct modes achieving this
minimal energy cost. Thus, we obtain a protocol independent lower bound on the
extraction of pure state entanglement from quadratic systems. Due to their
generality, our results apply to a large range of physical systems, as we
discuss with examples.
