  Conversions between the ground states in quantum critical systems via
entanglement-assisted local operations and classical communications (eLOCC) are
studied. We propose a new method to reveal the different convertibility by
local operations when a quantum phase transition occurs. We have studied the
ground state local convertibility in the one dimensional transverse field Ising
model, XY model and XXZ model. It is found that the eLOCC convertibility sudden
changes at the phase transition points. In transverse field Ising model the
eLOCC convertibility between the first excited state and the ground state are
also distinct for different phases. The relation between the order of quantum
phase transitions and the local convertibility is discussed.
