  The most basic local conversion is local operations and classical
communications (LOCC), which is also the most natural restriction in quantum
information processing. We investigate the conversions between the ground
states in quantum critical systems via LOCC and propose an novel method to
reveal the different convertibility via majorization relation when a quantum
phase transition occurs. The ground-state local convertibility in the
one-dimensional transverse field Ising model is studied. It is shown that the
LOCC convertibility changes nearly at the phase transition point. The relation
between the order of quantum phase transitions and the LOCC convertibility is
discussed. Our results are compared with the corresponding results using the
Renyi entropy and the LOCC convertibility with assisted entanglement.
