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 Rényi entropy and the LOCC convertibility with assisted entanglement. [10], have also been used for detecting quantum phase transitions. Other investigations in entanglement spectra [11][12][13] and fidelity [14] as well as the fidelity susceptibility of the ground state show their abilities in exploring numerous phase transition points in various critical systems, as well. These observations have achieved great success in understanding the deep nature of the different phase transitions with the tools of quantum information science harnessed in the analysis of quantum many-body systems. The reverse, however, seems unclear and tough.From a new point of view, Jian Cui and his collaborators reveal that the systems undergoing quantum phase transition will also exhibit different operational properties from the perspective of QIP [15,16]. They demonstrate with several models that nearly at the critical points, the entanglement-assisted local operations and classical communications (ELOCC) convertibility decided via the Rényi entropy interception changes * Electronic address: yrzhang@iphy.ac.cn † Electronic address: sxqu@snnu.edu.cn ‡ Electronic address: hfan@iphy.ac.cn suddenly, see Fig. 1. The Rényi entropy is defined as [17]and it tends to the von Neumann entropy as α → 1. The unexpected results suggest that not only are the tools of quantum information useful as alternative signatures of quantum phase transitions, but the study of quantum phase transitions may also offer additional insight into QIP. However, the most basic and natural local conversion is local operations and classical communications (LOCC), which has fabulous applications in QIP, such as teleportation [18], quantum states discrimination [19], and quantum channel corrections [20]. In this paper, 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 [21] when a quantum phase transition occurs. We study the one-dimensional transverse field Ising model and show that the LOCC convertibility changes nearly at the phase transition point. Presenting the relation between the order of quantum phase transitions and the LOCC convertibility, we compare our results with the corresponding resul...