  In this paper, we discuss the connection between two genuinely quantum
phenomena --- the discontinuity of quantum maximum entropy inference and
quantum phase transitions at zero temperature. It is shown that the
discontinuity of the maximum entropy inference of local observable measurements
signals the non-local type of transitions, where local density matrices of the
ground state change smoothly at the transition point. We then propose to use
the quantum conditional mutual information of the ground state as an indicator
to detect the discontinuity and the non-local type of quantum phase transitions
in the thermodynamic limit.
