  We study the entanglement wedge cross-section (EWCS) in holographic massive
gravity theory, in which a first and second-order phase transition can occur.
We find that the mixed state entanglement measures, the EWCS and mutual
information (MI) can characterize the phase transitions. The EWCS and MI show
exactly the opposite behavior in the critical region, which suggests that the
EWCS captures distinct degrees of freedom from that of the MI. More
importantly, EWCS, MI and HEE all show the same scaling behavior in the
critical region. We give an analytical understanding of this phenomenon. By
comparing the quantum information behavior in the thermodynamic phase
transition of holographic superconductors, we analyze the relationship and
difference between them, and provide two mechanisms of quantum information
scaling behavior in the thermodynamic phase transition.
