  We study the mixed state entanglement of an asymptotic AdS black hole system
with scalar hair. Through numerical calculations, we find that the holographic
entanglement entropy (HEE) presents a non-monotonic behavior with the system
parameter, depending on the size of the subregion. In addition, the mutual
information (MI) also shows non-monotonic behavior in certain ranges of system
parameters. However, the entanglement wedge minimum cross-section (EWCS), which
is a mixed state entanglement measure, increases monotonically with the AdS
radius; meanwhile, shows non-monotonic with the temperature. We also give
analytical understandings of the phenomena above.
