  We study the relationship between mixed state entanglement and thermal phase
transitions. As a typical example, we compute the holographic entanglement
entropy (HEE), holographic mutual information (MI) and the holographic
entanglement of purification (EoP) over the superconductivity phase transition.
We find that HEE, MI and EoP can all diagnose the superconducting phase
transition. They are continuous at the critical point, but their first
derivative with respect to temperature is discontinuous. MI decreases with
increasing temperature and exhibits a convex behavior, while HEE increases with
increasing temperature and exhibits a concave behavior. However, EoP can
exhibit either the same or the opposite behavior as MI, depending on the size
of the specific configuration. These results show that EoP captures more
abundant information than HEE and MI.
  We also provide a new algorithm to compute the EoP for general
configurations.
