Mixed State Entanglement and Thermal Phase Transitions

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.