Conditional and multipartite entanglements of purification and holography

In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purificiation is now optimized in either a constrained way or over multiple parties. In doing so, we separately derive new constraints on both the entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. Further, we also find inequalities that are true only for the holographic dual object, thus providing additional consistency checks for states dual to (semi)-classical bulk geometries.