  We introduce a new information-theoretic measure of multipartite
quantum/classical correlations $\Delta_P$, by generalizing the entanglement of
purification to multipartite states. We provide proofs of its various
properties, focusing on several entropic inequalities, in generic quantum
systems. In particular, it turns out that the multipartite entanglement of
purification gives an upper bound on multipartite mutual information, which is
a generalization of quantum mutual information in the spirit of relative
entropy. After that, motivated by a tensor network description of the AdS/CFT
correspondence, we also define a holographic dual of multipartite entanglement
of purification $\Delta_W$, as a sum of minimal areas of codimension-2 surfaces
which divide the entanglement wedge into multi-pieces. We prove that this
geometrical quantity satisfies all properties we proved for the multipartite
entanglement of purification. These agreements strongly support the
$\Delta_{P}=\Delta_{W}$ conjecture. We also show that the multipartite
entanglement of purification gives an upper bound on multipartite squashed
entanglement, which is a promising measure of multipartite quantum
entanglement. We discuss potential saturation of multipartite squashed
entanglement onto multipartite mutual information in holographic CFTs and its
applications.
