Entanglement of Purification for Multipartite States and its Holographic Dual

TL;DR

This paper introduces a new multipartite entanglement measure, proves its properties, and proposes a holographic dual, supporting the conjecture that these measures are equivalent in the context of AdS/CFT correspondence.

Contribution

It generalizes the entanglement of purification to multipartite states, establishes its properties, and defines a holographic dual based on minimal surface areas, supporting the Δ_P=Δ_W conjecture.

Findings

Multipartite entanglement of purification bounds multipartite mutual information.

The holographic dual Δ_W satisfies all properties of Δ_P.

Multipartite entanglement of purification bounds multipartite squashed entanglement.

Abstract

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.