Abstract
We introduce a new information-theoretic measure of multipartite correlations ΔP, 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 conjecture a holographic dual of multipartite entanglement of purification ΔW, 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 ΔP = ΔW conjecture. We also show that the multipartite entanglement of purification is larger than 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.
Highlights
Another important direction is to explore multipartite correlation measure and its geometric dual
After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification ∆W, as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces
We show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement
Summary
We will define a generalization of entanglement of purification for multipartite correlations and prove its various information-theoretic properties. Where the minimization is taken over purifications ρAB = TrA B [|ψ ψ|AA BB ] This is an information-theoretic measure of total correlations, namely, it captures both quantum and classical correlations between A and B. (VIIb) For a class of states that saturate the strong subadditivity, i.e. SAB + SAC = SB + SC, it reduces to the entanglement entropy, EP (A : B) = SA when SAB + SAC = SB + SC. These properties are not independent from each other and one can prove (VI) from (I) and (Va), and prove (VIIa) (or (VIIb)) from (III) and (Va) (or (Vb)). We refer to [23, 48] for detailed proofs and discussion
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