Entanglement of Purification for Multipartite States and its Holographic Dual

TL;DR

The article generalizes the entanglement of purification to multipartite systems via $Δ_P$, establishing its operational bounds against multipartite mutual information and relating it to multipartite squashed entanglement. It then proposes a holographic dual $Δ_W$, defined as a minimal-area construction within the entanglement wedge, and proves that $Δ_W$ satisfies the same information-theoretic properties as $Δ_P$, motivating the conjecture $Δ_P=Δ_W$ in large-$N$ theories. The authors explicitly compute $Δ_W$ in AdS$_3$/CFT$_2$ for pure AdS$_3$ and BTZ black holes, providing evidence in support of the conjecture and illustrating phase structure in holographic geometries. They discuss saturation and monogamy properties, potential operational interpretations, and future directions, including time-dependent backgrounds and connections to holographic squashed entanglement. Overall, the work offers a unified framework linking multipartite correlation measures to geometric duals, with implications for understanding how quantum information imprints onto spacetime geometry.

Abstract

We introduce a new information-theoretic measure of multipartite quantum/classical correlations $Δ_P$, 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 $Δ_W$, 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 $Δ_{P}=Δ_{W}$ 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.

Figures (11)

• Figure 1.1: An example of minimal surfaces which gives $\Delta_{W}$ for a tripartite setup.
• Figure 3.1: An example of tripartite entanglement wedge cross-section. The black bold dashed lines represents the minimal surface $\Gamma_{ABC}^{min}$, giving a part of the boundary of $M_{ABC}$. The yellow thin dashed lines represents $\Sigma_{ABC}^{min}$ whose area (divided by 4$G_{N}$) is $\Delta_{W}$.
• Figure 3.2: An example of tripartite entanglement wedge cross-section in a black hole geometry. Each surface of $\Sigma_{ABC}^{min}$ is doubled.
• Figure 3.3: The proof of an upper bound of $\Delta_{W}$. The sum of blue real lines is $S_{A}+S_{B}+S_{AB}$ and the sum of dashed yellow lines are $\Delta_{W}(\rho_{ABC})$. Clearly $\Delta_{W}(\rho_{ABC})\leq S_{A}+S_{B}+S_{AB}$ holds somewhat trivially since $S_{A}+S_{B}+S_{AB}$ has UV divergences while $\Delta_{W}$ does not. When two of subsystems share the boundary, $\Delta_{W}$ also diverges, but it is always weaker than $S_{A}+S_{B}+S_{AB}$ shown by a graph.
• Figure 3.4: The proof of a lower bound of $\Delta_{W}$. The sum of dashed yellow lines is $\Delta_{ABC}+S_{ABC}$ and the sum of real blue lines are $S_{A}+S_{B}+S_{C}$. Clearly $\Delta_{W}(\rho_{ABC})+S_{ABC}\geq S_{A}+S_{B}+S_{C}$ follows since the entanglement entropy are defined as minimal surfaces.

Theorems & Definitions (24)

• Definition 1
• Lemma 2
• proof
• Proposition 3
• proof
• Proposition 4
• Proposition 5
• proof
• Proposition 6
• proof