Entanglement Wedge Cross Sections Require Tripartite Entanglement

TL;DR

This work argues that holographic CFT states demand a large amount of tripartite entanglement, challenging the Mostly-Bipartite Conjecture (MBC). By examining two strong conjectures—the reflected entropy relation $S_R(A:B)=2EW(A:B)$ and the entanglement of purification conjecture $E_P(A:B)=EW(A:B)$—the authors show that the MBC state yields $S_R(\rho_{AB})\approx I(A:B)$ and $E_P(\rho_{AB})\approx \tfrac{1}{2}I(A:B)$ at leading order, whereas holographic expectations require $S_R$ and $E_P$ to differ from mutual information by $\mathcal{O}\left(\frac{1}{G_N}\right)$. They establish a new Fannes-type continuity bound for $S_R$ and a related bound for $E_P$, proving that small corrections to the MBC cannot reconcile these conjectures with holography. Consequently, holographic CFT states must harbor $\mathcal{O}\left(\frac{1}{G_N}\right)$ amounts of tripartite entanglement, prompting a reexamination of tensor-network models and the precise form of multipartite entanglement underpinning AdS/CFT. The results sharpen our understanding of multipartite entanglement in holography and point toward identifying the specific tripartite entanglement structure compatible with bulk reconstruction.

Abstract

We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well-supported conjectures about the entanglement wedge cross section surface $E_W$. If $E_W$ is related to either the CFT's reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $\mathcal{O}(\frac{1}{G_N})$. We prove that this implies holographic CFT states must have $\mathcal{O}(\frac{1}{G_N})$ amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.

Figures (5)

• Figure 1: The entanglement wedge of boundary subregion $AB$ is shaded blue, while the complementary entanglement wedge, corresponding to boundary subregion $C$, is shaded red. The RT surface is $\gamma_{AB}$ (solid line), and the minimal cross section of the entanglement wedge is $EW(A:B)$ (dashed line).
• Figure 2: Subregion $AB$ at the threshold of a mutual information phase transition. There are two competing RT surfaces, denoted by solid and dashed black lines. The area of the dashed lines is equal to the area of the solid lines. $EW(A:B)$ before the transition is denoted by a solid orange line, while it vanishes after the transition.
• Figure 3: A random stabilizer tensor network with subregion $AB$ in the connected phase. The green dotted line represents the RT surface for subregion $AB$, while the yellow dotted lines represent the RT surface of $A$ and $B$ respectively. The red dotted line represents $EW(A:B)$.
• Figure 4: (Left): A reduced tensor network corresponding to the entanglement wedge of $AB$ is obtained by using the isometry from the boundary legs of subregion $C$ to the legs at the RT surface (denoted black and green dotted lines). Two copies of this RSTN glued as shown prepare the canonically purified state. We call this doubled network TN'. (Right): Geometrically, this resembles the AdS/CFT construction discussed in Engelhardt:2017auxEngelhardt:2018kcsDutta:2019gen. If the RT formula holds, then $S_R(A:B)=2 EW(A:B)$.
• Figure 5: After applying local unitaries, the RSTN drastically simplifies to a combination of Bell pairs shared by the three parties. The Bell pairs then lead to a simple canonically purified state.

Theorems & Definitions (4)

• Theorem 2.1: Continuity of the Reflected Entropy
• proof
• Theorem 3.1: Continuity of the Entanglement of Purification
• proof