Entanglement Wedge Cross Sections Require Tripartite Entanglement

TL;DR

The paper demonstrates that holographic CFT states inherently possess significant tripartite entanglement, challenging the idea that their entanglement is primarily bipartite, based on relations involving the entanglement wedge cross section.

Contribution

It provides a proof that holographic CFT states must have large tripartite entanglement, using a new Fannes-type inequality for reflected entropy and analyzing the entanglement wedge cross section.

Findings

Holographic CFT states require $rac{1}{G_N}$-scale tripartite entanglement.

Relations between $E_W$, reflected entropy, and entanglement of purification imply tripartite entanglement.

New Fannes-type inequality for reflected entropy has broad applications.

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.