Purely GHZ-like entanglement is forbidden in holography

Abstract

We provide evidence that three-party entanglement signals in holography obey a relation that is not satisfied by generalized Greenberger-Horne-Zeilinger (GHZ) states. Using proposed holographic duals for these entanglement signals, we provide a geometric argument establishing this relation. This is the first known inequality on the structure of pure three-party holographic states, and shows that time-symmetric holographic states can never have purely GHZ-like entanglement. We also discuss similar relations for four parties.

Figures (5)

• Figure 1: The shaded blue region bounded by the boundary subregion $AB$ and the minimal surface $\Gamma_C = \Gamma_{AB}$ is the entanglement wedge $EW(AB)$. The entanglement wedge cross-section $\gamma_{A:B}$ is the minimal surface that separates $A$ and $B$ within $EW(AB)$.
• Figure 2: The multi-entropy $S^{(3)}(A:B:C)$ is computed by the minimal brane web $\mathcal{W}_{A:B:C}$ that separates all boundary subregions $A,B,C$ from each other.
• Figure 3: The upper-half plane representation of the hyperbolic disc, showing the minimal surfaces involved in the computation of $R^{(3)}(A:B)$ and the brane web $\mathcal{W}_{A:B:C}$ involved in the computation of $GM^{(3)}$.
• Figure 4: The top figure shows the minimal area brane web $\mathcal{W}_{A:B:C:D}$ that computes $S^{(4)}(A:B:C:D)$. The bottom figure shows a brane web consisting of the RT surface and the entanglement wedge cross-sections.
• Figure 5: A plot showing the tripartite and quadripartite entanglement quantities for vacuum AdS$_3$ as a function of the conformal cross ratio $\eta$. Clearly, the inequalities \ref{['eq:main_2']} and \ref{['eq:main_3']} are satisfied.