Tripartite Correlation Signal from Multipartite Entanglement of Purification

We propose a signal ${\mathrm{\Delta }}_p^{(3)}$ for genuine tripartite entanglement in finite-dimensional quantum systems and ${\mathrm{\Delta }}_w^{(3)}$ for holographic systems. We prove that ${\mathrm{\Delta }}_p^{(3)}$ is non-negative for any tripartite entangled mixed states. Based on the conjecture, the equality between an entanglement wedge cross section $E_w$ and entanglement of purification $E_p$, i.e., $E_w=E_P$ in the semiclassical limit, we apply the tripartite entanglement measure to study the structures of tripartite entanglement in AdS${}_3$/CFT${}_2$, especially for pure AdS${}_3$. We comment on a generalization to $n$-partite entanglement signals ${\mathrm{\Delta }}_p^{(n)}(A_1:\cdots :A_n)$.