Entanglement wedge cross section triangle information and holographic entanglement of assistance

We identify a non-negative and upper-bounded entanglement signal in holography which is defined as a combination of entanglement wedge cross sections (EWCS) for a tripartite mixed state $ABE$: ${\mathrm{E}\mathrm{I}}_{\mathrm{\Delta }}(A:B\mathrm{\mid }E)=\mathrm{E}\mathrm{W}\mathrm{C}\mathrm{S}(A:EB)+\mathrm{E}\mathrm{W}\mathrm{C}\mathrm{S}(B:EA)-\mathrm{E}\mathrm{W}\mathrm{C}\mathrm{S}(E:AB)$. This quantity is an analogue of conditional mutual information (CMI) and shares similar mathematical structures in both quantum information theory and holography. We show that CMI is upper bounded by a quantum information quantity, the entanglement of assistance, which quantifies the entanglement that can be generated between two parties $A$ and $B$, given assistance from a third party $E$. We prove that ${\mathrm{E}\mathrm{I}}_{\mathrm{\Delta }}$ is also upper bounded by the entanglement of assistance in the canonical purification state. We analyze its upper bound by maximizing ${\mathrm{E}\mathrm{I}}_{\mathrm{\Delta }}(A:B\mathrm{\mid }E)$ over all configurations of the auxiliary subsystem $E$ in AdS${}_3$/CFT${}_2$. The maximized ${\mathrm{E}\mathrm{I}}_{\mathrm{\Delta }}$ displays a rich phase structure governed by the cross ratio $X_{AB}$: it vanishes below a critical threshold and, beyond a second phase transition point, saturates the bound of entanglement of assistance. We comment on the interpretation of ${\mathrm{E}\mathrm{I}}_{\mathrm{\Delta }}$ as characterizing the assisted bipartite quantum entanglement between $A$ and $B$ with the help of $E$.