Entanglement wedge cross section triangle information and holographic entanglement of assistance

Abstract

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 $ABE$: $\mathrm{EI}_Δ(A:B|E) = \mathrm{EWCS}(A:EB) + \mathrm{EWCS}(B:EA) - \mathrm{EWCS}(E:AB)$. 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 $\mathrm{EI}_Δ$ is also upper bounded by the entanglement of assistance in the canonical purification state. We analyze its upper bound by maximizing $\mathrm{EI}_Δ(A:B|E)$ over all configurations of the auxiliary subsystem $E$ in AdS$_3$/CFT$_2$. The maximized $\mathrm{EI}_Δ$ 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 $\mathrm{EI}_Δ$ as characterizing the assisted bipartite quantum entanglement between $A$ and $B$ with the help of $E$.