Entanglement wedge cross section triangle information and holographic entanglement of assistance

TL;DR

This work defines the EWCS triangle information $EI_Δ(A:B|E)$ as a positive, upper-bounded holographic diagnostic of tripartite entanglement in a mixed state $ABE$, linking holographic EWCS to quantum information notions via the canonical purification: $EI_Δ(A:B|E)=\tfrac12 I(AA^*:BB^*)$ and $EI_Δ\le 2\,HE(A:B|E)=\mathrm{EoA}(AA^*:BB^*|EE^*)$. The authors prove upper bounds by translating CMI inequalities to EWCS language and show how the maximal $EI_Δ$ over auxiliary regions $E$ exhibits a phase structure controlled by the cross ratio $X_{AB}$ in AdS$_3$/CFT$_2$. They develop generalized multi-phase transition (MPT) diagrams to study $E$ consisting of $n$ intervals, revealing that the maximal EIΔ vanishes below a threshold and saturates the entanglement-of-assistance bound beyond a second phase point, with the number of phases growing with $n$. The results establish a concrete operational interpretation of EIΔ as an assisted bipartite entanglement measure and connect it to the entanglement of purification via the canonical purification geometry, suggesting directions for higher-dimensional and time-dependent explorations.

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 $E$.

Figures (19)

• Figure 1: An illustration of a multi-mouth wormhole that is dual to the state $\lvert \sqrt{\rho_{ABE}} \rangle$ in \ref{['sqrtrho']}, with region $E$ consisting of three intervals. The wormhole is constructed by doubling the entanglement wedge of $ABE$ and gluing them together along the minimal RT surfaces that separate the three boundary regions, i.e., the black curves. Three candidates of $\mathrm{EoA}(A:B|E)$ are shown in three different colors in the wormhole.
• Figure 2: Proof of EWCS triangle inequality in holography.
• Figure 3: Calculation of the EWCS triangle information $\mathrm{EI}_{\Delta}(A:B|E)$ when tuning E, in the case where $n=1$ and the entanglement wedge of $ABE$ is fully connected. Three EWCSs $L_A$, $L_B$ and $L_E$, which are relevant for the computation of $\mathrm{EI}_{\Delta}(A:B|E)$, are shown in the diagram.
• Figure 4: The division of the $(X_{AE}, X_{BE})$ parameter plane into regions where the functional forms of $\widehat{\mathrm{EI}}_{\Delta}(A:B|E)$ are distinct, with four relatively small values of $X_{AB}<1$. The boundaries of the connection conditions and the boundaries of the four possible values of $\widehat{\mathrm{EI}}_{\Delta}(A:B|E)$ are shown in solid and dashed curves respectively. The legends of all the 4 sub-figures are placed in the white region of panel (d). The orange regions violate at least one of the connection conditions such that $\mathrm{EW}(ABE)$ is not fully connected. The purple regions fulfill all the connection conditions as well as the condition $\widehat{L}_E>\widehat{L}_A \widehat{L}_B$, in which $\widehat{\mathrm{EI}}_{\Delta}(A:B|E)$ is 1 in \ref{['posval']}. $\mathrm{EI}_{\Delta}(A:B|E)$ vanishes in both the orange and purple regions while remaining finite in the white regions in which $\widehat{\mathrm{EI}}_{\Delta}(A:B|E)=(\widehat{L}_A \widehat{L}_B)/\widehat{L}_E$. $X_{AB}$ increases across subfigures (a)–(d). Panel (a) shows a representative parameter space of the first phase in which the orange and purple regions cover the whole plane such that $\mathrm{EI}_{\Delta}(A:B|E)$ vanishes in the entire space. In panel (b), $X_{AB}$ reaches the first critical point at which the curves $\widehat{L}_E=\widehat{L}_A \widehat{L}_B$(dashed purple), $X_{AB}/X_{D_1 D_2}=1$(solid red) and $X_{D_0 D_1}=1$(solid blue) (or $X_{D_0 D_2}=1$(solid orange)) intersect at a single point. Panel (c) and (d) depict the parameter space at selected $X_{AB}$ in the second phase where there exist regions(white) where $\mathrm{EI}_{\Delta}(A:B|E)$ does not vanish. The black dots mark the locations at which $\mathrm{EI}_{\Delta}(A:B|E)$ attains its maximum in each of these two sub-figures, namely the intersection points of the solid red and solid blue (or solid orange) curves. Since $\mathrm{\widehat{EI}}_{\Delta}(A:B|E)\neq 1$ for any totally connected configuration of $E$ when $X_{AB}>0.2071$, the purple region disappears in (d).
• Figure 6: $\mathrm{Max}(\mathrm{EI}_{\Delta}(A:B|E))$ as a function of $X_{AB}$ in the case that region $E$ has only one interval. The phase transition points are denoted as red dots.

Theorems & Definitions (2)

• Theorem 1
• proof