Quantum and Classical Correlations Inside the Entanglement Wedge

TL;DR

The paper investigates the entanglement wedge cross section $E_W$ as a bulk measure of correlations in holography and shows it can exceed standard quantum entanglement measures, highlighting a significant role for classical correlations inside the entanglement wedge. It introduces the entanglement wedge mutual information $E_M$ as the holographic dual of the $Q$-correlation and proves that $E_M$ obeys strong superadditivity and is bounded by $E_W$ and $\tfrac{1}{2}I$, thereby capturing both quantum and classical correlations. It is shown that two commonly studied optimized measures, the squashed entanglement $E_{sq}$ and the conditional mutual information $E_I$, reduce to $\tfrac{1}{2}I$ in holography, offering no new bulk geometry, while $E_R$ coincides with $E_W$. The results suggest a richer landscape of holographic correlations beyond quantum entanglement, with potential implications for information processing and tensor-network descriptions in AdS/CFT, and motivate further exploration of multipartite generalizations and bit-thread formulations. Overall, the work uncovers a dual bulk quantity (EWMI) tied to the $Q$-correlation that extends our understanding of classical versus quantum correlations in holographic spacetimes.

Abstract

We show that the entanglement wedge cross section (EWCS) can become larger than the quantum entanglement measures such as the entanglement of formation in the AdS/CFT correspondence. We then discuss a series of holographic duals to the optimized correlation measures, finding a novel geometrical measure of correlation, the \textit{entanglement wedge mutual information} (EWMI), as the dual of the $Q$-correlation. We prove that the EWMI satisfies the properties of the $Q$-correlation as well as the strong superadditivity, and that it can become larger than the entanglement measures. These results imply that both of the EWCS and the EWMI capture more than quantum entanglement in the entanglement wedge, which enlightens a potential role of classical correlations in holography.

Figures (12)

• Figure 1: The EWCS (red dashed lines) on a time slice of the entanglement wedge.
• Figure 2: A holographic configuration for which the Araki-Lieb inequality is saturated $S_{A}+S_{AB}=S_{B}$.
• Figure 3: The two configurations of the EWCS $E_{W}(A:B)=E_{W}(A:B_{1}B_{2})$, denoted by the orange dashed line, for the symmetric setup in the Poincaré ${\rm AdS}_{3}$. The left (right) configuration is preferred when the relative size $p<p_{{\rm EW}}^{*}$ ($p>p_{{\rm EW}}^{*}$). The primed symbols allocated on the upper semi-circle denote the partition in (\ref{['eq:EWCS']}) with $\mathcal{A}=A\cup A'$ and $\mathcal{B}=B_{1}\cup B_{2}\cup B_{1}'\cup B_{2}'$.
• Figure 4: Half of the mutual information and the EWCS for the Araki-Lieb transition (normalized by subtracting $S_{AB}$).
• Figure 5: The entanglement wedge mutual information $E_{M}$ in the entanglement wedge. In the above picture, $E_{M}$ is given by the area of red codimension-2 surfaces subtracted by the area of blue codimension-2 surface (divided by $2\cdot4G_{N}$), which may be understood as the mutual information $\frac{1}{2}I(A:M)$. The symmetry $E_{M}(A:B)=E_{M}(B:A)$ stems from the fact that the RT-surface of $S_{BA'}$ and $S_{AB'}$ have the same configurations. The optimal partition $A_{M}^{*}$ and $B_{M}^{*}$ of the EWMI located on the RT-surface of $S_{AB}$, are not necessarily equivalent to these $A_{W}^{*}$ and $B_{W}^{*}$ of the EWCS.