Entanglement Wedge Cross Section from the Dual Density Matrix

TL;DR

The paper introduces odd entanglement entropy (OEE) as a novel quantity derived from the partial transposition of a bipartite density matrix, and demonstrates that the entanglement wedge cross section (E_W) in holographic CFTs can be obtained from OEE via E_W = S_o − S. Through replica-trick calculations in AdS3 and planar BTZ backgrounds, it shows that the OEE reproduces the known holographic E_W for vacuum and thermal states, with explicit results for two disjoint intervals and thermal configurations in CFT2. The work provides evidence that OEE captures geometric information about the entanglement wedge and conjectures a general dimensional validity, while also discussing operational interpretations and the potential for negative E_W values. Overall, it links a boundary entanglement measure derived from partial transposition to a bulk geometric quantity, offering a new tool for understanding the entanglement-geometry correspondence.

Abstract

We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic CFTs. The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (AdS${}_3$ and planar BTZ blackhole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.

Figures (2)

• Figure 1: Left: An example of the entanglement wedge cross section $\Sigma^{\textrm{min.}}_{A_1A_2}$ (a blue dotted line). The vertical direction corresponds to the radial one of Poincare AdS${}_3$. The horizontal line coincides with a time slice of CFT${}_2$. Blue curved lines show the minimal surfaces $\Gamma^{\textrm{min.}}_A\, (A=A_1A_2)$ for $S(\rho_{A_1A_2})$ where the $\rho_{A_1A_2}$ is a state acting on bipartite Hilbert space $\mathcal{H}_{A_1}\otimes\mathcal{H}_{A_2}$ (associated with geometrical subregions $A_1$ and $A_2$) and is supposed to be dual to the entanglement wedge. A blue shaded region represents a time slice of the entanglement wedge. Right: If $A_1$ and $A_2$ are sufficiently distant, we have no connected entanglement wedge and $E_W(\rho_{A_1A_2})=0$.
• Figure 2: Calculation of $E_W$ for the static planar BTZ black hole. The inverse temperature $\beta$ is determined by the radius of the horizon. If the subsystem $A$ is sufficiently small $\ell\ll\beta$, the $E_W$ computes the geodesics anchored on the boundary of $A$ (black curve) which agrees with the \ref{['eq:ewcsbtz_x1']}. For $\ell\gg\beta$, the $E_W$ does the disconnected surfaces (dotted vertical lines) which is consistent with the \ref{['eq:ewcsbtz_x0']}.