Notes on entanglement wedge cross sections

TL;DR

This work extends the holographic entanglement wedge cross section $E_W$ beyond the familiar AdS$_{3}$ setups to higher dimensions, including slabs, concentric spheres, and creases in Minkowski space, and analyzes its behavior in pure AdS, AdS black branes, confining geometries, and ABJM-like RG flows. The authors derive analytic expressions for $E_W$ in several geometries, uncovering non-monotonic scale dependence, phase transitions between connected and disconnected entanglement wedges, and even discontinuous jumps in $E_W$ under RG flow in nonconformal backgrounds. They demonstrate that $E_W$ generally tracks qualitative features of entanglement measures such as the mutual information, while exhibiting unique signatures (e.g., non-monotonicity, cusps, and jumps) that support its candidacy as a holographic proxy for $E_P$ or negativity in mixed states. The results provide evidence that purification-like behavior can occur through scale expansion or contraction and offer concrete, testable predictions across a range of holographic geometries with potential implications for quantum information in strongly coupled field theories.

Abstract

We consider the holographic candidate for the entanglement of purification $E_P$, given by the minimal cross sectional area of an entanglement wedge $E_W$. The $E_P$ is generally very complicated quantity to obtain in field theories, thus to establish the conjectured relationship one needs to test if $E_W$ and $E_P$ share common features. In this paper the entangling regions we consider are slabs, concentric spheres, and creases in field theories in Minkowski space. The latter two can be mapped to regions in field theories defined on spheres, thus corresponding to entangled caps and orange slices, respectively. We work in general dimensions and for slabs we also consider field theories at finite temperature and confining theories. We find that $E_W$, similarly to holographic mutual information, is not a monotonic function of a scale. We also study a full ten-dimensional string theory geometry dual to a non-trivial RG flow of a three-dimensional Chern-Simons matter theory coupled to fundamentals. We show that also in this case $E_W$ behaves non-trivially, which if connected to $E_P$, lends further support that the system can undergo purification simply by expansion or reduction in scale.

Figures (11)

• Figure 1: Sketch of the different entanglement entropy phases available for parallel strip configurations of width $l$ and separated by a distance $s$. The disconnected phase (left) has zero entanglement wedge cross section. The connected phase (right) does have a non-zero $E_W$, given by the area of $\Gamma$, the surface with the minimal area out of the infinite set of surfaces $\Sigma_{AB}^\text{min}$.
• Figure 2: Comparison of $E_W$ and $I/2$ in pure $AdS_4$. Entanglement regions are parallel strips with width $l$ and separation $s$ fixed such that $s/l=0.4$. In both quantities we have omitted a prefactor of $\frac{V L_{AdS}^2}{4 G_N^{(4)}}$.
• Figure 3: Critical separation between parallel strips in $AdS_d-BH$ for $d=2,\dots,10$. Circles denote numerical results. Green and orange crosses correspond to analytical approximations (\ref{['eq:crit_s_1']}) and (\ref{['eq:crit_s_2']}), respectively.
• Figure 4: Comparison of $E_W$ and $I/2$ in an asymptotically $AdS_4$ black brane geometry. We have again fixed $s/l=0.4$ and omitted a common prefactor of $V L_{AdS}^2 z_h/4 G_N^{(4)}$. A qualitative difference to the pure $AdS_4$ (Fig. \ref{['fig:ads_poincare']}) is the appearance of a phase transition of $S(AB)$ to the disconnected phase marked by the point where both $E_W$ and $I/2$ become zero.
• Figure 5: Phase diagram for a bipartite system composed out of two parallel, infinitely long strips for $d=3$Ben-Ami:2014gsa. There are four possible bulk surfaces for $S(AB)$, where $A$ and $B$ have widths $l$ and are separated by $s$. Only two of these give rise to a connected entanglement wedge and thus a non-zero $E_W$, denoted by a red line in the sketches. The black dashed line shows the parameter space of Fig. \ref{['fig:confining_EW']}.