Entanglement Wedge Reconstruction and Entanglement of Purification

TL;DR

The paper investigates bulk curve reconstruction within entanglement wedges in AdS3, revealing that entanglement entropy alone cannot reconstruct generic curves inside an AdS-Rindler wedge due to an entanglement shade. It introduces a generalized entanglement of purification and its differential form, differential purification, and demonstrates that pairing differential entropy with differential purification yields complete reconstruction of all spacelike curves in any three-dimensional bulk geometry. The authors show how nonshade segments are captured by boundary entanglement entropies, while shade segments are encoded through purified boundary data on the entanglement wedge's RT surface, enabling a unified, boundary-based reconstruction protocol. This work clarifies the role of purification in subregion duality, extends hole-ography to covariant and purified contexts, and offers a pathway toward higher-dimensional generalizations.

Abstract

In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recently shown to suffice for reconstruction of arbitrary spacelike curves in the Poincare patch. This limitation is an analog of the familiar phenomenon of entanglement shadows, which we call 'entanglement shade'. We overcome it by showing that the information about the nonreconstructible curve segments is encoded in a slight generalization of the concept of entanglement of purification, whose holographic dual has been discussed very recently. We introduce the notion of 'differential purification', and demonstrate that, in combination with differential entropy, it enables the complete reconstruction of all spacelike curves within an arbitrary entanglement wedge in any 3-dimensional bulk geometry.

Figures (5)

• Figure 1: Schematic depiction of the entanglement wedge ${\mathcal{E}}_{A}$ and causal wedge ${\mathcal{C}}_{A}$ for a boundary subregion $A$, or equivalently, for its boundary domain of dependence ${\mathcal{D}}_A$. See the main text for the explicit definitions. For arbitrary bulk geometries, the entanglement wedge, bounded by null geodesics that are shot towards the boundary from the Ryu-Takayanagi surface $\Gamma_A$, is larger than the causal wedge, bounded by null geodesics that are shot into the bulk from the edge of ${\mathcal{D}}_A$. The spatial surface $\Xi_A$ on which the latter geodesics intersect is the causal information surface defined in hr. In a few situations $\Xi_A=\Gamma_A$, and the two types of wedges coincide. This happens in particular when $A$ is a spherical region in the vacuum of a $d$-dimensional conformal field theory, which for $d=2$ gives rise to the anti-de-Sitter-Rindler wedge considered throughout most of this paper.
• Figure 2: Each of these solid cylinders is a Penrose diagram for AdS$_3$, covered in full by the global coordinates $(\varrho,\tau,\theta)$, but only in part by the Poincaré coordinates $(t,x,r)$ on the left, or the Rindler coordinates $({\mathbf{t}},{\mathbf{x}},{\mathbf{r}})$ on the right. a) Generic spatial bulk curves in the Poincaré wedge (such as the circle shown in red) have segments whose tangent geodesics (shown in orange) are not fully contained within the wedge. In spite of this, a variant of hole-ography that employs 'null alignment' allows their reconstruction with entanglement entropies in the CFT poincarepoint. b) A Rindler wedge covers a smaller portion of global AdS, and in particular, it does not contain a full Cauchy slice. A priori, it is not clear if the 'null alignment' variant of hole-ography is sufficient to reconstruct arbitrary bulk curves within the Rindler wedge (such as the circle shown in red).
• Figure 3: Entanglement shade for a Rindler wedge in AdS$_3$, in the range $0<u^{{\mathbf{t}}}<10$, $0<u^{{\mathbf{r}}}<10$, having chosen the parametrization $\lambda={\mathbf{x}}$ (which implies $u^{{\mathbf{x}}}=1$). The shaded region indicates the radial depths that cannot be penetrated by geodesics with the indicated tangent vector $u$, or with any other vector $U$ obtained from it by null alignment ($U=u+n$ with $n\cdot n=n\cdot u=0$). As expected from the analysis in the main text, when we consider larger values of $u^{{\mathbf{r}}}$, corresponding to steeper curves, the shade grows larger. On the other hand, the figure shows that upon increasing the value of $u^{{\mathbf{t}}}$ the shade is reduced. By symmetry, the radial position where the shade begins is independent of the sign of $u^{{\mathbf{t}}}$ and $u^{{\mathbf{r}}}$, and of course, it is also independent of the values of ${\mathbf{t}}$ and ${\mathbf{x}}$. The entire region shown corresponds to spacelike $u$.
• Figure 4: An example of a closed spacelike curve: a circle at constant time ${\mathbf{t}}$, centered at ${\mathbf{x}}=1$, ${\mathbf{r}}=1.4$, with coordinate radius $a=1$. The top and bottom, shown in solid red, have tangent geodesics of the type (\ref{['staticgeodesicvpm']}), lying fully within the Rindler wedge. A sample such geodesic is shown in orange, with both of its endpoints extending up to the boundary at ${\mathbf{r}}\to\infty$. This is not true for the segments on the sides, shown in dashed black, which violate condition (\ref{['criterium1']}) and therefore cannot be reconstructed using entanglement entropies. Geodesics tangent to them, such as the one shown in blue, are of the type (\ref{['staticgeodesicvhvinfty']}), and have one endpoint on the boundary but cross the horizon ${\mathbf{r}}=0$ on the other side. If we parametrize the circle by $\lambda\in [0,1)$, with $\lambda=0$ located at the top, the gluing between the four segments occurs at the values $\lambda=0.138, 0.278, 0.722, 0.862$. If we wished, we could use null alignment (\ref{['nullalignment']}) to reduce the size of the dashed segments, but as discussed in the main text, no choice of $n$ can make them disappear completely.
• Figure 5: Ingredients for the holographic computation of the entanglement of purification $P$ and its generalization $P'$. The disk represents a constant-time slice of a static geometry dual to some pure state. Upon restricting the field theory to the region $A=BC$, we are left in the gravity description with the corresponding spatial slice of the entanglement wedge of $BC$, shown as the shaded region. a) In the generic case where $B$ and $C$ are not contiguous, the Ryu-Takayanagi surface $\Gamma_{BC}$ has two disconnected components, indicated in green. Running between them at the narrowest part of the shaded region we see the entanglement wedge cross section, $\Sigma$, whose area encodes, according to (\ref{['eoph']}), the entanglement of purification (\ref{['eopdef']}) for the bipartition $BC$ of the given state. The corresponding minimal surface in the overall geometry would include the dotted segments as well, but these are excluded from the definition of $P$. The degrees of freedom $A'$ of the purification 'live on' $\Gamma_{BC}$, and $\Sigma$ partitions them into a specific choice of $B'$ and $C'$. b) In the particular case where $B$ and $C$ are contiguous, one of the components of $\Gamma_{BC}$ shrinks down to the transition point between $B$ and $C$, and $\Sigma$ is seen to extend from there to the closest point in the remaining, finite component. If the overall geometry is global AdS, the shaded region is an AdS-Rindler wedge. c) If in the setup of b) we consider instead a minimal surface $\Sigma'\neq\Sigma$, we obtain a different, suboptimal partition of $A'$ into $B'$ and $C'$, and the area of $\Sigma'$ is then expected to yield via (\ref{['eoph2']}) the entanglement of purification (\ref{['eopdef2']}) associated with that specific partition.