---
title: Holographic Entanglement in dS₂ Space
url: https://www.emergentmind.com/topics/holographic-entanglement-entropy-for-ds-_2-space
type: topic
---

# Holographic Entanglement in dS₂ Space

Holographic entanglement entropy in two-dimensional de Sitter space (dS$_2$) probes the interplay between quantum information, gravity, and higher-dimensional holographic dualities. The study of EE in dS$_2$ is sharply distinct from standard anti–de Sitter (AdS) holography, exposing the peculiar structure of extremal surfaces, non-trivial phase transitions, and subtle stability phenomena tied to the signature and extrinsic curvature of de Sitter branes. A variety of approaches—analytic continuation from AdS, direct holographic calculations in AdS-dS braneworlds, double-scaled SYK constructions, and covariant QES functionals—collectively demonstrate the nuances of holographic entropy in dS$_2$, the emergence of non-extremal “islands,” and the realization of the Gibbons–Hawking entropy as an entanglement entropy.

## 1. Doubly Holographic Formulation and Brane Embedding

The central challenge in dS$_2$ arises because the conventional island formula, successful in AdS and quantum black hole studies, fails to yield a satisfactory quantum extremal surface (QES) when naively transplanted to de Sitter space. In [2407.21617], this issue is resolved by embedding a dS$_2$ “end-of-world” (EOW) brane into a three-dimensional AdS$_3$ bulk, constructing a doubly holographic setup.

The AdS$_3$ geometry in a dS$_2$ slicing,
\[
ds^2_3 = L_{\rm AdS}^2 [d\eta^2 + \sinh^2\!\eta\, (-dt^2 + \cosh^2 t\,d\theta^2)],
\]
admits a brane at fixed $\eta=\eta_b$, its location and tension $\mathcal{T}$ determined by the Neumann boundary condition:
\[
K_{ij} - K\,g_{ij} + \mathcal T\,g_{ij} = 0,\qquad
\mathcal T = \frac{\coth\eta_b}{L_{\rm AdS}}.
\]
The intrinsic brane geometry is
\[
ds^2_{\rm brane} = L^2 (-dt^2 + \cosh^2 t\,d\theta^2),\quad L = L_{\rm AdS} \sinh\eta_b,
\]
realizing dS$_2$ (up to overall length scale).

## 2. Generalized Entropy Functional and Non-Extremal Islands

In the AdS/braneworld setup, the generalized entropy functional for a region $\mathcal A$ in the non-gravitating “bath” is
\[
S_{\rm gen}\{I\} = \frac{\operatorname{Area}(\partial I)}{4 G_N^{(3)}}
 + S_{\rm bulk}(\mathcal A \cup I),
\]
with $\partial I$ a codimension-2 locus (in dS$_2$-brane: a point or two points) and $S_{\rm bulk}$ the CFT von Neumann entropy of $\mathcal A \cup I$.

The RT “area-term” is set by the length of a geodesic in AdS$_3$ stretching from the AdS boundary to the brane,
\[
\cosh D_{PX} =
\cosh\eta_\infty\,\cosh\eta_b [\cosh t_A\,\cosh t_I\,\cos(\theta_A-\theta_I) - \sinh t_A\,\sinh t_I]
- \sinh\eta_\infty\,\sinh\eta_b,
\]
with
\[
\frac{\operatorname{Area}(\Gamma)}{4G_N^{(3)}} =
\frac{L_{\rm AdS}}{4G_N^{(3)}} D_{PX}.
\]
The bulk CFT entropy on intervals $[\theta_A,\theta_I]\cup[-\theta_I,-\theta_A]$ is
\[
S_{\rm bulk} = \frac{c}{3}\,
\ln \frac{2(1 + \sinh t_A\,\sinh t_I - \cosh t_A\,\cosh t_I\,\cos(\theta_A-\theta_I))}{\epsilon},
\]
with UV cutoff $\epsilon$.

Extremizing $S_{\rm gen}$, one finds the only saddle:
\[
\theta_I = \theta_A - \pi,\quad t_I = -t_A,
\]
is not a true minimum for the spatial direction on the dS brane; it is a local maximum instead. Bulk correlation functions instead pick out a non-extremal saddle at the spatial **edge** of the dS brane, i.e. $\theta_I = \pm\pi/2$, where the “island” appears. The corresponding time is fixed by
\[
\tan t_I^\ast = \frac{\tan t_A}{\lvert \sin\theta_A \rvert}.
\]
For a symmetric interval, entanglement entropy is:
\[
S_{\mathcal A}  = \min\left\{
\frac{c}{3}\ln\!\left[\frac{2\cosh t_A\,\sin\theta_A}{\epsilon}\right]
,\;
\frac{c}{3}\ln\!\left(\frac{2}{\epsilon}\left[
\cosh\eta_b - \sinh\eta_b \sqrt{\cosh^2 t_A \sin^2\theta_A - \sinh^2 t_A}
\right]\right)
\right\}.
\]

## 3. Local Instability, Second Variation, and the Distinction to AdS

The physical difference between AdS$_2$ and dS$_2$ branes is rooted in their extrinsic curvature $K$. For a surface to be a local minimum of the generalized entropy functional, the quadratic variation of the geodesic length must be positive:
\[
\delta^2 L = -K(v,v)|_{\rm brane} + \int_0^1 \{\|\nabla_t v\|^2 - R(v,t,t,v)\} ds.
\]
Since $K_{ij} = \mathcal T g_{ij}$ and $\mathcal T > 1/L_{\rm AdS}$ for dS brane, $K > 0$ and consequently surfaces orthogonally ending on the brane are **unstable** in the spatial direction; they are not quantum extremal in the QES sense. In contrast, AdS branes ($K<0$) permit standard quantum extremal surfaces. Thus, in dS$_2$, the “island” is a **non-extremal** one—pinned to the edge of the gravitational region, extremal only in time.

## 4. Connections to Alternative dS$_2$ Holography Approaches

Several independent constructions provide insight and confirm this picture:

- **Analytic continuation from AdS$_2$:** Under double Wick-rotation, the RT geodesic length in EAdS$_2$ becomes complex, rendering the dS$_2$ entropy formula $S_A^{(\mathrm{dS}_2)} = -i \frac{c}{3}\ln(\Delta x / \epsilon)$. The negative imaginary factor and lack of a known unitary dual signals peculiarities in dS/CFT frameworks [1501.04903].
- **Double-scaled SYK and edge modes:** In the DSSYK model, subsystem definition ties to quantum reference frames and edge modes. Here, the holographic EE recovers a real-valued, unitary result even in dS$_2$, and the area term matches a generalized RT formula with entangling surfaces at $\mathcal{I}^\pm$. The entropy depends on Krylov complexity and at $t=0$ reproduces Gibbons–Hawking entropy [2511.03779].
- **Static patch and horizon calculations:** Using Fefferman–Graham slicing and regulating with a UV cutoff, the RT prescription gives $S_{\rm EE} \to \frac{1}{2G_2}$ at the dS$_2$ horizon, exactly matching Wald entropy [1904.13119, 2108.07674].

| Approach            | RT Surface Location      | Stability      | Entropy Formula                  |
|---------------------|-------------------------|---------------|----------------------------------|
| AdS$_3$ brane [2407.21617] | Edge of dS brane           | Non-extremal (maximal bend) | $S_{\mathcal A}$ as above        |
| EAdS$_2$ Wick rotation [1501.04903] | Usual RT geodesic (complexified) | Not unitary          | $S_A^{(\mathrm{dS}_2)} = -i \frac{c}{3}\ln(\Delta x/\epsilon)$ |
| SYK/dS JT [2511.03779]   | dS$_2$ horizon / $\mathcal{I}^\pm$ | Unitary (Krylov)    | $S_{EE}(t) = (1/4G_N) e^{i\ell_{dS}(t)/2}$ |
| Static patch[1904.13119, 2108.07674] | dS horizon                   | N/A              | $S_{\rm EE} = \frac{1}{2G_2}$      |

## 5. Phase Transitions and Multi-Saddle Structures

In closed dS$_2$ spacetimes, doubly holographic models reveal the existence of several distinct extremal and non-extremal surfaces. Three entropic saddles control EE:
- **Connected–Hartman–Maldacena:** Geodesics connecting bath endpoints.
- **New doubly-connected saddle:** Includes segments ending on dS brane, which may have negative length inside bulk horizons.
- **Island saddle:** Geodesics stretching between bath and brane endpoints, saturating late-time entropy.

As a function of temperature and time, there is a tri-critical phase structure. At low temperatures, the new saddle dominates early times, with transitions to connected and finally island saddles as time increases. The negative-entropy regions introduced by behind-horizon geodesics are regulated using a “multiplicity one” rule: only a single branch is counted per endpoint [2502.08380].

## 6. Quantum Extremal Surface (QES) Prescriptions and the Role of Matter

Covariant bilayer QES functionals in dS$_2$ clarify the emergent structure of entanglement wedges and their reconstructibility. Since all points on the dS$_2$ horizon are classically degenerate (yielding the same RT length), quantum corrections from bulk matter, notably the CFT contribution,
\[
S_{\rm semicl}(\Delta \theta) = \frac{c}{3} \ln\left(\frac{1}{\epsilon} \sin(\Delta \theta/2)\right),
\]
select between candidate surfaces. At the critical slice, there is a first-order phase transition in the entanglement wedge, corresponding to changing reconstructibility between antipodal boundary screens [2403.14889]. The inclusion of higher-order corrections such as conformal anomaly terms leads to subleading log corrections in higher dimensions, but dS$_2$ remains dominated by the leading order.

## 7. Entropy Bounds, Swampland Criteria, and Ambiguities

In the DS$_3$/dS$_2$ correspondence, a one-parameter family of minimal geodesics (both "horizon-hugger" and "slice" geodesics) yield the same dS entropy, $S_{dS} = {\pi L}/{2G_3}$. When matter is introduced, only the "horizon-hugger" surface maintains this equality; others can exceed it, potentially violating entropy bounds. This suggests that dS entropy serves as a "swampland" bound constraining semiclassical matter configurations [1904.02170]. The redundancy of minimal surfaces persists due to the degeneracy arising from the dS$_2$/CFT$_1$ or DS$_3$/dS$_2$ geometries.

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Collectively, these results establish that holographic entanglement entropy in dS$_2$ cannot be formulated solely in terms of conventional quantum extremal surfaces. The correct recipe picks out non-extremal, edge-pinned islands, or relies on careful inclusion of quantum and matter contributions, with the Gibbons–Hawking entropy emerging as an entanglement entropy in several distinct constructions [2407.21617, 2511.03779, 1904.13119, 2502.08380, 2403.14889]. This web of results reflects the broader ambiguity, non-uniqueness, and non-unitarity challenges at the heart of de Sitter holography.

Source: https://www.emergentmind.com/topics/holographic-entanglement-entropy-for-ds-_2-space