Cosmological Entanglement Entropy and Edge Modes from Double-Scaled SYK \& Its Connection with Krylov Complexity

TL;DR

This work develops a first-principles framework to define and compute entanglement entropy in the DSSYK model by introducing a unique isometric factorization of the one-particle chord space into left and right zero-particle sectors. By constructing gauge-invariant partial traces and reduced density matrices, the authors reveal edge-mode contributions that act as quantum reference frames and connect these boundary constructions to a generalized bulk horizon entropy, realized as a RT-like area term in dS$_2$/CFT$_1$. In the triple-scaling limit, they establish a concrete RT formula for dS$_2$ holography that reproduces the Gibbons-Hawking entropy for certain entangling regions and remains real-valued due to boundary unitarity, while also tying the entropy to Krylov complexity growth. The results provide a detailed dictionary linking boundary entanglement, edge modes, and bulk geometric data, offering a coherent picture of holographic entanglement in a dS-like setting and highlighting the interplay between quantum reference frames and bulk reconstruction. Overall, the paper advances our understanding of holographic entanglement beyond AdS, clarifies the role of edge modes in the DSSYK bulk dual (sine dilaton gravity), and uncovers a direct connection between entanglement entropy and Krylov complexity within a well-defined quantum-gravitational framework.

Abstract

We investigate entanglement entropy in the double-scaled SYK (DSSYK) model, its holographic interpretation in terms of edge modes (acting as quantum reference frames); particularly its de Sitter (dS) space limit; and its connection with Krylov complexity. We define subsystems relative to a particle insertion in the boundary theory. This leads to a natural notion of partial trace and reduced density matrices. The corresponding entanglement entropy takes the form of a generalized horizon entropy in the bulk dual; revealing the emergence of edge modes in the entangling surfaces. We match the entanglement entropy of the DSSYK in an appropriate limit to an area computed through a \emph{Ryu-Takayanagi formula} in dS$_2$ space with entangling surfaces at $\mathcal{I}^{\pm}$; providing a first principles example of holographic entanglement entropy for dS$_2$ space. This formula reproduces the Gibbons-Hawking entropy for specific entangling regions points; while it decreases for others. This construction does not display some of the puzzling features in dS holography. The entanglement entropy remains real-valued (since the boundary theory is unitary), and it depends on Krylov state complexity in this limit.

Figures (8)

• Figure 1: Effective AdS$_2$ black hole geometry in the bulk dual Blommaert:2024ydx proposal of the DSSYK (reviewed in App. \ref{['app:sine dilaton']}), where we trace out part of the geometry (represented in gray) with respect to a particle excitation (red solid line) inside the bulk Stanford:2014jdaAguilar-Gutierrez:2025mxf due to operator insertions $\hat{\mathcal{O}}_\Delta$ in the boundary theory \ref{['eq:factorization map']}. The edge modes are QRFs in the asymptotic boundaries with respect to which relational (i.e. gauge-invariant) observables can be defined, such as the two-sided minimal geodesic lengths Aguilar-Gutierrez:2025sqh (represented $L_{L/R}(t_{L/R})$, solid blue lines) evolving through boundary time (blue arrows, which is gauge-fixed to be the same value $t_{L}=t_{R}$ in the figure), which corresponds to Krylov complexity in the boundary theory (Sec. \ref{['ssec:Lloyd']}). The dashed line represents the effective AdS$_2$ black hole horizon.
• Figure 2: Example of a chord diagram with $2m$ operator insertions $\hat{\mathcal{O}}_{\Delta_i}$.
• Figure 3: Representation of reduced density matrices from the boundary perspective. Two operator insertions ($\hat{\mathcal{O}}_\Delta$ in red) are inserted thermal circle, which defines a factorization map and traces in chord space. By tracing out the right chord sector, we generate a reduced density matrix $\hat{\rho}_L$ to evaluate its von Neumann entropy.
• Figure 4: DS$_2$ geometry where minimal length geodesic dressings (blue) connect the entangling regions (black dots) at $\mathcal{I}^\pm$ to the RT surfaces (cyan dots, denoted $\gamma$\ref{['eq:RT surface dS']}), where the dilaton reaches is minimal value with respect to the homology constraint. The timelike geodesics serve to gauge-invariantly define the location of RT surface in the static patch as we spatially translate the location of the entangling surface points (which are gauge-fixed at spatially symmetric points). The dash black lines represent the cosmological horizon, red solid line the particle location (mapped from the effective geometry Fig. \ref{['fig:bulk_picture']}). The Milne patch is the expanding region outside the cosmological horizon and bounded by $\mathcal{I}^\pm$; while the static patch is its complement.
• Figure 5: Representation of the semiclassical evolution of the chord number \ref{['eq:length dS']} in the chord diagram (similar to Blommaert:2024ydxBlommaert:2023opb) prepared from the reference state $\ket{\Omega}$ and evolved in a HH preparation of state, where $\ell_{\rm dS}(t)$\ref{['eq:length dS exp ']} (blue) evolves in real time, and $\beta(\theta)$ is the periodicity of the thermal circle \ref{['eq:semiclassicao']}.