Cosmological Entanglement Entropy and Edge Modes from Double-Scaled SYK

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 Ryu-Takayanagi formula in dS_2 space with entangling surfaces at future and past infinity; 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.