Bulk reconstruction using timelike entanglement in (A)dS

Abstract

It is well-known that the entanglement entropies for spacelike subregions, and the associated modular Hamiltonians play a crucial role in the bulk reconstruction program within Anti de-Sitter (AdS) holography. Explicit examples of HKLL map exist mostly for the cases where the emergent bulk region is the so-called entanglement wedge of the given boundary subregion. However, motivated from the complex pseudo-entropy in Euclidean conformal field theories (CFT), one can talk about a `timelike entanglement' in Lorentzian CFTs dual to AdS spacetimes. One can then utilize this boundary timelike entanglement to define a boundary `timelike modular Hamiltonian'. We use constraints involving these Hamiltonians in a manner similar to how it was used for spacelike cases, and write down bulk operators in regions which are not probed by an RT surface corresponding to a single CFT. In the context of two dimensional CFT, we re-derive the HKLL formulas for free bulk scalar fields in three examples: in AdS Poincar\'{e} patch, inside and outside of the AdS black hole, and for de Sitter flat slicings. In this method, one no longer requires the knowledge of bulk dynamics for sub-horizon holography.