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 the Hamilton-Kabat-Lifschytz-Lowe (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 pseudoentropy 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 rederive the HKLL formulas for free bulk scalar fields in three examples: in AdS Poincaré 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 subhorizon holography. Published by the American Physical Society 2024
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