Petz Map, Black Hole Interior Reconstruction, and Papadodimas-Raju Proposal

The black hole information paradox is one of the important questions of physics in the last few decades. We study the reconstruction of the bulk operators in AdS/CFT when the geometry contains a black hole. The black hole exterior can be mapped to the CFT via a very simple Petz map which coincides with the HKLL map reconstruction of the black hole exterior. For the interior modes of the bulk theory, using the definition of the Petz recovery channel in modular theory, we can find the mapping from the black hole interior to the dual boundary theory. In the case of the evaporating black hole, it is expected that the interior modes map to some operators that have support only on the bath system, the cavity that absorbs the Hawking radiation. The most important observation that we have here is that in the case that we have a typical black hole microstate in the bulk, the CFT dual of the interior modes that we can find using the Petz recovery channel are exactly the operators that so-called "mirror operator " in the Papadodimas-Raju proposal. The main idea of the PR proposal is to focus on a code subspace of the CFT theory, which is created by acting with a small algebra on the corresponding pure state and then fnd the CFT description of the interior operator in a state-dependent manner just in the chosen code subs. Therefore, we can interpret Papadodimas- Raju proposal as an example of the Petz map reconstruction. It may help us answer some open questions about their procedure.