Quantum Error Correction in SYK and Bulk Emergence

We will discuss the price of the quantum error correcting codes, defined as the number of physical qubits needed to reconstruct whether a given operator has been acted upon the thermal state or not. By thinking about reconstruction via quantum teleportation, we will argue for a bound that relates the price to the ordinary operator size in systems that display so-called detailed size winding of Nezami et al. (2021). We then discuss how in the specific example of SYK this bound is roughly saturated. Computing the price requires computing modular flowed correlators with respect to the density matrix associated to a subset of fermions. We offer an interpretation of these correlators as probing a quantum extremal surface in the AdS dual of SYK. In the large N limit, the operator algebras associated to subsets of fermions in SYK satisfy half-sided modular inclusion, which is indicative of an emergent Type III1  von Neumann algebra. We will mention the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.