Quantum error correction in SYK and bulk emergence

Abstract

We analyze the error correcting properties of the Sachdev-Ye-Kitaev model, with errors that correspond to erasures of subsets of fermions. We study the limit where the number of fermions erased is large but small compared to the total number of fermions. We compute the price of the quantum error correcting code, 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 argue for a bound that relates the price to the ordinary operator size in systems that display so-called detailed size winding [1]. We then find that in SYK the price roughly saturates this bound. 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 discuss the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.

The bulk Hilbert space of double scaled SYK

“…Along these lines, tensor networks are meant to be a toy model of the bulk-toboundary map that illustrate certain information theoretic properties. But now that we have concrete formulas for the bulk-to-boundary map, it would be nice to directly characterize its error-correcting properties [43,44].…”

Large $N$ von Neumann algebras and the renormalization of Newton's constant

“…• It also seems very important to see how this new equivalence between large N entanglement and areas of surfaces in spacetime arises in explicit models of quantum gravity. The SYK model [43], matrix quantum mechanics [44], or maybe some supersymmetric field theories seem to be cases in which one could try to implement a renormalization scheme akin to the one identified in this paper.…”

The bulk Hilbert space of double scaled SYK

“…Along these lines, tensor networks are meant to be a toy model of the bulk-to-boundary map that illustrate certain information theoretic properties. But now that we have concrete formulas for the bulk-to-boundary map, it would be nice to directly characterize its errorcorrecting properties [41,42].…”

“…11 It would be interesting to find the N = 2 generalization of (42). For m = 0, H L = H R and we get N = 4 Liouville quantum mechanics [17,28] with a single bound state (q = 1).…”