Large and Small Corrections to the JLMS Formula from Replica Wormholes

The JLMS formula relates the bulk and boundary relative entropies and is fundamental to the holographic dictionary, providing justification for entanglement wedge reconstruction. We revisit the replica trick for relative entropy and find corrections of the JLMS formula in a variety of scenarios, even after accounting for effects of quantum extremality. We analyze the problem in the PSSY model, a model of Jackiw-Teitelboim gravity coupled to end-of-the-world branes. We find non-perturbative (in $G$) corrections that are always present, arising from subdominant replica wormhole gravitational saddles that indicate the approximate error-correcting nature of AdS/CFT. Near entanglement phase transitions, these saddles can get enhanced to large corrections. We find $O\left(G^{-1/2}\right)$ corrections arising from area fluctuations and $O\left(G^{-1}\right)$ corrections from incompressible bulk quantum states. Lastly, we find our most surprising result, an infinite violation of the JLMS formula after the Page time arising from a rank deficiency in the bulk entanglement spectrum. We discuss similar calculations in tensor networks and comment on the implications for bulk reconstruction.

Comment: v1: 23 pages, 6 figures, lots of mini-figures; v2: minor clarifications, references added