# Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula The JLMS formula can be written as follows: $P_C \hat{K}_R P_C = \frac{\hat{A}}{4 G} + \hat{K}_r,$where $P_C$ is a projector onto the code subspace, $\hat K_R$ is the modular Hamiltonian of a boundary region $R$, and $\hat A$ denotes the area operator. The entanglement wedge of $R$, or more precisely a partial Cauchy surface of the entanglement wedge, is denoted by $r$. It also has the following compact version:$D\left(\rho_{R} \| \sigma_{R}\right)=D_{\text {bulk }}\left(\rho_{r} \| \sigma_{r}\right),$i.e., the bulk [[0199 Relative entropy|relative entropy]] equals boundary relative entropy. It is derived using Euclidean path integral, but has a Lorentzian interpretation from [[0146 Quantum error correction|quantum error correction]]. [[2016#Harlow]] demonstrated that the existence of an exact QEC imply the JLMS formula. JMLS also implies the existence of a bulk reconstruction map, the [[0413 Petz map|Petz map]]. ## Refs - original - [[2016#Jafferis, Lewkowycz, Maldacena, Suh]] - more detailed derivation and extension to higher orders - [[2016#Dong, Harlow, Wall]] - quantum JLMS - [[2017#Dong, Lewkowycz]] ## Properties - corrections from replica wormholes (which can be large) - [[2022#Kudler-Flam, Rath (Mar)]] ## Applications - bulk locality - [[2017#Faulkner, Lewkowycz]] - extracting information from the [[0213 Islands|islands]] - [[2019#Chen]] ## Related - [[0416 Modular Hamiltonian]]