# Holographic entropy cone The holographic entropy cone is a subspace of all quantum boundary states for a holographic system. Via [[0007 RT surface|RT/HRT formula]], boundary entanglement entropy translates to bulk geometric quantities. By studying the geometry of the bulk metric, there are relations that the RT/HRT surfaces have to satisfy, therefore constraining the holographic states. ## Refs - original: [[BaoNezamiOoguriStoicaSullyWalter2015]] - 5 regions: [[2019#Hernandez-Cuenca]] - 6-region and general discussion about multipartite information measures: [[2023#Hernandez-Cuenca, Hubeny, Jia]] - a 7-region inequality: [[CzechWang2022]][](https://arxiv.org/pdf/2209.10547.pdf) - two infinite families of inequalities: [[2023#Czech, Shuai, Wang, Zhang]] - deterministic method for finding all inequalities: [[2024#Bao, Furuya, Naskar]] - relation to erasure correction: [[2025#Czech, Shuai, Wang]] ## Examples of entropy inequalities - subadditivity (true for all quantum systems): $S_A+S_B \geq S_{A B}$ - [[0218 Strong subadditivity|strong subadditivity]] (true for all quantum systems): $S_{A B C}+S_C \ge S_{A C}+S_{B C}$ - monogamy of [[0300 Mutual information|mutual information]] (MMI): $S_{A B}+S_{B C}+S_{A C} \geq S_A+S_B+S_C+S_{A B C}$ - MMI with two extra regions $D$ and $E$ adjoined to two distinct regions (here $BD$ and $CE$): $S_{A B D}+S_{B C D E}+S_{A C E} \geq S_A+S_{B D}+S_{C E}+S_{A B C D E}$ - cyclic inequalities. See [[BaoNezamiOoguriStoicaSullyWalter2015]] ## Related topics - [[2018#Bao, Chatwin-Davies, Remmen]] has extra conditions on the entropy structure of holographic CFT (in terms of entanglement of purification) - [[2024#Bao, Naskar]]: deterministic way of finding contraction maps - [[2024#Naskar, Samal]]: relation to topological entanglement entropy ## Extensions - for averaged entropy - [[CzechShuai2021]][](https://arxiv.org/pdf/2112.00763.pdf) - quantum - [[2021#Akers, Hernandez-Cuenca, Rath]] - dynamic (needs HRT) - [[2012#Wall]]: SSA and MMI - [[2019#Czech, Dong]]: (2+1) dims, all inequalities, but contains a mistake for multiply connected spacetimes - [[2024#Grado-White, Grimaldi, Headrick, Hubeny]]: strong evidence for all inequalities - beyond holography - [[2024#Bousso, Kaya]]: MMI is correct for generalised entanglement wedge - [[2025#Bousso, Kaya]]: in static generalised entanglement wedges of bulk regions in arbitrary spacetimes