# Renyi entropy The **Renyi entropy** associated with a density matrix $\rho$ is defined for positive integers $n$ as $S_n=-\frac{1}{n-1} \log \operatorname{Tr}\left[\rho^n\right].$Here $\rho$ can be taken to be the reduced density matrix associated with a subregion of interest by tracing over degrees of freedom in the complement region. ## Properties - $S_n=0$ if $\rho$ is pure; $S_n>0$ otherwise - for a given state, $S_n$ is non-increasing in $n$ ## Special values of $n$ - $S_0=\ln(\operatorname{rank}(\rho))$ (Hartley or max entropy) - $S_1 =$ [[0301 Entanglement entropy|von Neumann entropy]] - $S_2=-\ln(\operatorname{Tr}(\rho^2))=-\ln(\text{purity})$ - most accessible experimentally - $S_\infty=-\ln(\text{prob. of most likely outcome})$ ## Holographic dual - [[2011#Hung, Myers, Smolkin, Yale]]: before cosmic branes - [[2016#Dong]]: (Euclidean) cosmic brane - [[2016#Dong, Lewkowycz, Rangamani]]: Lorentzian "conical" geometry - [[2021#Colin-Ellerin, Dong, Marolf, Rangamani, Wang]]: explicit low-dimensional calculations - [[2023#Dong, Kudler-Flam, Rath]]: modified cosmic brane; works for with more than two subregions - [[2023#Kastikainen, Svesko (Dec, a)]]: relation to corner terms - [[2024#Harper, Takayanagi, Tsuda]]: Renyi multi-entropy - [[2024#Penington, Rath]]: derives the diagonal approximation used in [[2023#Dong, Kudler-Flam, Rath]] ## Renyi second law - [[2018#Bernamonti, Galli, Myers, Oppenheim]]: minimally coupled gravity-scalar theory - [[2024#Bernamonti, Galli, Myers, Reyes]] ## Extensions - charged Renyi: [[BaigueraBianchiChapmanGalante2022]][](https://arxiv.org/pdf/2203.15028.pdf) ## Calculations - single interval in 2d CFT - [[HolzheyLarsenWilczek1994]][](https://arxiv.org/abs/hep-th/9403108) - [[2004#Calabrese, Cardy]] - two intervals - [[CalabreseCardyTonni2009]][](https://arxiv.org/abs/0905.2069): free 2d CFT - [[2010#Headrick]]: first holographic Renyi calculation - $N$ intervals - [[2013#Faulkner]]: constructing bulk handlebody solutions - [[2013#Hartman]]: using CFT [[0032 Virasoro algebra|Virasoro]] vacuum [[0031 Conformal block|blocks]] - [[2024#Shi, Zhang]]: proposes a universal approach using swap operations - $n\to0$ limit in higher-dimensional CFT - [[2025#Kusuki, Ooguri, Pal]] - other - [[IrakleousTomarasToumbas2021]][](https://arxiv.org/pdf/2105.11858.pdf): direct calculation for scattering processes in QED ## Connections to other topics - [[2024#Iizuka, Nishida]]: Renyi as a sign of chaos