# Entanglement entropy The entanglement entropy is the quantum analogue of the Shannon entropy. It is a quantifier of the entanglement between two sub-systems of a pure state. It is defined as ${S}_A=-\operatorname{Tr}\left[\rho_A \log \rho_A\right],$where $\rho_A$ is the reduced density matrix obtained by tracing over $B$. It is often computed by taking the $n\to1$ limit of the [[0293 Renyi entropy|Renyi entropy]], $S_n=-\frac{1}{n-1} \log \operatorname{Tr}\left[\rho^n\right],$a procedure referred to as the *replica trick*. The entanglement entropy typically contains two parts: an area term and a log term. The area term depends on the regularisation but the log term is universal. ## The log term - universal - with straight edges: a sum over vertices (kinks of the entangling surface): $S_{\log}=\sum_i c(\theta_i)\log(L/\epsilon)$ where $\epsilon$ is the UV cut-off of the lattice - More generally it is related to the curvature of the surface. E.g. for a sphere, $S=c\frac{\int_\Sigma d\sigma}{\epsilon^2}+2A\int_\Sigma d\sigma.\mathcal{R}\log(\epsilon)+\text{finite}$ ## Properties - symmetry - so can talk about entanglement *between* A and B - see e.g. [[Rsc0031 TASI 2021]] Hayden for proof ## Refs - replica trick - [[2004#Calabrese, Cardy]] - topological entanglement entropy - [[2005#Kitaev, Preskill]] - [[2005#Levin, Wen]] - [[1991#Affleck, Ludwig]] - curved background - [[2024#Shekar, Taylor]] - explicit computations - [[BenedettiDaguerre2023]][](https://arxiv.org/abs/2307.00057): [[0531 Rarita-Schwinger fields|Rarita-Schwinger field]] - universal contribution in even dim CFT - [[2006#Ryu, Takayanagi (May)]] - [[Solodukhin2008]][](https://arxiv.org/abs/0802.3117) - twist-field correlator - [[2025#Estienne, Lin]] and refs therein - BCFT method - [[2014#Ohmori, Tachikawa]]: original proposal - [[2025#Roy, Lukyanov, Saleur]]: numerical test ## Related - [[0145 Generalised area]] - parent [[0290 Quantum information measures]]