# Relative entropy Relative entropy, defined via $S(\rho| \sigma)=\operatorname{Tr}[\rho \log \rho-\rho \log \sigma],$where $\sigma$ is a reference "vacuum state" and $\rho$ is an arbitrary state we care about, is a measure of distinguishability between the two states. ## Properties - asymmetric in $\rho \leftrightarrow \sigma$ - non-negative - if two states are the same, 0 - if two states are both pure and orthogonal, infinite ## Significance - hypothesis testing - see [[Rsc0031 TASI 2021]] Hayden notes