Many applications of diversity indices are only valid if they are first transformed into their equivalent number of species. These equivalent numbers of species can be multiplicatively partitioned into independent alpha, beta and gamma components, and can be formed into mathematically consistent similarity measures. The utility of beta diversity and similarity measures that incorporate information about the degree of ecological dissimilarity between species is becoming increasingly recognized. The concept of equivalent number of species is here extended to Rao's quadratic entropy, opening the way to methods of diversity partitioning that take into account taxonomic or ecological differences between species.