Generalized entanglement entropies in two-dimensional conformal field theory

Abstract

We introduce and study generalized Rényi entropies defined through the traces of products of ${\rm Tr}_B (|Ψ_i\rangle\langle Ψ_j|)$ where $|Ψ_i\rangle$ are eigenstates of a two-dimensional conformal field theory (CFT). When $|Ψ_i\rangle=|Ψ_j\rangle$ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four-point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.