Genuine Multi-Entropy of Fully Symmetric Gaussian States

Abstract: We study the tri-partite genuine Rényi multi-entropy of fully symmetric Gaussian states as a function of the squeezing parameter for different total number of bosonic modes under different subsystem configurations. Exact results for a few modes are obtained, and we conjecture an asymptotic formula at the limit of large squeezing parameter as a function of the replica index $n$, while also finding subleading subsystem-size dependent terms for $n=3$ under a specific tripartition. We analytically show that the genuine $n=2$ Rényi multi-entropy of any pure bosonic Gaussian states is zero for any number of modes and for any tripartition. Furthermore, to investigate the dependence on subsystem size, we plot the curve of genuine Rényi multi-entropy as a function of the number of modes in the subsystems while keeping the total number of modes fixed. We discuss differences and similarities between this result and studies of the multi-entropy curve for Haar random states. In particular, we identify a subsystem size configuration related with the multi-entropy time in an evaporating black hole for which the behavior of the genuine multi-entropy changes.