Comment on “Separability of completely symmetric states in a multipartite system”

Abstract

In their interesting paper [Phys. Rev. A 99, 032312 (2019)], Chen et al. consider the completely symmetric (CS) states, which are invariant under any index permutation. They proved that the CS states are separable if and only if they are a convex combination of real symmetric pure product states in Theorem 14, and claimed that any CS state in a two-qutrit system is separable in Theorem 20. We point out that their proof of Theorem 20 is flawed because of invalid assumptions. Furthermore, we present a sufficient and necessary condition for the separability of the CS states in two-qutrit systems.