Separability of completely symmetric states in a multipartite system

Symmetry plays an important role in the field of quantum mechanics. We consider a subclass of symmetric quantum states in the multipartite system N - d, namely, the completely symmetric states, which are invariant under any index permutation. It was hypothesized by L. Qian and D. Chu (arXiv:1810.03125 [quant-ph]) that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. We prove that this conjecture is true for the both bipartite and multipartite cases. Further, we prove that the completely symmetric state ρ is separable if its rank is at most 5 or N+1. For the states of rank 6 or N+2, they are separable if and only if their range contains a product vector. We apply our results to a few widely useful states in quantum information, such as symmetric states, edge states, extreme states, and non-negative states. We also study the relation of completely symmetric states to Hankel and Toeplitz matrices.