Separability of a mixture of Dicke states

The structural relation between multipartite entanglement and symmetry is one of the central mysteries of quantum mechanics. In this paper, we study the separability of quantum states in the bosonic system. We show that a mixture of multiqubit Dicke states is separable if and only if its partial transpose is positive semidefinite, which confirms the hypothesis of Wolfe and Yelin [E. Wolfe and S. F. Yelin, Phys. Rev. Lett. 112, 140402 (2014)]. We generalize this result to a class of bosonic states in the d circle times d system; and for general d, we determine its separability is NP-hard although verifiable conditions for separability are easily derived when d = 3,4.