Decomposition of completely symmetric states states

In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state $|x,x\rangle\langle x,x|$, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or $N$. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to $4$ or $N+1$. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.