  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.
