Decomposition of completely symmetric states

Symmetry is a fundamental milestone of quantum physics, and the relation between entanglement is one of the central mysteries of quantum mechanics. In this paper, we consider a subclass of symmetric quantum states in the bipartite system, namely the completely symmetric states, which is invariant under the index permutation. We investigate the separability of these states. After studying some examples, we conjecture that the completely symmetric state is separable if and only if it is S-separable, i.e., each term in this decomposition is a symmetric pure product state $|x,x⟩⟨x,x|$. It was proved to be true when the rank does not exceed $max\{4,N+1\}$. After studying the properties of these state, we propose a numerical algorithm which is able to detect S-separability. This algorithm is based on the best separable approximation, which furthermore turns out to be applicable to test the separability of quantum states in bosonic system. Besides, we analyse the convergence behaviour of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.