Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group $S_N$. With the observation that symmetric property under the exchange of two constituent states and their separability are intimately linked, we show that antisymmetric (fermionic) states are necessarily globally entangled, while symmetric (bosonic) states are either globally entangled or fully separable and possess essentially identical states in all the constituent systems. It is also shown that there cannot exist a fully separable state which is orthogonal to all symmetric states, and that full separability of states does not survive under total symmetrization unless the states are originally symmetric. Besides, anyonic states permitted under the braid group $B_N$ must also be globally entangled. Our results reveal that exchange symmetry is actually sufficient for pure states to become globally entangled or fully separable.

• Received 17 June 2008

DOI:https://doi.org/10.1103/PhysRevA.78.052105