Exchange Symmetry and Multipartite Entanglement

TL;DR

This paper explores how exchange symmetry influences multipartite entanglement, revealing that symmetric and anti-symmetric states have distinct entanglement properties, with implications for bosonic, fermionic, and anyonic systems.

Contribution

It establishes a direct link between exchange symmetry and entanglement, showing that symmetry properties determine whether states are globally entangled or separable, and extends results to anyonic states.

Findings

Anti-symmetric states are necessarily globally entangled.

Symmetric states are either globally entangled or fully separable.

Full separability does not survive symmetrization unless states are originally symmetric.

Abstract

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 anti-symmetric (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 should also be globally entangled. Our results reveal that exchange symmetry is actually sufficient for pure states to become globally entangled or fully separable.