Classification scheme of pure multipartite states based on topological phases

We investigate the connection between the concept of affine balancedness (a-balancedness) introduced in \[Phys. Rev A. {\bf 85}, 032112 (2012)\] and polynomial local SU invariants and the appearance of topological phases respectively. It is found that different types of a-balancedness correspond to different types of local SU invariants analogously to how different types of balancedness as defined in \[New J. Phys. {\bf 12}, 075025 (2010)\] correspond to different types of local SL invariants. These different types of SU invariants distinguish between states exhibiting different topological phases. In the case of three qubits the different kinds of topological phases are fully distinguished by the three-tangle together with one more invariant. Using this we present a qualitative classification scheme based on balancedness of a state. While balancedness and local SL invariants of bidegree (2n,0) classify the SL-semistable states \[New J. Phys. {\bf 12}, 075025 (2010), Phys. Rev. A {\bf 83} 052330 (2011)\], a-balancedness and local SU invariants of bidegree (2n-m,m) gives a more fine grained classification. In this scheme the a-balanced states form a bridge from the genuine entanglement of balanced states, invariant under the SL-group, towards the entanglement of unbalanced states characterized by U invariants of bidegree (n,n). As a by-product we obtain generalizations to the W-state, states that are entangled, but contain only globally distributed entanglement of parts of the system.