The invariant-comb approach and its relation to the balancedness of multipartite entangled states

The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call comb in reference to the hairy-ball theorem. An appealing feature of this approach is that, for qubits (or spins 1/2), the combs are automatically invariant under SL(2,C), which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial SL(2,C) invariant, we find that it is the presence of a balanced part that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of irreducibly balanced states. The latter indicates a tight connection with stochastic local operations and classical communication (SLOCC) classifications of qubit entanglement. Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs, such that it is nontrivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the SL(2,C), we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.