Constructing $N$-qubit entanglement monotones from antilinear operators

We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call “comb”. For qubits (or spin $1∕2$) the combs are automatically invariant under $\mathrm{SL}(2,\mathbb{C})$. This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulas for the concurrence and the three-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-qubit entanglement.