A $(k+1)$-partite entanglement measure of $N$-partite quantum states

The concept of \textquotedblleft the permutationally invariant part of a density matrx\textquotedblright constitutes an important tool for entanglement characterization of multiqubit systems. In this paper, we first present $(k+1)$-partite entanglement measure of $N$-partite quantum system, which possesses desirable properties of an entanglement measure. Moreover, we give strong bounds on this measure by considering the permutationally invariant part of a multipartite state. We give two definitions of efficient measurable degree of $(k+1)$-partite entanglement. Finally, several concrete examples are given to illustrate the effectiveness of our results.