Necessary conditions for classifying $$\mathbf {m}$$m -separability of multipartite entanglements

We study the norms of the Bloch vectors for arbitrary n-partite quantum states. A tight upper bound of the norms is derived for n-partite systems with different individual dimensions. These upper bounds are used to deal with the separability problems. Necessary conditions are presented for $$\mathbf {m}$$-separable states in n-partite quantum systems. Based on the upper bounds, classification of multipartite entanglement is illustrated with detailed examples.