We show that all the N-qubit states can be classified as N entanglement classes each of which has an entanglement index E=N−p=0,1,...,N−1(E=0 corresponds to a fully separate class) where p denotes number of groups for a partition of the positive integer N. In other words, for any partition (n1,n2,...,np) of N with nj≥1 and N=∑j=1pnj, the entanglement index for the corresponding state ρn1⨂ρn2...⨂ρnp with ρnj denoting a fully entangled state of nj−qubits is E(ρn1⨂ρn2...⨂ρnp)=∑j=1p(nj−1)=N−p.
In tri-partite systems, there are three basic biseparability, A-BC, B-CA, and C-AB, according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three-qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members to see that there are infinitely many kinds of three-qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three-qubit Greenberger–Horne–Zeilinger diagonal states distinguishing those convex sets in the chain.