N-qubit Entanglement Index and Classification

Abstract

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 $(n_1,n_2,...,n_p)$ of N with $n_j\ge 1$ and $N=\sum_{j=1}^{p}n_j$, the entanglement index for the corresponding state $ρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p}$ with $ρ_{n_j}$ denoting a fully entangled state of $n_j-$qubits is $E(ρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p})=\sum_{j=1}^{p}(n_j-1)=N-p$.