Entanglement of Four-Qubit Rank-$2$ Mixed States

It is known that there are three maximally entangled states $\ket{Φ_1} = (\ket{0000} + \ket{1111}) / \sqrt{2}$, $\ket{Φ_2} = (\sqrt{2} \ket{1111} + \ket{1000} + \ket{0100} + \ket{0010} + \ket{0001}) / \sqrt{6}$, and $\ket{Φ_3} = (\ket{1111} + \ket{1100} + \ket{0010} + \ket{0001}) / 2$ in four-qubit system. It is also known that there are three independent measures ${\cal F}^{(4)}_j \hspace{.2cm} (j=1,2,3)$ for true four-way quantum entanglement in the same system. In this paper we compute ${\cal F}^{(4)}_j$ and their corresponding linear monotones ${\cal G}^{(4)}_j$ for three rank-two mixed states $ρ_j = p \ket{Φ_j}\bra{Φ_j} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$, where $\ket{\mbox{W}_4} = (\ket{0111} + \ket{1011} + \ket{1101} + \ket{1110}) / 2$. We discuss the possible applications of our results briefly.

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