REE From EOF

It is well-known that entanglement of formation (EOF) and relative entropy of entanglement (REE) are exactly identical for all two-qubit pure states even though their definitions are completely different. We think this fact implies that there is a veiled connection between EOF and REE. In this context, we suggest a procedure, which enables us to compute REE from EOF without relying on the converse procedure. It is shown that the procedure yields correct REE for many symmetric mixed states such as Bell-diagonal, generalized Vedral-Plenino, and generalized Horodecki states. It also gives a correct REE for less symmetric Vedral-Plenio-type state. However, it is shown that the procedure does not provide correct REE for arbitrary mixed states.Comment: 17 pages, 1 figure, several typos corrected, final version to appear in Quantum Information Processin

Entanglement of Four-Qubit Rank-22 Mixed States

It is known that there are three maximally entangled states $\ket{\Phi_1} = (\ket{0000} + \ket{1111}) / \sqrt{2}$, $\ket{\Phi_2} = (\sqrt{2} \ket{1111} + \ket{1000} + \ket{0100} + \ket{0010} + \ket{0001}) / \sqrt{6}$, and $\ket{\Phi_3} = (\ket{1111} + \ket{1100} + \ket{0010} + \ket{0001}) / 2$ 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 \rho_j = p \ket{\Phi_j}\bra{\Phi_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.Comment: 20 pages, 5 eps figures, will appear in Quantum Information Processin