Difficulties in analytic computation for relative entropy of entanglement

It is known that relative entropy of entanglement for entangled state $ρ$ is defined via its closest separable (or positive partial transpose) state $σ$. Recently, it has been shown how to find $ρ$ provided that $σ$ is given in two-qubit system. In this paper we study on the inverse process, i.e. how to find $σ$ provided that $ρ$ is given. It is shown that if $ρ$ is one of Bell-diagonal, generalized Vedral-Plenio and generalized Horodecki states, one can always find $σ$ from a geometrical point of view. This is possible due to the following two facts: (i) The Bloch vectors of $ρ$ and $σ$ are identical with each other (ii) The qubit-interaction vector of $σ$ can be computed from a crossing point between minimal geometrical object, in which all separable states reside in the presence of Bloch vectors, and a straight line, which connects the point corresponding to the qubit-interaction vector of $ρ$ and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these nice properties are not maintained for the arbitrary two-qubit states.

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