Difficulties in analytic computation for relative entropy of entanglement

It is known that relative entropy of entanglement for an entangled state rho is defined via its closest separable (or positive partial transpose) state sigma. Recently, it has been shown how to find rho provided that sigma is given in a two-qubit system. In this article we study the reverse process, that is, how to find sigma provided that rho is given. It is shown that if rho is of a Bell-diagonal, generalized Vedral-Plenio, or generalized Horodecki state, one can find sigma from a geometrical point of view. This is possible due to the following two facts: (i) the Bloch vectors of rho and sigma are identical to each other; (ii) the correlation vector of sigma can be computed from a crossing point between a 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 correlation vector of rho and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these properties are not maintained for the arbitrary two-qubit states.