As a special case we consider the Bell-diagonal case by letting R$_{1}$ = R$_{4}$ = Y = 2 ¯ R$_{1}$ = 2 ¯ R$_{4}$ = − 2 ¯ R$_{2}$ = − 2 ¯ R$_{3}$ = a , R$_{2}$ = R$_{3}$ = − 2 ¯ Y = b , R ′ 1 = R ′ 4 = Y ′ = 2 ¯ R ′ 1 = 2 ¯ R ′ 4 = − 2 ¯ R ′ 2 = − 2 ¯ R ′ 3 = a ′ and R ′ 2 = R ′ 3 = − 2 ¯ Y ′ = b $^{′}$. Of course, one can show directly that ρ$_{Z}$ ( x ) and ξ$_{Z}$ ( x $^{′}$) are really Bell-diagonal states. Using the normalization conditions 2( a + b ) = 2( a $^{′}$+ b $^{′}$) = 1, it is easy to verify that the crossing point ( µ$_{1}$, µ$_{2}$, µ$_{3}$ ) is simply µ$_{1}$ = µ$_{2}$ = 1 and µ$_{3}$ = − 1, which is one of the vertices of T . It is worthwhile noting that the crossing point is independent of particular choice of Bell-diagonal states ρ$_{Z}$ ( x ) and ξ$_{Z}$ ( x $^{′}$). This fact implies that all straight lines, which connect τ and t , pass one of the vertices of T , which is consistent with theorem 2. However for the arbitrary mixtures Eq.(6.4) implies that the crossing point ( µ$_{1}$, µ$_{2}$, µ$_{3}$ ) is dependent on the choice of the entangled states ρ$_{Z}$ ( x ) and ξ$_{Z}$ ( x $^{′}$). This is why the nice property (ii), which holds for the Bell-diagonal, generalized VP, and generalized Horodecki states, does not hold for the arbitrary mixture as Fig. 7 has indicated. In order to, there- fore, derive the closed formula of E$_{R}$ ( ρ ) for the arbitrary two-qubit mixtures ρ , we have to understand how the property (ii) is modified when r , s and t are arbitrary. Unfortunately, still this is an unsolved problem too. In this paper we have considered how to find the CSS in the two-qubit system from the geometrical point of view. Of course, one can straightforwardly compute the REE of the state ρ if its CSS is found. Therefore, it is important to develop a technique for finding CSS to overcome the calculational difficulty of the REE. Since, furthermore, the REE is a tight upper bound of the distillable entanglement, finding CSS is also important to understand the nature of the optimal (or near-optimal) purification protocols. If ρ is one of Bell-diagonal, generalized VP, and generalized Horodecki states, we have shown how to find the CSS of ρ , say σ , systematically by proving the following nice two properties: (i) The Bloch vectors of σ are identical with those of ρ . (ii) The correlation vector of σ exactly corresponds to the crossing point between the line ℓ and the geometrical object L $_{r,s}$. Using these two properties it is straightforward to find the CSS of ρ . As we have shown in the previous section, however, these two nice properties are not maintained for the general two-qubit states. Therefore, in order to derive the closed formula VII. CONCLUSION 20