Therefore, the set of the entangled states, which have ˜ π$_{vp}$ as CSS, can be represented by Finally, the set of the entangled states, which have π$_{vp}$ as CSS, can be derived by taking the inverse LU transformation, i.e. It is easy to show that ρ$_{Z}$ ( x ) reduces to ρ$_{vp}$ in Eq.(4.1) when x = x$_{vp}$ = λ$_{1}$L/ | λ$_{2}$ − λ$_{3}$ | ≥ 0, which completes the proof. FIG. 5: Fig. 5 shows how to find the CSS for the VP states. Fig. 5a and Fig. 5b correspond to r = s = 0 . 3 and r = s = 0 . 5, respectively. To find a CSS make a straight line ℓ first, which connects the nearest vertex of T and a point t$_{vp}$ . Second, compute the coordinate for the intersection point between ℓ and L $_{r,s}$. Thirdly, identify the crossing point with τ$_{vp}$ . Keeping u$_{vp}$ = r$_{vp}$ and v$_{vp}$ = s$_{vp}$ , one can find the CSS of the VP state. Fig. 5 shows how to find the CSS for the VP state geometrically when r = s = 0 . 3 (Fig. 5a) and r = s = 0 . 5 (Fig. 5b). Fig. 5 indicates that the generalized VP states are on the edges of T . First we make a line, which connects the nearest vertex of T and ρ$_{vp}$ . Then, we compute the coordinate of the crossing point τ between the line and L $_{r,s}$. Finally, the CSS π$_{vp}$ of ρ$_{vp}$ can be computed by Eq.(4.3) with keeping the Bloch vectors. In this section we discuss how to derive the CSS of the generalized Horodecki states. The Horodecki states are defined as mixture of one Bell state and separable states, which are orthogonal to the Bell state. One but most general example of the VP state is ρ$_{Z}$ ( x ) = ( 1 ⊗ σ$_{x}$ ) $^{†}$˜ ρ$_{Z}$ ( x )( 1 ⊗ σ$_{x}$ ) . (4.14) ˜ ρ$_{Z}$ ( x ) =    0 0 0 0 0 λ 1 2 + λ$_{2}$ x | λ 2 − λ 3 | 2 L 0 0 x | λ 2 − λ 3 | 2 L λ 1 2 + λ$_{3}$ 0 0 0 0 0    . (4.13) ρ$_{H}$ = λ$_{1}$ | β$_{1}$ 〉〈 β$_{1}$ | + λ$_{2}$ | 01 〉〈 01 | + λ$_{3}$ | 10 〉〈 10 | (5.1) V. CSS FOR THE GENERALIZED HORODECKI STATES 14