Since we have to choose the nearest point from t$_{H}$ , the solution we want is the former. Therefore, τ$_{H}$ becomes ( q$_{1}$, − q$_{1}$, 2 q$_{1}$ − 1). Then theorem 4 claims that the CSS of ρ$_{H}$ is Now, we will show that π$_{H}$ in Eq.(5.8) is really CSS of ρ$_{H}$ by making use of Eq.(1.6). In order to show this we define ˜ π$_{H}$ = ( 1 ⊗ σ$_{x}$ ) π$_{H}$ ( 1 ⊗ σ$_{x}$ ) $^{†}$. Then by making use of Eq.(4.8) and Eq.(4.9) it is straightforward to find a set of the entangled quantum states ˜ ρ ( x ), whose CSS are ˜ π$_{H}$ . After taking the inverse LU transformation one can derive ρ ( x ) = ( 1 ⊗ σ$_{x}$ ) $^{†}$˜ ρ ( x )( 1 ⊗ σ$_{x}$ ). The expression of ρ ( x ) is π$_{H}$ = 1 4 [ I ⊗ I + ( λ$_{2}$ − λ$_{3}$ ) { σ$_{3}$ ⊗ I − I ⊗ σ$_{3}$ } (5.8) + q$_{1}$σ$_{1}$ ⊗ σ$_{1}$ − q$_{1}$σ$_{2}$ ⊗ σ$_{2}$ + (2 q − 1) σ$_{3}$ ⊗ σ$_{3}$ ] = 1 4    ( λ$_{1}$ + 2 λ$_{2}$ )( λ$_{1}$ + 2 λ$_{3}$ ) 0 0 ( λ$_{1}$ + 2 λ$_{2}$ )( λ$_{1}$ + 2 λ$_{3}$ ) 0 ( λ$_{1}$ + 2 λ$_{2}$ ) 2 0 0 0 0 ( λ$_{1}$ + 2 λ$_{3}$ ) 2 0 ( λ$_{1}$ + 2 λ$_{2}$ )( λ$_{1}$ + 2 λ$_{3}$ ) 0 0 ( λ$_{1}$ + 2 λ$_{2}$ )( λ$_{1}$ + 2 λ$_{3}$ )    . ρ ( x ) =    Y + xη 0 0 Y + xη 0 R$_{1}$ − xη 0 0 0 0 R$_{4}$ − xη 0 Y + xη 0 0 Y + xη    (5.9) R$_{1}$ = 1 4 ( λ$_{1}$ + 2 λ$_{2}$ ) 2 R$_{4}$ = 1 4 ( λ$_{1}$ + 2 λ$_{3}$ ) 2 (5.10) Y = 1 4 ( λ$_{1}$ + 2 λ$_{2}$ )( λ$_{1}$ + 2 λ$_{3}$ ) η = Y 2 R$_{1}$ + R$_{4}$ . x = x$_{H}$ = 1 η ( λ$_{1}$ 2 − Y ) , (5.11) where x ≥ 0 and When ρ ( x ) reduces to ρ$_{H}$ in Eq.(5.1). It is easy to prove that x$_{H}$ ≥ 0 if λ 2 1 ≥ 4 λ$_{2}$λ$_{3}$ , which is an entangled condition for ρ$_{H}$ . Therefore, theorem 4 is completely proved. Fig. 6 shows how to find the CSS for the generalized Horodecki state ρ$_{H}$ when r = − s = 0 . 3 (Fig. 6a) and r = − s = 0 . 5 (Fig. 6b). If ρ$_{H}$ is explicitly given, compute r$_{H}$ , s$_{H}$ and t$_{H}$ . Then make a straight line which connects a point t$_{H}$ and the nearest vertex of T . Find the crossing points between the line and L $_{r,s}$. As Fig. 3 shows, there are two intersection 16