and its CSS be The following theorem shows how to compute u$_{vp}$ , v$_{vp}$ and τ$_{vp}$ from ρ$_{vp}$ . Theorem 3. If π$_{vp}$ is the CSS of ρ$_{vp}$ , u$_{vp}$ = r$_{vp}$ and v$_{vp}$ = s$_{vp}$ . Let ℓ be a straight line, which connects t$_{vp}$ and the nearest vertex of T . Then, τ$_{vp}$ is a crossing point between ℓ and L $_{r,s}$. Proof. We will prove this theorem as following procedure. First, we assume that this theorem is correct. Then, following this theorem one can derive the trial CSS state of ρ$_{vp}$ . Next, by making use of Eq.(1.6) we will show that this trial CSS state is a really CSS state of ρ$_{vp}$ . Since other VP states can be derived from Eq.(4.1) by local-unitary (LU) transformation, it is sufficient to show that the CSS of ρ$_{vp}$ in Eq.(4.1) satisfies this theorem. The other case can be proven similarly. For ρ$_{vp}$ in Eq.(4.1) r$_{vp}$ , s$_{vp}$ and t$_{vp}$ become r$_{vp}$ = (0 , 0 , r ), s$_{vp}$ = (0 , 0 , s ) and t$_{vp}$ = ( t$_{1}$, t$_{2}$, t$_{3}$ ) where Then, a point P = ( q$_{1}$, q$_{2}$, q$_{3}$ ) on the line ℓ satisfies q$_{2}$ = − q$_{1}$ and q$_{3}$ = 1. If the point P = ( q$_{1}$, q$_{2}$, q$_{3}$ ) is a crossing point between ℓ and L $_{r,s}$, the corresponding separable state satisfies where µ Γ − and ν Γ − are defined at Table II. Therefore, the CSS condition (3.7) implies q$_{1}$ = 0, which results in τ$_{vp}$ = (0 , 0 , 1). If, therefore, this theorem is correct, the CSS of ρ$_{vp}$ is In order to show that π$_{vp}$ in Eq.(4.6) is really CSS of ρ$_{vp}$ , it is convenient to define another π$_{vp}$ = 1 4 ( I ⊗ I + u$_{vp}$ · σ ⊗ I + I ⊗ v$_{vp}$ · σ + 3 ∑ n =1 ( τ$_{vp}$ )$_{n}$ σ$_{n}$ ⊗ σ$_{n}$ ) . (4.3) r = s = λ$_{2}$ − λ$_{3}$ t$_{1}$ = − t$_{2}$ = λ$_{1}$ t$_{3}$ = 1 . (4.4) µ Γ − = − 1 2 | q$_{1}$ | , ν Γ − = 1 2 (1 − | λ$_{2}$ − λ$_{3}$ | ) , (4.5) π$_{vp}$ = 1 4 [ I ⊗ I + ( λ$_{2}$ − λ$_{3}$ ) ( σ$_{3}$ ⊗ I + I ⊗ σ$_{3}$ ) + σ$_{3}$ ⊗ σ$_{3}$ ] (4.6) =    λ 1 2 + λ$_{2}$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 1 2 + λ$_{3}$    . 12