content_image/362700.jpg content_image/362701.jpg content_image/362702.jpg content_image/362703.jpg
r = − s = 0 . 3 (Fig. 7b), r = s = 0 . 5 (Fig. 7c) and r = − s = 0 . 5 (Fig. 7d). For convenience, we plot T and L r,s together in each figure. Each line in Fig. 7 represents a set of t , whose CSS has same τ . As Fig. 7 exhibits, not all lines do pass one of the vertices of T . This fact indicates that unfortunately the nice property (ii) is not maintained for the arbitrary states. 

The non-maintenance of the property (ii) can be proved on the analytical ground by making use of the simpler model. Let us consider σ$_{Z}$ in Eq.(4.8) and ρ$_{Z}$ ( x ) in Eq.(4.9). Then, the Bloch vectors r , s and the correlation vector t of ρ$_{Z}$ ( x ) are r = (0 , 0 , r ), s = (0 , 0 , s ) and t = ( t$_{1}$, t$_{2}$, t$_{3}$ ), where 

<img src='content_image/362700.jpg'>

Of course, if we take x → 0 limit in Eq.(6.1), the corresponding quantities are the Bloch vectors and the correlation vector of σ$_{Z}$ . Now, let us consider another state π$_{Z}$ , which can be obtained from σ$_{Z}$ by changing Y → Y ′ and R$_{i}$ → R ′ $_{i}$( i = 1 , · · · , 4). In order to ensure that π$_{Z}$ is CSS we require Y ′ = √ R ′ $_{1}$R ′ 4 and R ′ $_{2}$R ′ 3 ≥ R ′ $_{1}$R ′ $_{4}$. Then, the set of the entangled states ξ$_{Z}$ ( x $^{′}$), whose CSS are π$_{Z}$ , can be obtained from ρ$_{Z}$ ( x ) by changing Y → Y $^{′}$, R$_{i}$ → R ′ $_{i}$( i = 1 , · · · , 4) and x → x $^{′}$. Thus, Bloch vectors r $^{′}$, s ′ and correlation vector t ′ of ξ$_{Z}$ ( x $^{′}$) are r ′ = (0 , 0 , r $^{′}$), s ′ = (0 , 0 , s $^{′}$) and t ′ = ( t ′ $_{1}$, t ′ $_{2}$, t ′ $_{3}$), where 

<img src='content_image/362701.jpg'>

Then, it is straightforward to show that the condition t = t ′ imposes 

<img src='content_image/362702.jpg'>

where ˜ r = R$_{1}$ − R$_{2}$ − R$_{3}$ + R$_{4}$ and ˜ r ′ = R ′ 1 − R ′ 2 − R ′ 3 + R ′ $_{4}$. Thus, one can compute the crossing point t = t ′ = ( µ$_{1}$, µ$_{2}$, µ$_{3}$ ), where µ$_{i}$ becomes 

<img src='content_image/362703.jpg'>

