content_image/425631.jpg content_image/425632.jpg content_image/425645.jpg content_image/425646.jpg content_image/425629.jpg content_image/425630.jpg content_image/425644.jpg content_image/425643.jpg content_image/425641.jpg content_image/425640.jpg content_image/425635.jpg content_image/425642.jpg content_image/425639.jpg content_image/425638.jpg content_image/425637.jpg content_image/425636.jpg content_image/425634.jpg
0. Since each of the states 

for | l − l $^{′}$| ≥ 2 has no component in the eigenspace of ˆ M$_{z}$ corresponding to the eigenvalue M$_{z}$ = 0, they are orthogonal to 〈 Ψ | . In the same way, 

for | l − l $^{′}$| ≥ 2 are orthogonal to 〈 Ψ | since each of them has no component in the eigenspace of ˆ M$_{x}$ corresponding to the eigenvalue M$_{x}$ = 0. Hence we have shown Eq. (7). 

Before showing the equation, let us note some useful relations. First, the projection operator ˆ t$_{2}$$_{,}$$_{3}$ “swaps” the entanglement 

where singlet pairs are schematically represented: sites represented by the circle of the same color make a singlet pair (see Fig. 8). 

## 2. Proof of Eq. (8) 

<img src='content_image/425631.jpg'>

<img src='content_image/425632.jpg'>

<img src='content_image/425645.jpg'>

<img src='content_image/425646.jpg'>

swaps the 

Second, by using this equation, we obtain 

<img src='content_image/425629.jpg'>

and 

<img src='content_image/425630.jpg'>

Third, by iterating the swap process, 

<img src='content_image/425644.jpg'>

and therefore 

<img src='content_image/425643.jpg'>

which gives 

Let us define 

<img src='content_image/425641.jpg'>

<img src='content_image/425640.jpg'>

Then, 

which gives 

Therefore, 

Also, we can show 

which gives 

Therefore, 

Hence we have shown Eq. (8). 

<img src='content_image/425635.jpg'>

<img src='content_image/425642.jpg'>

<img src='content_image/425639.jpg'>

<img src='content_image/425638.jpg'>

<img src='content_image/425637.jpg'>

<img src='content_image/425636.jpg'>

<img src='content_image/425634.jpg'>

