0. Since each of the states Second, by using this equation, we obtain and Third, by iterating the swap process, 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). swaps the Hence we have shown Eq. (8). Therefore, which gives Also, we can show Therefore, which gives Then, Let us define which gives and therefore 8 〈◦ • • ◦ | ◦ ◦ • •〉 = 〈◦ • • ◦ | ˆ t$_{2}$$_{,}$$_{3}$ | ◦ ◦ • •〉 = − 1 2 〈◦ • • ◦ | ◦ • • ◦〉 = − 1 2 〈◦ ◦ • • | ˆ t$_{2}$$_{,}$$_{3}$ | ◦ ◦ • •〉 = − 1 2 〈◦ ◦ • • | ◦ • • ◦〉 = 1 4 . ˆ t$_{2}$$_{,}$$_{3}$ | ◦ ◦ • •〉 = − 1 2 | ◦ • • ◦〉 , GLYPH<j49> GLYPH<j50> GLYPH<j51> GLYPH<j52> GLYPH<j49> GLYPH<j50> GLYPH<j51> GLYPH<j52> GLYPH<j116> GLYPH<j50>GLYPH<j44>GLYPH<j51> FIG. 8. (Color online) The projection operator ˆ t 2 , 3 entanglement. Red lines represent singlet pairs. 〈 Ψ | ˆ T $^{2}$| Ψ 〉 = O ( N $^{2}$) . 〈 VB$_{i}$ | ˆ T $^{2}$| VB$_{j}$ 〉 ≃ δ$_{i,j}$ ( 9 N 2 64 + 3 N 32 ) . 〈 φ$_{i}$ | φ$_{j}$ 〉 ≃ δ$_{i,j}$ ( N 2 64 + 3 N 32 ) , 〈 Ψ | ˆ T | Ψ 〉 ≃ 0 . 〈 VB$_{i}$ | ˆ T | VB$_{j}$ 〉 ≃ δ$_{i,j}$ ( − 1) $_{i}$3 N 8 . 〈 VB$_{i}$ | φ$_{j}$ 〉 ≃ δ$_{i,j}$ ( − 1) i $_{+1}$N 8 , | φ$_{2}$ 〉 ≡ ˆ T | VB$_{2}$ 〉 − N 2 | VB$_{2}$ 〉 = − ∑ l =odd ˆ t$_{l,l}$$_{+1}$ | VB$_{2}$ 〉 . | φ$_{1}$ 〉 ≡ ˆ T | VB$_{1}$ 〉 + N 2 | VB$_{1}$ 〉 = ∑ l =even ˆ t$_{l,l}$$_{+1}$ | VB$_{1}$ 〉 〈 VB$_{2}$ | VB$_{1}$ 〉 = ( − 1 $_{2}$) N/ 2 − 1 ≃ 0 . 〈 VB$_{2}$ | VB$_{2}$ 〉 = ( − 2) N/ 2 − $^{1}$〈 VB$_{2}$ |$^{N/}$$^{2}$$^{−}$$^{1}$ ∏ l =1 ˆ t$_{2}$$_{l,}$$_{2}$$_{l}$$_{+1}$ | VB$_{1}$ 〉 , | VB$_{2}$ 〉 = ( − 2) N/ 2 − 1 N/ 2 − 1 ∏ l =1 ˆ t$_{2}$$_{l,}$$_{2}$$_{l}$$_{+1}$ | VB$_{1}$ 〉 , ˆ σ$_{x}$ ( l ) | Ψ 〉 , ˆ σ$_{y}$ ( l ) Ψ , ˆ σ$_{x}$ ( l )ˆ σ$_{x}$ ( l $^{′}$) | Ψ 〉 , ˆ σ$_{x}$ ( l )ˆ σ$_{y}$ ( l $^{′}$) Ψ , ˆ σ$_{x}$ ( l )ˆ σ$_{z}$ ( l $^{′}$) | Ψ 〉 , ˆ σ$_{y}$ ( l )ˆ σ$_{y}$ ( l $^{′}$) Ψ , ˆ σ$_{y}$ ( l )ˆ σ$_{z}$ ( l $^{′}$) | Ψ 〉 , ˆ σ$_{z}$ ( l ) | Ψ 〉 , ˆ σ$_{z}$ ( l )ˆ σ$_{z}$ ( l $^{′}$) Ψ , 2. Proof of Eq. (8)