6 Using this result ρ$_{A}$ can be simplified to Notice that Y$_{B}$ = i | B | ( | B |− 1) / $^{2}$∏ j ∈ B η$_{j}$ is actually the fermion number parity operator in B region, we see that the operator 1+ x$_{B}$ ( w ) Y$_{B}$ 2 is actually the projector of the fermion number parity Y$_{B}$ = x$_{B}$ ( w ). For convenience, we define ρ$_{A}$ = 1 2 N + L +1 ∑ g$_{A}$,g ′ $_{A}$,w ∑ g$_{B}$ X$_{g}$$_{A}$ | u$_{A}$, w 〉 〈 u$_{A}$, w | X † g ′ A · Y$_{g}$$_{A}$ Tr$_{B,F}$ [ | φ ( u ) 〉 〈 φ ( u ) | (1 + Y$_{B}$ )] Y † g ′ $_{A}$, = 1 2 N$_{A}$ + L ∑ g$_{A}$,g ′ $_{A}$,w X$_{g}$$_{A}$ | u$_{A}$, w 〉 〈 u$_{A}$, w | X † g ′ A · Y$_{g}$$_{A}$ Tr$_{B,F}$ [ | φ ( u ) 〉 〈 φ ( u ) | ( 1 + x$_{B}$ ( w ) Y$_{B}$ 2 )] Y † g ′ $_{A}$. (A7) P x$_{B}$ ( w ) B,F = 1 + x$_{B}$ ( w ) Y$_{B}$ 2 (A8) Tr$_{B,F}$ [ | φ ( u ) 〉 〈 φ ( u ) | P x$_{B}$ ( w ) B,F ] = ρ x$_{B}$ ( w ) A,F (A9) and similar for A region. The reduced density matrix is the fermion density matrix with restriction to the fixed fermion number parity x$_{B}$ ( w ) in the B region. By using similar interproduct formula (A5) for A region, ρ 2 A can be computed as ρ 2 A = 1 2 N$_{A}$ +2 L − 1 ∑ g$_{A}$,g ′ $_{A}$,w X$_{gA}$ | u$_{A}$, w 〉 〈 u$_{A}$, w | X † g ′ A · Y$_{gA}$ρ x$_{B}$ ( w ) A,F P x$_{A}$ ( w ) A,F ρ x$_{B}$ ( w ) A,F Y$_{g}$ ′ A (A10) ρ n A = 1 2 N$_{A}$ + nL − n +1 ∑ g$_{A}$,g ′ $_{A}$,w X$_{gA}$ | u$_{A}$, w 〉 〈 u$_{A}$, w | X$_{g}$ ′ A · Y$_{gA}$ρ x$_{B}$ ( w ) A,F ( P x$_{A}$ ( w ) A,F ρ x$_{B}$ ( w ) A,F ) n − $^{1}$Y$_{g}$ ′ A (A11) Tr$_{A}$ [ ρ n $_{A}$] = 1 2 n ( L − 1) ∑ w Tr$_{A}$ ( P x$_{A}$ ( w ) A,F ρ x$_{B}$ ( w ) A,F ) n (A12) Finally we take the trace of ρ n A by using the same Eq. (A5) to obtain The same calculation can be repeated to obtain For Eq. (A6), we see that flipping the sign of one link w$_{i}$ , for some integer 1 ≤ i ≤ L , changes the sign of both x$_{A}$ ( w ) and x$_{B}$ ( w ). If we define ∏ ij ∈ A ( B $_{)}$u$_{ij}$ = p$_{A}$$_{(}$$_{B}$$_{)}$ , then x$_{A}$$_{(}$$_{B}$$_{)}$ ( w ) = p$_{A}$$_{(}$$_{B}$$_{)}$ ∏ L n $_{=1}$w$_{n}$ , and the summation over all 2 L configurations of w = { w$_{i}$, i = 1 , 2 , ..., L } leads to Tr$_{A}$ [ ρ n $_{A}$] = 1 2 ( n − 1)( L − $_{1)}$Tr$_{A}$ [( P p$_{A}$ $_{A,F}$ρ p$_{B}$ A,F ) n + ( P − p$_{A}$ A,F ρ − p$_{B}$ A,F ) $^{n}$] (A13) ρ p$_{B}$ $_{A,F}$P − p$_{A}$ A,F = Tr$_{B,F}$ [ | φ ( u ) 〉 〈 φ ( u ) | P p$_{B}$ B,F ] P − p$_{A}$ A,F = 0 (A14) Tr$_{A}$ [ ρ n $_{A}$] = 1 2 ( n − 1)( L − $_{1)}$Tr$_{A}$ [( P p$_{A}$ $_{A,F}$ρ p$_{B}$ A,F ) + ( P − p$_{A}$ A,F ρ − p$_{B}$ A,F )] $^{n}$, = 1 2 ( n − 1)( L − $_{1)}$Tr$_{A}$ [ ρ n A,F ] , (A15) so that In the main text we have discussed that the ground state | u 〉 | φ ( u ) 〉 must satisfy the constraint D | u 〉 | φ ( u ) 〉 = | u 〉 | φ ( u ) 〉 . Since D = X$_{A}$X$_{B}$Y$_{A}$Y$_{B}$ and X$_{A}$X$_{B}$ | u 〉 = p$_{A}$p$_{B}$ | u 〉 , we obtain Y$_{A}$Y$_{B}$ | φ ( u ) 〉 = p$_{A}$p$_{B}$ | φ ( u ) 〉 . In other words, the total fermion parity is fixed, so that P p$_{A}$ $_{A,F}$P − p$_{B}$ A,F | φ ( u ) 〉 = P − p$_{A}$ A,F P p$_{B}$ A,F | φ ( u ) 〉 = 0. Thus we have where ρ$_{A,F}$ = ρ + A,F + ρ − A,F = Tr$_{B}$ [ | φ ( u ) 〉 〈 φ ( u ) | ] is the free fermion density matrix without fermion number parity constraint in the B region. To understand this result more intuitively, we note that Tr$_{A,G}$[ ρ n $_{A,G}$] = 1 2 ( n − 1)( L − $_{1)}$, (A16)