the objective value of (\[SDP\_Z2Synch00\]) cannot exceed that of (\[SDP\_Z2Synch2dual\]). Thus, if there exists $Q$, feasible solution of (\[SDP\_Z2Synch2dual\]), such that ${\operatorname{Tr}}(Q) = {\operatorname{Tr}}\left[\left(A_G -2A_H\right)11^T\right]$, then $X=11^T$ is an optimal solution of (\[SDP\_Z2Synch00\]). Moreover, $Q$ and $11^T$ have to satisfy complementary slackness: ${\operatorname{Tr}}(11^T(Q-\left(A_G -2A_H\right)))=0$. Given these constraints, one can ask that the equality holds for each row partial sum and construct the natural candidate $Q=D_G - 2D_H$. Indeed, it is easy to see that ${\operatorname{Tr}}(D_G-2D_H) = {\operatorname{Tr}}\left[\left(A_G -2A_H\right)11^T\right]$. Hence, if $$\label{conditionneededdual0}
 L_G - 2L_H = D_G - 2D_H -\left(A_G -2A_H\right) \succeq 0,$$ i.e., the dual variable is positive-semidefinite (PSD), then $11^T$ must be an optimal solution of (\[SDP\_Z2Synch00\]). Additionally, if $L_G - 2L_H$ is not only PSD but also its second smallest eigenvalue is non-zero, since the complementarity conditions guarantee that any optimal solution $X'$ needs to satisfy ${\operatorname{Tr}}\left(X'(L_G - 2L_H)\right)=0$, it is not difficult to show that any optimal solution needs to be a multiple of $11^T$. As one can easily see from the constraints of the SDP that no other multiple of $11^T$ is a feasible solution, $11^T$ must be the unique optimal solution. Since the success of (\[SDP\_Z2Synch00\]) does not depend on the value $g_i = (-1)^{x_{i}}$ 