convex, one can derive a formal solution to the $N$-representability problem. This necessary and sufficient condition states that a $2$DM is $N$-representable if and only if $$\mathrm{Tr}~\Gamma~H^{(2)}_\nu \geq E^N_0\left(H^{(2)}_\nu\right)~,
\label{eq_dual_n_rep}$$ for all $2$-particle Hamiltonians $H^{(2)}_\nu$, with corresponding ground-state energies $E^N_0(H^{(2)})$. These conditions are obviously not practically implementable, but can be used to derive a set of necessary constraints. Consider the class of positive Hamiltonians: $$\hat{H} = B^\dagger B\qquad\text{with}\qquad B^\dagger = \sum_{\alpha\beta}g_{\alpha\beta}a^\dagger_\alpha a_\beta~.$$ Inserting this class of Hamiltonians into Eq. (\[eq\_dual\_n\_rep\]) leads to the inequality: $$\sum_{\alpha\beta\gamma\delta}g_{\alpha\beta}{\langle \Psi^N|}a^\dagger_\alpha a_\beta a^\dagger_\delta a_\gamma{|\Psi^N\rangle}g_{\gamma\delta} \geq 0~,$$ from which the matrix-positivity condition [@garrod] $$\mathcal{G}(\Gamma) \succeq 0 \qquad\text{with}\qquad\mathcal{G}(\Gamma)_{\alpha\beta;\gamma\delta}=\delta_{\beta\delta}\rho_{\alpha\gamma}-\Gamma_{\alpha\delta;\gamma\beta}~,$$ follows. In this fashion one can derive two more necessary conditions [@coleman], one is simply the non-negativity condition on the 2DM itself, another requires the non-negativity of: $$\mathcal{Q}(\Gamma) \succeq 0 \qquad\text{with}\qquad \mathcal{Q}(\Gamma)_{\alpha\beta;\gamma\delta} = {\langle \Psi^N|}a_\alpha a_\beta a^\dagger_\delta a^\dagger_\gamma{|\Psi^N\rangle}~.$$ These three combined form the so-called two-index conditions, often referred to as $\mathcal{PQG}$.

Another class of constraints is derived using the positive Hamiltonian [@zhao; @hammond; @erdahl]: $$\hat{H} = B^\dagger B + B B^\dagger~,$$ in which the $B^\dagger$ is a three-body operator. Two conditions are derived in this way: $$\begin{aligned}
\mathcal{T}_1\left(\Gamma\right) \succeq& 0 \qquad \text{using} \qquad B^\dagger = \sum_{\alpha\beta\gamma}t^1_{\alpha\beta\gamma}a^\dagger_\alpha a^\dagger_\beta a^\dagger_\gamma~,\qquad\text{and}\\
\label{T2p}\mathcal{T}_2'\left(\Gamma\right) 