of the SD space $\{\Phi\}$ and restricting the trial function space $\{\Psi\}$ to $\{\Psi\}_{\text{PDC}}$. The OC is a necessary and sufficient condition for $N$-representability, whereas the PDC is only sufficient. We impose the PDC on the $\Phi$ as a simplifying variational approximation.

Under the OC or PDC, $\rho(x',x)$ takes the diagonal form $$\rho(x',x) = \sum_k p_1(k) \, \psi_k(x')\psi_k(x).
\label{rho1diag}$$ Here, the $p_1(k) = \sum_{{{\bf{n}}}} C_{{{\bf{n}}}}^2 \, \nu_{k,{{\bf{n}}}}$, where $\nu_{k,{{\bf{n}}}}
= 1$ if $k \in {{\bf{n}}}$ and $0$ otherwise, are the eigenvalues of $\rho(x',x)$, the occupation numbers or occupation probabilities (1-OP) of its eigenfunctions $\psi_k$. They satisfy the necessary and sufficient conditions $0 \le p_1(k) \le 1$ and $\sum_k p_1(k) = N$. In general, only $M > N$ occupation numbers $p_1(k)$ are non-negligible, and only the corresponding active NSOs need be included in the representation of any trial function, providing a natural cutoff. The 1-DM is thus of algebraic complexity in the 1-OPs and the NSOs, as $M$ scales linearly with $N$.

The 2-DM, the sign conjecture, the $\xi$-approximation
------------------------------------------------------

The 2-DM of $\Psi$, $$\begin{aligned}
\pi(x'_1x'_2;x_1x_2) = N (N-1) \int dx_3\cdots dx_N \\
\Psi(x'_1,x'_2,x_3,\cdots,x_N) \Psi^*(x_1,x_2,x_3,\cdots,x_N),\end{aligned}$$ becomes $$\begin{aligned}
\pi(x'_1x'_2;x_1x_2) &=& 
\sum_{\scriptscriptstyle
\overset{i<i', j<j',{{\bf{k}}}}
{i,i',j,j' \not\in {{\bf{k}}}}}
C_{ii'{{\bf{k}}}} C_{jj'{{\bf{k}}}}\\
&&\left(
\psi_i(x'_1)\psi_{i'}(x'_2)-\psi_{i'}(x'_1)\psi_i(x'_2)\right)\\
&&\left(
\psi_j(x_1)\psi_{j'}(x_2)-\psi_{j'}(x_1)\psi_j(x_2)\right).\end{aligned}$$ 